wiki143 - User contributions [en]

Rearrangement inequality

2025-09-27T00:38:47Z

173.79.19.248: /* Three or more sequences */ remove pointless "note that"

{{Short description|Theorem in mathematics}}
In [[mathematics]], the '''rearrangement inequality'''<ref>{{Citation | last1 = Hardy | first1 = G.H. | author-link = G. H. Hardy | last2 = Littlewood | first2 = J.E. | author2-link = John Edensor Littlewood | last3 = Pólya | first3 = G. | author3-link = George Pólya | title = Inequalities | publisher = [[Cambridge University Press]] | series = Cambridge Mathematical Library | edition = 2. | year = 1952 | location = [[Cambridge]] | isbn = 0-521-05206-8 | mr = 0046395 | zbl = 0047.05302}}, Section 10.2, Theorem 368</ref> states that for every choice of [[real number]]s
<math display=block>x_1 \le \cdots \le x_n \quad \text{ and } \quad y_1 \le \cdots \le y_n</math>
and every [[permutation]] <math>\sigma</math> of the numbers <math>1, 2, \ldots n</math> we have
{{NumBlk|:|<math> x_1 y_n + \cdots + x_n y_1
\le x_1 y_{\sigma(1)} + \cdots + x_n y_{\sigma(n)}
\le x_1 y_1 + \cdots + x_n y_n</math>.|{{EquationRef|1}}}}
Informally, this means that in these types of sums, the largest sum is achieved by pairing large <math>x</math> values with large <math>y</math> values, and the smallest sum is achieved by pairing small values with large values. This can be formalised in the case that the <math>x_1, \ldots, x_n</math> are distinct, meaning that
<math>x_1 < \cdots < x_n,</math> then:
# The upper bound in ({{EquationNote|1}}) is attained only for permutations <math>\sigma</math> that keep the order of <math>y_1, \ldots, y_n,</math> that is, <math display=block>y_{\sigma(1)} \le \cdots \le y_{\sigma(n)},</math> or equivalently <math>(y_1,\ldots,y_n) = (y_{\sigma(1)},\ldots,y_{\sigma(n)}).</math> Such a <math>\sigma</math> can permute the indices of <math>y</math>-values that are equal; in the case <math>y_1 = \cdots = y_n</math> every permutation keeps the order of <math>y_1, \ldots, y_n.</math> If <math>y_1 < \cdots < y_n,</math> then the only such <math>\sigma</math> is the identity.
# Correspondingly, the lower bound in ({{EquationNote|1}}) is attained only for permutations <math>\sigma</math> that reverse the order of <math>y_1, \ldots, y_n,</math> meaning that <math display=block>y_{\sigma(1)} \ge \cdots \ge y_{\sigma(n)}.</math> If <math>y_1 < \cdots < y_n,</math> then <math>\sigma(i) = n - i + 1</math> for all <math>i = 1, \ldots, n,</math> is the only permutation to do this.
Note that the rearrangement inequality ({{EquationNote|1}}) makes no assumptions on the signs of the real numbers, unlike inequalities such as the [[AM-GM Inequality|arithmetic-geometric mean inequality]].

{{TOC right}}

==Applications==

Many important inequalities can be proved by the rearrangement inequality, such as the [[Inequality of arithmetic and geometric means|arithmetic mean – geometric mean inequality]], the [[Cauchy–Schwarz inequality]], and [[Chebyshev's sum inequality]].

As a simple example, consider real numbers <math>x_1 \le \cdots \le x_n</math>: By applying ({{EquationNote|1}}) with <math>y_i := x_i</math> for all <math>i = 1,\ldots,n,</math> it follows that
<math display=block>x_1 x_n + \cdots + x_n x_1
\le x_1 x_{\sigma(1)} + \cdots + x_n x_{\sigma(n)}
\le x_1^2 + \cdots + x_n^2</math>
for every permutation <math>\sigma</math> of <math>1, \ldots, n.</math>

==Intuition==

The rearrangement inequality can be regarded as intuitive in the following way. Imagine there is a heap of $10 bills, a heap of $20 bills and one more heap of $100 bills. You are allowed to take 7 bills from a heap of your choice and then the heap disappears. In the second round you are allowed to take 5 bills from another heap and the heap disappears. In the last round you may take 3 bills from the last heap. In what order do you want to choose the heaps to maximize your profit? Obviously, the best you can do is to gain <math>7\cdot100 + 5\cdot20 + 3\cdot10</math> dollars. This is exactly what the upper bound of the rearrangement inequality ({{EquationNote|1}}) says for the sequences <math>3 < 5 < 7</math> and <math>10 < 20 < 100.</math> In this sense, it can be considered as an example of a [[greedy algorithm]].

==Geometric interpretation==

Assume that <math>0 < x_1 < \cdots < x_n</math> and <math> 0< y_1 < \cdots < y_n.</math> Consider a rectangle of width <math>x_1 + \cdots + x_n</math> and height <math>y_1 + \cdots + y_n,</math> subdivided into <math>n</math> columns of widths <math>x_1,\ldots,x_n</math> and the same number of rows of heights <math>y_1,\ldots,y_n,</math> so there are <math>\textstyle n^2</math> small rectangles. You are supposed to take <math>n</math> of these, one from each column and one from each row. The rearrangement inequality ({{EquationNote|1}}) says that you optimize the total area of your selection by taking the rectangles on the diagonal or the antidiagonal.

==Proofs==

===Proof by contradiction===

The lower bound and the corresponding discussion of equality follow by applying the results for the upper bound to
<math display="inline">- y_n \le \cdots \le - y_1</math>, that is, let <math>\upsilon</math> be any permutation of the numbers <math display="inline">1, 2, \ldots n</math> we have <math display="inline">x_1 (-y_{\upsilon(1)})+\cdots +x_n (-y_{\upsilon (n)})\leq x_1 (-y_{n})+\cdots +x_n (-y_{1})</math>. Then, <math display="inline">x_1 y_{n}+\cdots +x_n y_{1}\leq x_1 y_{\upsilon(1)}+\cdots +x_n y_{\upsilon (n)}</math> for every permutation <math>\upsilon</math> of <math>1, \dots ,n</math>.

Therefore, it suffices to prove the upper bound in ({{EquationNote|1}}) and discuss when equality holds.
Since there are only finitely many permutations of <math>1,\ldots,n,</math> there exists at least one <math>\sigma</math> for which the middle term in ({{EquationNote|1}})
<math display="block">x_1 y_{\sigma(1)} + \cdots + x_n y_{\sigma(n)}</math>
is maximal. In case there are several permutations with this property, let σ denote one with the highest number of integers <math>i</math> from <math>\{1,\ldots,n\}</math> satisfying <math>y_i = y_{\sigma(i)}.</math>

We will now [[reductio ad absurdum|prove by contradiction]], that <math>\sigma</math> has to keep the order of <math>y_1,\ldots,y_n</math> (then we are done with the upper bound in ({{EquationNote|1}}), because the identity has that property). Assume that there exists a <math>j\in\{1,\ldots,n-1\}</math> such that <math>y_i = y_{\sigma(i)}</math> for all <math>i\in\{1,\ldots,j-1\}</math> and <math>y_j \neq y_{\sigma(j)}.</math> Hence <math>y_j < y_{\sigma(j)}</math> and there has to exist a <math>k\in\{j+1,\ldots,n\}</math> with <math>y_j = y_{\sigma(k)}</math> to fill the gap. Therefore,
{{NumBlk|:|<math> x_j \le x_k
\qquad\text{and}\qquad
y_{\sigma(k)} < y_{\sigma(j)},</math>|{{EquationRef|2}}}}
which implies that
{{NumBlk|:|<math> 0 \le (x_k - x_j)(y_{\sigma(j)} - y_{\sigma(k)}).</math>|{{EquationRef|3}}}}
Expanding this product and rearranging gives
{{NumBlk|:|<math> x_j y_{\sigma(j)} + x_k y_{\sigma(k)} \le x_j y_{\sigma(k)} + x_k y_{\sigma(j)}\,,</math>|{{EquationRef|4}}}}
which is equivalent to ({{EquationNote|3}}). Hence the permutation
<math display=block>\tau(i):=\begin{cases}\sigma(i)&\text{for }i \in \{1,\ldots,n\}\setminus\{j,k\},\\
\sigma(k)&\text{for }i = j,\\
\sigma(j)&\text{for }i = k,\end{cases}</math>
which arises from <math>\sigma</math> by exchanging the values <math>\sigma(j)</math> and <math>\sigma(k),</math> has at least one additional point which keeps the order compared to <math>\sigma,</math> namely at <math>j</math> satisfying <math>y_j=y_{\tau(j)},</math> and also attains the maximum in ({{EquationNote|1}}) due to ({{EquationNote|4}}). This contradicts the choice of <math>\sigma.</math>

If <math>x_1 < \cdots < x_n,</math> then we have strict inequalities in ({{EquationNote|2}}), ({{EquationNote|3}}), and ({{EquationNote|4}}), hence the maximum can only be attained by permutations keeping the order of <math>y_1 \le \cdots \le y_n,</math> and every other permutation <math>\sigma</math> cannot be optimal.

===Proof by induction===

As above, it suffices to treat the upper bound in ({{EquationNote|1}}). For a proof by [[mathematical induction]], we start with <math>n=2.</math> Observe that
<math display=block>x_1 \le x_2 \quad \text{ and } \quad y_1 \le y_2</math>
implies that
{{NumBlk|:|<math>0 \le (x_2 - x_1) (y_2 - y_1),</math>|{{EquationRef|5}}}}
which is equivalent to
{{NumBlk|:|<math>x_1 y_2 + x_2 y_1 \le x_1 y_1 + x_2 y_2,</math>|{{EquationRef|6}}}}
hence the upper bound in ({{EquationNote|1}}) is true for <math>n = 2.</math>
If <math>x_1 < x_2,</math> then we get strict inequality in ({{EquationNote|5}}) and ({{EquationNote|6}}) if and only if <math>y_1 < y_2.</math> Hence only the identity, which is the only permutation here keeping the order of <math>y_1 < y_2,</math> gives the maximum.

As an induction hypothesis assume that the upper bound in the rearrangement inequality ({{EquationNote|1}}) is true for <math>n-1</math> with <math>n\ge3</math> and that in the case <math>x_1 <\cdots <x_{n-1}</math> there is equality only when the permutation <math>\sigma</math> of <math>1,\ldots,n-1</math> keeps the order of <math>y_1,\ldots,y_{n-1}.</math>

Consider now <math>x_1 \le \cdots \le x_n</math> and <math>y_1 \le \cdots \le y_n.</math> Take a <math>\sigma</math> from the finite number of permutations of <math>1,\ldots,n</math> such that the rearrangement in the middle of ({{EquationNote|1}}) gives the maximal result. There are two cases:

* If <math>\sigma(n)=n,</math> then <math>y_n=y_{\sigma(n)}</math> and, using the induction hypothesis, the upper bound in ({{EquationNote|1}}) is true with equality and <math>\sigma</math> keeps the order of <math>y_1,\ldots,y_{n-1},y_n</math> in the case <math>x_1 < \cdots < x_n.</math>
* If <math>k := \sigma(n) < n,</math> then there is a <math>j\in\{1,\dots,n-1\}</math> with <math>\sigma(j)=n.</math> Define the permutation <math display=block>\tau(i)=\begin{cases}\sigma(i)&\text{for }i \in \{1,\ldots,n\}\setminus\{j,n\},\\
k&\text{for }i = j,\\
n&\text{for }i = n,\end{cases}</math> which arises from <math>\sigma</math> by exchanging the values of <math>j</math> and <math>n.</math> There are now two subcases:
# If <math>x_j = x_{\sigma_j} = x_n</math> or <math>y_n = y_{\sigma_n} = y_k,</math> then this exchange of values of <math>\sigma</math> has no effect on the middle term in ({{EquationNote|1}}) because <math>\tau</math> gives the same sum, and we can proceed by applying the first case to <math>\tau.</math> Note that in the case <math>x_1 < \cdots < x_n,</math> the permutation <math>\tau</math> keeps the order of <math>y_1,\ldots,y_n</math> if and only if <math>\sigma</math> does.
# If <math>x_j < x_n</math> and <math>y_k < y_n,</math> then <math>0 < (x_n - x_j) (y_n - y_k),</math> which is equivalent to <math>x_j y_{\sigma_j}+ x_n y_{\sigma_n} < x_j y_{\tau_j} + x_n y_{{\tau_n}}</math> and shows that <math>\sigma</math> is not optimal, hence this case cannot happen due to the choice of <math>\sigma.</math>

==Generalizations==

===Three or more sequences===
A straightforward generalization takes into account more sequences. Assume we have finite ordered sequences of nonnegative real numbers
<math display=block>0 \le x_1\le\cdots\le x_n\quad\text{and}\quad 0\le y_1\le\cdots\le y_n\quad\text{and}\quad 0\le z_1\le\cdots\le z_n</math>
and a permutation <math>y_{\sigma(1)},\ldots,y_{\sigma(n)}</math> of <math>y_1,\dots,y_n</math> and another permutation <math>z_{\tau(1)},\dots,z_{\tau(n)}</math> of <math>z_1,\dots,z_n.</math> Then
<math display=block> x_1 y_{\sigma(1)} z_{\tau(1)} + \cdots + x_n y_{\sigma(n)} z_{\tau(n)} \le x_1 y_1 z_1 + \cdots + x_n y_n z_n.</math>

Unlike the standard rearrangement inequality ({{EquationNote|1}}), this statement requires the numbers to be nonnegative. A similar statement is true for any number of sequences with all numbers nonnegative.

===Functions instead of factors===
Another generalization of the rearrangement inequality states that for all [[real number]]s <math>x_1 \le \cdots \le x_n</math> and every choice of continuously [[differentiable function]]s <math>f_i: [x_1,x_n] \to \R</math> for <math>i = 1, 2, \ldots, n</math> such that their derivatives <math>f'_1,\ldots,f'_n</math> satisfy
<math display=block>f'_1(x) \le f'_2(x) \le \cdots \le f'_n(x) \quad \text{ for all } x \in [x_1,x_n],</math>
the inequality
<math display=block>\sum_{i=1}^n f_{n-i+1}(x_i) \le \sum_{i=1}^n f_{\sigma(i)}(x_i) \le \sum_{i=1}^n f_i(x_i)</math>
holds for every [[permutation]] <math>f_{\sigma(1)}, \ldots, f_{\sigma(n)}</math> of <math>f_1, \ldots, f_n.</math><ref>{{citation | year=2017 | last=Holstermann | first=Jan | title=A Generalization of the Rearrangement Inequality | periodical=Mathematical Reflections | issue = 5 (2017) | url=https://www.awesomemath.org/wp-pdf-files/math-reflections/mr-2017-05/rearrangement_inequality.pdf}}</ref>
Taking real numbers <math>y_1 \le \cdots \le y_n</math> and the linear functions <math>f_i(x) := x y_i</math> for real <math>x</math> and <math>i = 1,\ldots,n,</math> the standard rearrangement inequality ({{EquationNote|1}}) is recovered.

==See also==
* [[Hardy–Littlewood inequality]]
* [[Chebyshev's sum inequality]]

==References==
{{reflist}}

[[Category:Rearrangement inequalities| ]]
[[Category:Inequalities (mathematics)]]
[[Category:Articles containing proofs]]

Whipping knot

2025-06-30T12:21:34Z

173.79.19.248: Undid revision 1297889458 by Umabruka (talk) this is not a reliable source and does not in any case support the content of the sentence to which it was added

{{Short description|Binding around the end of a rope to prevent it from fraying}}
{{For|the punishment tool occasionally featuring knots at the end of its lashing ropes|Cat-o'-nine-tails}}
[[Image:Common-whipping.png|upright=1.4|[[Common whipping]] knot|right|thumb]]

A '''whipping knot''' or '''whipping''' is a [[binding (knot)|binding]] of {{linktext|marline}} [[twine]] or [[Whipcord#Cord|whipcord]] around the end of a [[rope]] to prevent its natural tendency to fray.

Some whippings are finished cleanly, as by drawing the bitter end of the cordage beneath the whipping itself. Others are tied off or have the end(s) of the twine sewn through the rope. According to ''[[The Ashley Book of Knots]]'', "The purpose of a ''whipping'' is to prevent the end of a rope from fraying ... A whipping should be, in width, about equal to the diameter of the rope on which it is put ... [Two sailmaker's whippings], a short distance apart, are put in the ends of every reef point, where the constant 'whipping' against the sail makes the wear excessive; this is said to be the source of the name ''whipping''."<ref name="Ashley547">Ashley, Clifford W. (1944). ''The Ashley Book of Knots'', p.547. Doubleday. {{ISBN|0-385-04025-3}}.</ref> The other type of stopping knot is a [[seizing knot]].

Whipping is suitable for synthetic and natural stranded and braided lines, including 3-strand rope, 4-strand cable and 8-strand multiplait, as well as concentric and braided constructions.{{cn|date=September 2020}}

==Tying==
[[File:Frayed.png|thumb|Unsecured end of double braid rope.]]
Multiple [[turn (knot)|turns]] of twine (sometimes called [[small stuff]] for smaller lines) or heavier whipcord (for large diameter cables and ropes) are tightly wrapped around a rope's cut end to prevent its fibers from unlaying.

Usually one end of the whipping cord is looped along the rope to be whipped, and the remaining cord wound tightly over the loop. Finally the loose end of the wound whipping is passed through the loop so that both ends may be drawn securely inside the winding.<ref>[https://www.youtube.com/watch?v=YMq9KdOtSJ0 YouTube video:How to Whip the Ends of Rope by TIAT]</ref>

Whippings may also be applied by hand or using a palm and needle, and either simply tied off or made neat and permanent by reeving the twine's cut ends into or behind the whipping, sewing them to adjacent strands, or through the rope itself.

In applications where a lot of flexing is expected, the whipping may be impregnated with dilute [[Varnish#Spar varnish|spar varnish]] or [[Cyanoacrylate|superglue]].

==Types==
{{main|Common whipping|Sailmaker's whipping|West Country whipping}}
{{multiple image
| total_width= 220
| image1= Sailmakers.jpg
| caption1= [[Sailmaker's whipping]]
| image2= Westcountry.jpg
| caption2= [[West Country whipping]]
}}

===French whipping===

{{Quote|''French whipping'' is merely a series of [[half hitch]]es. Start with a [[running eye]] and finish up with the end tucked back under the last few hitches. The ridge of the hitches should follow the lay of the rope.|''[[The Ashley Book of Knots]]''<ref name="Ashley547"/>}}

'''French whipping''' is a whipping knot that consists of a series of [[half hitch]]es. It is used to stop unraveling of rope ends as well as to provide a grip over railings.<ref>{{cite web |url=http://www.marinews.com/knots/rope-knots/household/stoppers-whippings/how-to-tie-a-french-whipping-knot/1/8/13/532/ |title=French Whipping |publisher=marinews.com |access-date=2012-10-29}}</ref>
{{clear}}

===Portuguese whipping===

{{Knot-details
| name=Portuguese whipping
| image=
| names=
| type= whipping
| strength=
| origin=
| related= [[Reef knot]]
| releasing=
| uses= To prevent a rope from fraying
| caveat=
| abok_number=
}}

{{Quote
| ''Portuguese whipping'' is the quickest of all to apply; the ends are merely [[reef knot]]ted together. It is given by Esparteiro in his ''Dicionario de Marinharia'' (Lisboa, 1936).
| ''[[The Ashley Book of Knots]]''<ref>Ashley, Clifford W. (1944). ''The Ashley Book of Knots'', p.548. Doubleday. {{ISBN|0-385-04025-3}}.</ref>
}}

The '''Portuguese whipping''' is a type of whipping knot. To make it you take the small diameter string and lay one end against the rope. Wrap backwards up the rope until you have both ends side by side, finish by tying a [[reef knot]]. This is the quickest of the seizings, but is not as secure as some.<ref>{{Cite web |title=Common Whipping {{!}} How to tie a Common Whipping Knot animated and illustrated. |url=https://www.netknots.com/rope_knots/common-whipping |access-date=2024-09-20 |website=www.netknots.com}}</ref>

==Alternatives==

===Constrictor knot===
A [[constrictor knot]] can be used temporarily to hold the fibres of a cut line until a final whipping can be applied.

===Tape===
Several turns of self-adhesive plastic tape may form a temporary or emergency substitute for whipping.

===Fusion===
[[File:FusedRopeEnds.png|thumb|Rope ends heat sealed with electric knife.]]
The ends of some man-made fibers such as [[Dacron]], [[Nylon]], [[polyethylene]], [[polyester]], and [[polypropylene]] (but not [[aramid]] fibers) may be melted to fuse their fibers to prevent fraying. However, the rope and knotting expert Geoffrey Budworth warns against this practice for boat operators thus:<ref>{{cite book |last=Budworth |first=Geoffrey |publisher=Sterling Publishing Co., Inc. |location=New York |title=The Knot Book |url=https://archive.org/details/isbn_9780806979441 |url-access=registration |isbn=0-8069-7944-5 |page=[https://archive.org/details/isbn_9780806979441/page/37 37] |year=1985}}</ref>

<blockquote>
''Sealing rope ends this way is lazy and dangerous.'' A tugboat operator once sliced the palm of his hand open down to the sinews after the hardened (and obviously ''sharp'') end of a rope that had been heat-sealed pulled through his grasp. There is no substitute for a properly made whipping.
</blockquote>

Among the methods of fusing are using an electrically heated rope cutter, heating the blade of a knife, or melting cut ends in a flame. The cool (transparent) part of a butane lighter flame works best.

It is helpful to wrap the end of a line to be fused with several turns of plastic tape first. The finished end will be neater and narrower if a cut is made through the tape.

===Back splice===
[[Back splice|Back splicing]] uses a stranded rope's own fibres to prevent fraying. A back splice adds extra thickness to the rope end, preventing it from running through [[block (sailing)|blocks]] and sheaves. It can also be of benefit when a user needs to feel the end of the rope, as on a bucket lanyard.<ref>{{Cite web |last=Knotter |date=2017-02-21 |title=Back Splice Rope How to |url=https://igkt-solent.co.uk/splice-rope/ |access-date=2025-06-29 |language=en-US}}</ref>

===Liquid===
Liquid whipping is a semi-permanent rubbery coating applied by dipping the cut end of a line into a container of the product. When the coating sets it is flexible but solid enough to keep the rope together. Liquid whipping can be used on both natural and synthetic fibers.

===Aglet===

An [[aglet]] is a permanent ending applied mechanically to bind the end of the rope. A typical example is the plastic aglet at the end of a shoelace. Metal aglets may be [[crimp (joining)|crimp]]ed onto ropes or cables. Aglets may also be made by melting a softer metal to cap the end of the cable.{{cn|date=October 2019}}

==See also==
* [[Bend knot]]
* [[List of knots]]
* [[Rope splicing]]
* [[Strangle knot]]

==References==
{{reflist}}

==External links==
*{{Animated Knots|sailmakers|Sailmaker's Whipping}}
*{{Animated Knots|sailors|Sailor's (Simple) Whipping}}
*{{Animated Knots|westcountry|West Country Whipping}}

{{Knots}}

[[Category:Whipping knots| ]]

Wiki hosting service

2025-06-27T12:03:33Z

173.79.19.248: /* Comparison of wiki hosting services */ inappropriate in-text EL

{{Short description|Server hosting independent wikis}}
{{Primary sources|date=February 2023}}
{{internet hosting}}

A '''wiki hosting service''', or '''wiki farm''', is a [[Server (computing)|server]] or an [[Server farm|array of servers]] that offers users tools to simplify the creation and development of individual, independent [[wiki]]s.

Prior to wiki farms, someone who wanted to operate a wiki had to install the software and manage the server(s) themselves. With a wiki farm, the farm's administration installs the core wiki code once on its own servers, centrally maintains the servers, and establishes unique space on the servers for the content of each individual wiki with the shared core code executing the functions of each wiki.

Both commercial and non-commercial wiki farms are available for users and online communities. While most of the wiki farms allow anyone to open their own wiki, some impose restrictions. Many wiki farm companies generate revenue through the insertion of advertisements, but often allow payment of a monthly fee as an alternative to accepting ads.

[[Fandom (website)|Fandom]], created by [[Jimmy Wales]] and [[Angela Beesley Starling]] in 2004 and formerly known as Wikia, is a well-known example of a wiki-hosting service.<ref>{{Cite web |title=About Us |url=https://about.fandom.com/about |access-date=2023-07-11 |website=Fandom |language=en-US |archive-date=2023-07-14 |archive-url=https://web.archive.org/web/20230714063351/https://about.fandom.com/about |url-status=live }}</ref><ref>{{Cite web |last=Chmielewski |first=Dawn |date=2016-01-25 |title=Wikia Launches Fandom, a New Place to Get Your Nerd On |url=https://www.vox.com/2016/1/25/11589030/wikia-launches-fandom-a-new-place-to-get-your-nerd-on |access-date=2023-07-11 |website=Vox |language=en |archive-date=2022-03-12 |archive-url=https://web.archive.org/web/20220312104519/https://www.vox.com/2016/1/25/11589030/wikia-launches-fandom-a-new-place-to-get-your-nerd-on |url-status=live }}</ref>

==Comparison of wiki hosting services==
This '''comparison of wiki hosting services''' or '''wiki farms''' is not comprehensive, it details only those '[[Notability in the English Wikipedia|notable]]' enough (in Wikipedia terms) to be included. A useful comprehensive comparison of wiki farms can be found on [[MediaWiki]]'s site, at [[mw:Hosting services]].

Online services which host [[wiki]]-style editable [[web page]]s. General characteristics of cost, presence of advertising, licensing are compared, as are technical differences in editing, features, wiki engine, multilingual support and syntax support.


This table compares general information for several of the more than 100 wiki hosting services that exist.<ref name="wikmat">"WikiMatrix - Compare them all" (selectable table),
WikiMatrix, 2007, webpage:
[http://www.wikimatrix.org/ wikimatrix-org-main] {{Webarchive|url=https://web.archive.org/web/20160131033544/http://www.wikimatrix.org/ |date=2016-01-31 }}.</ref>

All the mentioned services have [[WYSIWYG]] editing.

{{mw-datatable}}{{sticky header}}
{| class="wikitable sortable mw-datatable sticky-header" style="font-size:90%;" width=100%
|+ Wiki hosting services
|- style=vertical-align:bottom;
! {{vert header|Hosting service}}
! {{vert header|Start}}
! {{vert header|Cost}}
! {{vert header|Ad supported}}
! {{vert header|Content license}}
! {{vert header|[[Subdomain]]}}
! {{vert header|Custom themes}}
! {{vert header|Download, backup}}
! {{vert header|Other Features}}
! {{vert header|Base wiki engine}}
! {{vert header|Language}}
! {{vert header|Syntax support}}
|-
| '''[[Confluence (software)|Confluence]]'''
| {{center|2004}}
| {{Nonfree}}{{efn|Confluence has no free hosted option, but offers free software for charitable nonprofits and open source projects.}}
| {{dunno}}
|
| {{No}}
| {{dunno}}
| {{dunno}}
| Plugins, SSL, file storage, permissioning, WebDAV.
|
|
| WYSIWYG Rich-text editor<ref>{{Cite web|url=https://support.atlassian.com/confluence-cloud/docs/insert-confluence-wiki-markup/|title=Confluence wiki markup syntax | Confluence Cloud|access-date=2024-02-01|archive-date=2023-12-01|archive-url=https://web.archive.org/web/20231201232149/https://support.atlassian.com/confluence-cloud/docs/insert-confluence-wiki-markup/|url-status=live}}</ref> script plugin No formulas
|-
| '''[[Central Desktop]]'''
| {{center|2005}}
| {{Nonfree}}
| {{dunno}}
|
| {{No}}
| {{dunno}}
| {{dunno}}
| Access control, full-text search, calendaring, single sign-on to multiple projects, project templates, [[RSS]] enabled.
| [?] (custom)
|
| HTML,<ref name="wikmat" /> CSS/templating No formulas
|-
| '''[[Ourproject.org]]'''
| {{center|2002}}
| {{Free}}
| {{Yes|No}}<ref>{{cite web |url=http://ourproject.org/docman/view.php/1/19/file19.html#bases |title=Ourproject.org Manifesto |date=July 2006 |publisher=[[Ourproject.org]] |access-date=2019-11-25 |archive-date=2019-12-04 |archive-url=https://web.archive.org/web/20191204211344/http://ourproject.org/docman/view.php/1/19/file19.html#bases |url-status=live }}</ref>
| [[Copyleft]] (choice of [[Creative Commons license|Creative Commons]], [[GNU FDL]], other licenses)
| {{Yes}}
| {{dunno}}
| {{dunno}}
| Mailing lists, FTP, SSH, ddbb, email alias, backups, CVS/SVN, forums, task management.
| [[MoinMoin]] by default; custom supported
| Supports English, Spanish, French, and many other languages.
|
|-
| '''[[PBworks]]'''
| {{center|2005}}
| {{Nonfree}}
| {{Yes|No}}
|
| {{Yes}}
| {{Yes}}
| {{dunno}}
| No page limits, SSL, RSS & Atom, email notifications, file management, page access settings.
| [?] (custom)
|
| All HTML,<ref name="wikmat" /> [[JavaScript]], [[LaTeX]] formulas
|-
| '''[[Fandom (website)|Fandom]]'''
| {{center|2004}}
| {{Free}}
| {{No|Yes}}
| [[Creative Commons license|Creative Commons]]
| {{sometimes|Yes, but not for multilingual wikis}}
| {{Yes}}
| {{Yes}}
| Common login and common preferences to all wikis of FANDOM for the same user. Blog, User Page, and User Talk pages for users.
| [[MediaWiki]]
| All languages Wikipedia supports (and some more); Community Support in English, Chinese, German, Japanese, Spanish, Portuguese, Italian, French and several other languages.{{Citation needed|date=August 2021}}
| [[Wikitext]] as used by MediaWiki, Some HTML,<ref name="wikmat" /> JavaScript, Math formula, Lua
|-
| {{Visible anchor|Miraheze|text='''Miraheze'''}}<ref>{{Cite web |date=2021-11-16 |title=Monster Girl Encyclopedia Wiki Will Be Sacked By Host Due To ‘Fetish’ Content |url=https://kotaku.com/monster-girl-encyclopedia-wiki-will-be-sacked-by-host-d-1848068972 |access-date=2025-02-18 |website=Kotaku |language=en}}</ref><ref>{{Cite web |last=Bak |first=Jesica |date=2022-11-01 |title=Student-created Huskypedia aims to provide resources, information to students |url=https://huntnewsnu.com/69493/campus/student-created-huskypedia-aims-to-provide-resources-information-to-students/ |access-date=2025-02-18 |website=The Huntington News}}</ref><ref>{{Cite web |title=Build Your Own Wiki Platform And Grow Your Community With Miraheze |url=https://www.websiteplanet.com/blog/miraheze-interview/ |url-status=live |archive-url=https://web.archive.org/web/20230620154605/https://www.websiteplanet.com/blog/miraheze-interview/ |archive-date=2023-06-20 |access-date=2023-07-11 |website=Website Planet |language=en}}</ref>
| {{center|2015}}
| {{Free}}
| {{Yes|No}}
| [[Creative Commons license|Creative Commons]]
| {{Yes}}
| {{Yes}}
| {{Yes}}
| Common login, custom extensions and skins and custom domains
| [[MediaWiki]]
| All languages Wikipedia supports (and some more)
| WYSIWYG Rich-text editor script plugin
|}

=== Deprecated wiki hosts ===
This section is for hosts that were previously in the list above but no longer available:
* [[Wikispaces]]

== Notes ==
{{Notelist}}

== References ==
{{reflist}}

== External links ==
* [https://www.curlie.org/Computers/Software/Groupware/Wiki/Wiki_Farms/ Computers/Software/Groupware/Wiki/Wiki Farms] {{Webarchive|url=https://web.archive.org/web/20221005141739/https://www.curlie.org/Computers/Software/Groupware/Wiki/Wiki_Farms/ |date=2022-10-05 }} at [[DMOZ|Curlie]]
* [https://web.archive.org/web/20160131033544/http://www.wikimatrix.org/ Wikimatrix, with interactive selection of wikifarms based on user preference], now only available through the [[Wayback Machine]].

{{Wiki topics}}

{{DEFAULTSORT:Wiki Farms}}
[[Category:Wiki farms| ]]
[[Category:Web hosting]]

Jacob Aaron Estes

2025-06-24T23:37:54Z

173.79.19.248: Undid revision 1293326300 by 2603:6011:59F0:21E0:8522:739F:2039:23AB (talk) rv uncited WP:BLP

{{short description|American screenwriter and film director|bot=PearBOT 5}}
{{Use mdy dates|date=August 2014}}
{{Infobox person
| name = Jacob Aaron Estes
| image =
| caption =
| birth_date = {{birth-date and age|September 6, 1972}}
| birth_place = [[Tulare County, California]], U.S.
| occupation = [[Screenwriter]], [[film director]]
| years_active =
| known_for = ''[[Mean Creek]]''
| spouse = [[Gretchen Lieberum]]
}}

'''Jacob Aaron Estes''' (born September 6, 1972) is an American [[screenwriter]] and [[film director]] known for his films ''[[Mean Creek]]'', ''[[The Details (film)|The Details]]'' and [[Don't Let Go (2019 film)|''Don't Let Go'']].

==Career==
Estes was a theatre major at the [[University of California, Santa Cruz]].<ref name="Mind Control">{{cite web|last1=Kushner|first1=Zach|title=GET THE DETAILS FROM DIRECTOR JACOB AARON ESTES: EXCLUSIVE INTERVIEW & TRAILER|url=http://www.standbyformindcontrol.com/2012/09/get-the-details-from-director-jacob-aaron-estes-exclusive-interview-trailer/|website=Stand By for Mind Control|accessdate=26 December 2017|date=24 September 2012}}</ref> After that, he went to film school at [[American Film Institute|AFI]].<ref name="Mind Control"/> Estes also attended the National Playwrights Conference at the [[Eugene O'Neill Theater Center]], which produced two of his plays, ''Free Lessons'' and ''Mean Creek''.<ref name="Mind Control"/><ref>{{cite web|title=Jacob Aaron Estes|url=https://www.doollee.com/PlaywrightsE/estes-jacob-aaron.html|website=doollee.com|accessdate=26 December 2017}}</ref> Estes turned ''[[Mean Creek]]'' into a screenplay, which became his feature film debut as writer and director in 2004.<ref name="Armstrong">{{cite web|url=http://www.allmovie.com/work/mean-creek-301376/review|title=Review: ''Mean Creek''|last=Armstrong|first=Derek|publisher=[[AllMovie]]|accessdate=February 24, 2011}}</ref> In his first offering as writer-director, he gained critical acclaim, receiving nominations at various film festivals and winning the [[Independent Spirit John Cassavetes Award]].<ref>{{cite web|url=https://www.latimes.com/archives/la-xpm-2005-feb-27-me-spirit27-story.html|title=Spirit Awards Toasts 'Sideways' With Sweep of Six Major Categories|date=February 27, 2005|work=[[Los Angeles Times]]}}</ref> 

His second film as writer/director came with the 2011 film ''[[The Details (film)|The Details]]'',<ref name="Holden">{{cite web|url=https://www.nytimes.com/2012/11/02/movies/tobey-maguire-stars-in-the-details.html |title=Relentless Raccoons and Other Suburban Problems Money Can't Always Fix |last=Holden|first=Stephen|work=The New York Times |date=November 2012 |accessdate=November 1, 2012}}</ref> which premiered at the [[2011 Sundance Film Festival]]<ref>{{Cite web |last=Greenberg |first=James |date=2011-01-26 |title=SUNDANCE REVIEW: 'The Details' Is a Rom-com That Becomes Something Much Darker |url=https://www.hollywoodreporter.com/movies/movie-news/sundance-review-details-rom-becomes-76271/ |access-date=2023-07-12 |website=The Hollywood Reporter |language=en-US}}</ref> and stars [[Tobey Maguire]], [[Laura Linney]], [[Ray Liotta]], [[Dennis Haysbert]], [[Kerry Washington]], and [[Elizabeth Banks]]. In 2019, Estes co-wrote and directed the [[Blumhouse]] thriller [[Don't Let Go (2019 film)|''Don't Let Go'']] starring [[David Oyelowo]] and [[Storm Reid]].

==Personal life==
Estes is married to musician [[Gretchen Lieberum]].<ref name="Mind Control"/>

==Filmography==
{|class="wikitable"
|-
! Year !! Film !! Director !! Writer !! Producer !! Notes
|-
| 2001 || ''Summoning'' || {{Yes}} || {{Yes}} || {{No}} || Short film
|-
| 2004 || ''[[Mean Creek]]'' || {{Yes}} || {{Yes}} || {{No}} || [[John Cassavetes Award]] winner<ref>{{cite web|url=http://www.indiewire.com/article/sideways_wins_six_at_independent_spirit_awards|title=Sideways Wins Six at Independent Spirit Awards|date=February 28, 2005|work=[[IndieWire]]|archive-url=https://archive.today/20130126075107/http://www.indiewire.com/article/sideways_wins_six_at_independent_spirit_awards|archive-date=January 26, 2013|url-status=live}}</ref>
|-
| 2005 || ''[[Nearing Grace]]'' || {{No}} || {{Yes}} || {{No}} ||
|-
| 2011 || ''[[The Details (film)|The Details]]'' || {{Yes}} || {{Yes}} || {{No}} ||
|-
| 2014 || ''[[7 Minutes (2014 film)|7 Minutes]]'' || {{No}} || {{No}} || {{Yes}} ||
|-
| 2015 || ''Bromance'' || {{No}} || {{No}} || {{No}} || Special thanks
|-
| 2017 || ''[[Rings (2017 film)|Rings]]'' || {{No}} || {{Yes}} || {{No}} || Co-write with [[David Loucka]] and [[Akiva Goldsman]]
|-
| 2019 || ''[[Don't Let Go (2019 film)|Don't Let Go]]'' || {{Yes}} || {{Yes}} || {{No}} ||
|-
| 2022 || ''[[He's Watching]]'' || {{Yes}} || {{Yes}} || {{Yes}} ||
|}

== References ==
{{Reflist}}

== External links ==
*{{IMDb name|nm1096524}}
*[https://www.youtube.com/watch?v=zMbKDWX9hqM AFI Movie Club interview]
*[https://www.allmovie.com/artist/jacob-aaron-estes-p386003 AllMovie]

{{Jacob Aaron Estes}}

{{Authority control}}

{{DEFAULTSORT:Estes, Jacob Aaron}}
[[Category:Film directors from California]]
[[Category:Independent Spirit Award winners]]
[[Category:1972 births]]
[[Category:Living people]]

Eric Rudolph

2025-06-24T23:36:12Z

173.79.19.248: Undid revision 1296306497 by Lightiggy (talk) communism does not appear to be mentioned anywhere on this page

{{short description|American domestic terrorist (born 1966)}}
{{about|the American domestic terrorist|the American racing driver|Erick Rudolph}}
{{Use mdy dates|date=June 2021}}
{{Infobox FBI Ten Most Wanted
| name = Eric Rudolph
| image = EricRudolphFull.jpg
| image_size =
| caption =
| birth_date = {{Birth date and age|1966|9|19|mf=y}}
| birth_place = [[Merritt Island, Florida]], U.S.
| birth_name = Eric Robert Rudolph
| cause =
| alias = Bob Randolph, Robert Randolph, Bobby Rudolph
| charge =
| conviction = [[Title 18 of the United States Code|Maliciously damaging by means
of an explosive a building and property used in an activity affecting interstate
commerce resulting in death (18 U.S.C. § 844)]] [[Criminal possession of a weapon|Using a destructive device during a crime of violence (18 U.S.C. § 924)]]
| conviction_penalty = Four consecutive [[Life imprisonment|life sentences]] without the possibility of [[parole]]
| imprisoned = [[ADX Florence]]
| occupation =
{{flatlist|
* Carpenter
* roofer
* handyman
}}
| spouse =
| children =
| added_date = May 5, 1998
| caught_date = May 31, 2003
| remove_date =
| number = 454
| status = Captured
}}
'''Eric Robert Rudolph''' (born September 19, 1966), also known as the '''Olympic Park Bomber''', is an American [[domestic terrorist]] convicted of a series of bombings across the [[Southern United States]] between 1996 and 1998, which killed two people and injured over 100 others,<ref name="FBI announcement">{{cite web |url=http://www.usdoj.gov/opa/pr/1998/October/477crm.htm |title=Eric Rudolph charged in Centennial Olympic Park bombing |website=Viceland News |access-date=September 19, 2019|archive-date=June 18, 2003 |url-status=live |archive-url=https://web.archive.org/web/20030618150204/http://www.usdoj.gov/opa/pr/1998/October/477crm.htm }}</ref><ref>{{cite web|url=http://www.cnn.com/2005/LAW/04/08/rudolph.plea/|title=Rudolph agrees to plea agreement|last1=Schuster|first1=Henry|date=April 12, 2005|website=[[CNN]]|access-date=January 8, 2016|quote="The many victims of Eric Rudolph's terrorist attacks in Atlanta and Birmingham can rest assured that Rudolph will spend the rest of his life behind bars," [U.S. Attorney General Alberto] Gonzales said in press release.|archive-url=https://web.archive.org/web/20050409061947/http://www.cnn.com/2005/LAW/04/08/rudolph.plea/|archive-date=April 9, 2005|url-status=live}}</ref> including the [[Centennial Olympic Park bombing]] at the [[1996 Summer Olympics]] in [[Atlanta]]. His stated motive was an opposition to "the ideals of global socialism" and to "[[abortion law|abortion on demand]]", both of which he claimed were condoned by the United States government.<ref>{{cite news |url=https://www.npr.org/2005/04/14/4600480/full-text-of-eric-rudolphs-confession |archive-url= https://web.archive.org/web/20131203065049/http://www.npr.org/templates/story/story.php?storyId=4600480 |archive-date=December 3, 2013 |title=Full Text of Eric Rudolph's Confession |publisher= [[National Public Radio|NPR]] |date=April 14, 2005 |access-date=July 3, 2024 |url-status=live }}{{void|Fabrickator|comment|alternate archive url:https://web.archive.org/web/20240524131433/https://www.npr.org/2005/04/14/4600480/full-text-of-eric-rudolphs-confession}}</ref> For five years, Rudolph was listed as one of the [[FBI Ten Most Wanted Fugitives]] until he was caught in 2003.

In 2005, as part of a [[plea bargain]], Rudolph pleaded guilty to numerous state and federal [[homicide]] charges and accepted four [[consecutive life sentences]] in exchange for avoiding a trial and a potential [[Capital punishment in the United States|death sentence]]. He remains incarcerated at the [[ADX Florence]] [[supermax prison]] near [[Florence, Colorado]].

==Early life==
Rudolph was born in [[Merritt Island, Florida]], in 1966.<ref name="TruTV">{{cite web|last=Noe|first=Denise|title=Eric Rudolph: Serial Bomber|url=http://www.trutv.com/library/crime/terrorists_spies/terrorists/eric_rudolph/6.html|publisher=TruTV|access-date=January 1, 2011|archive-url=https://web.archive.org/web/20110305022651/http://www.trutv.com/library/crime/terrorists_spies/terrorists/eric_rudolph/6.html|archive-date=March 5, 2011|url-status=live}}</ref> After his father Robert died in 1981, he moved with his mother and siblings to [[Nantahala, North Carolina|Nantahala]], [[Macon County, North Carolina|Macon County]], in western [[North Carolina]].<ref name="SPLC">{{Citation | year = 2001 | title = Tim and Sarah Gayman Discuss Growing Up in the Anti-Semitic Christian Identity Movement | periodical = Intelligence Report | edition = Summer 2001 | issue = 102 | publisher = Southern Poverty Law Center | url = http://www.splcenter.org/get-informed/intelligence-report/browse-all-issues/2001/summer/coming-out?page=0,1 | access-date = August 16, 2012 | archive-url = https://web.archive.org/web/20110712035613/http://www.splcenter.org/get-informed/intelligence-report/browse-all-issues/2001/summer/coming-out?page=0,1 | archive-date = July 12, 2011 | url-status = live | df = mdy-all }}</ref><ref>{{cite news |title=Deborah Rudolph Speaks Out About Her Former Brother-In-Law, Olympic Park Bomber Eric Robert Rudolph|url=https://www.splcenter.org/resources/reports/deborah-rudolph-speaks-out-about-her-former-brother-law-olympic-park-bomber-eric-robert/|website=SPLCenter |access-date=May 20, 2025|date=November 29, 2001 |archive-url=https://web.archive.org/web/20250123034040/https://www.splcenter.org/resources/reports/deborah-rudolph-speaks-out-about-her-former-brother-law-olympic-park-bomber-eric-robert/|archive-date=January 23, 2025 |url-status=live}}</ref>

Rudolph attended ninth grade at the [[Nantahala School]] but dropped out after that year and worked as a carpenter with his older brother Daniel. When Rudolph was 18, he spent time with his mother at a [[Christian Identity]] compound in [[Schell City, Missouri]] known as the [[Church of Israel]].<ref name="CNN">{{cite news |title=Rudolph's mother: Son not a 'monster' |url=https://www.cnn.com/2005/LAW/08/22/rudolph.mother/index.html |website=[[CNN]] |access-date=February 14, 2011 |date=April 4, 2005 |archive-url=https://web.archive.org/web/20140715230302/http://www.cnn.com/2005/LAW/08/22/rudolph.mother/index.html?_s=PM%3ALAW |archive-date=July 15, 2014 |url-status=live }}</ref>

After Rudolph received his [[General Educational Development|GED]], he enlisted in the U.S. Army, undergoing basic training at [[Fort Benning]] in [[Georgia (U.S. state)|Georgia]]. He was [[Military discharge|discharged]] in January 1989, due to [[marijuana]] use, while serving with the [[101st Airborne Division]] at [[Fort Campbell]], [[Kentucky]].<ref>{{cite news|first1=Jeffrey|last1=Gettleman|author-link1=Jeffrey Gettleman|first2=David M.|last2=Halbfinger|newspaper=[[The New York Times]]|url=https://www.nytimes.com/2003/06/01/us/suspect-in-96-olympic-bombing-and-3-other-attacks-is-caught.html|title=Suspect in '96 Olympic Bombing And 3 Other Attacks Is Caught|archive-url=https://web.archive.org/web/20170204132818/http://www.nytimes.com/2003/06/01/us/suspect-in-96-olympic-bombing-and-3-other-attacks-is-caught.html |date=June 1, 2003|access-date=December 26, 2014|archive-date=February 4, 2017 }}</ref> In 1988, the year before his discharge, Rudolph had attended the [[United States Army Air Assault School|Air Assault School]] at Fort Campbell. He attained the rank of [[Specialist (rank)|Specialist]]/[[Uniformed services pay grades of the United States|E-4]].

Rudolph joined several [[White supremacy|white supremacist groups]] in the years before he perpetrated the bombings.<ref>{{cite web |title=WashingtonPost.com: Eric Rudolph |url=https://www.washingtonpost.com/wp-srv/national/longterm/rudolph/bio.htm |website=www.washingtonpost.com |access-date=10 September 2022 |date=December 1998 |archive-date=August 29, 2000 |archive-url=https://web.archive.org/web/20000829040809/http://washingtonpost.com/wp-srv/national/longterm/rudolph/bio.htm |url-status=live }}</ref><ref>{{cite web |title=Why did Rudolph do it? - Apr 13, 2005 |url=https://www.cnn.com/2005/US/04/11/schuster.column/index.html |website=www.cnn.com |access-date=10 September 2022 |date=12 April 2005 |archive-date=July 8, 2022 |archive-url=https://web.archive.org/web/20220708054044/http://www.cnn.com/2005/US/04/11/schuster.column/index.html |url-status=live }}</ref>

==Bombings==
{{further|Centennial Olympic Park bombing|Otherside Lounge bombing}}
At age 29, Rudolph perpetrated the Centennial Olympic Park bombing in [[Atlanta]], which occurred on July 27, 1996, during the [[1996 Summer Olympics]]. He made two anonymous 911 calls, warning about the bomb before it detonated.<ref name=fulltext>[http://www.armyofgod.com/EricRudolphStatement.html Full text of Eric Rudolph's written statement] {{Webarchive|url=https://web.archive.org/web/20061028003642/http://www.armyofgod.com/EricRudolphStatement.html |date=October 28, 2006 }} Army of God website</ref> The blast killed one spectator and wounded 111 others.<ref name="bonvillian">{{cite news|first=Crystal|last=Bonvillian|url=https://www.ajc.com/news/national/serial-bomber-eric-rudolph-targeted-olympics-gay-clubs-abortion-clinics/2Pm8Uoj3XucMLRowsheGtL/|title=Serial bomber Eric Rudolph targeted Olympics, gay club, abortion clinics|newspaper=[[Atlanta Journal-Constitution]]|date=March 19, 2018|access-date=April 13, 2018|archive-date=April 17, 2021|archive-url=https://web.archive.org/web/20210417170646/https://www.ajc.com/news/national/serial-bomber-eric-rudolph-targeted-olympics-gay-clubs-abortion-clinics/2Pm8Uoj3XucMLRowsheGtL/|url-status=live}}</ref> A 40-year-old [[Turkey|Turkish]] news [[videographer|cameraman]] suffered a fatal [[myocardial infarction|heart attack]] while running to the scene.<ref name=Times-Uzunyol>{{cite news |url=https://www.nytimes.com/1996/07/28/us/bomb-at-the-olympics-heart-ailment-kills-war-survivor-in-altanta.html |title=BOMB AT THE OLYMPICS; Heart Ailment Kills War Survivor in Altanta [sic] |date=July 28, 1996 |access-date=August 26, 2020 |newspaper=[[The New York Times]] |archive-date=November 6, 2020 |archive-url=https://web.archive.org/web/20201106064814/https://www.nytimes.com/1996/07/28/us/bomb-at-the-olympics-heart-ailment-kills-war-survivor-in-altanta.html |url-status=live }}</ref><ref>{{cite news |title= In Atlanta, Fear Roams Hand In Hand With Anger |first= Jeff |last= Jacobs |date= July 28, 1996 |url= https://www.courant.com/1996/07/28/in-atlanta-fear-roams-hand-in-hand-with-anger/ |work= [[Hartford Courant]] |access-date= August 27, 2020 |archive-date= October 19, 2013 |archive-url= https://web.archive.org/web/20131019152603/http://articles.courant.com/1996-07-28/news/9607280085_1_alice-s-hawthorne-explosion-centennial-olympic-park |url-status= live }}</ref><ref name="bonvillian" />

Rudolph's motive for the bombing, according to his April 13, 2005 statement, was political:
<blockquote>In the summer of 1996, the world converged upon Atlanta for the Olympic Games. Under the protection and auspices of the regime in Washington millions of people came to celebrate the ideals of global [[socialism]]. Multinational corporations spent billions of dollars, and Washington organized an army of security to protect these best of all games. Even though the conception and the purpose of the so-called Olympic movement is to promote the values of global socialism as perfectly expressed in the song "[[Imagine (John Lennon song)|Imagine]]" by [[John Lennon]], which was the theme of the 1996 Games—even though the purpose of the Olympics is to promote these ideals, the purpose of the attack on July 27 was to confound, anger and embarrass the Washington government in the eyes of the world for its abominable sanctioning of [[abortion law|abortion on demand]].

The plan was to force the cancellation of the games, or at least create a state of insecurity in order to empty the streets around the venues and thereby eat into the vast amounts of money that had been invested in them.<ref name="fulltext"/></blockquote>

Rudolph's statement cleared [[Richard Jewell]], a Centennial Olympic Park security guard, of any involvement in the bombing. Despite having been initially hailed as a hero for being the first one to spot Rudolph's explosive device and helping to clear the area, Jewell came under FBI scrutiny in the days following the attack, ultimately becoming the prime suspect and the subject of international media attention.

Rudolph confessed to three other bombings: 1) an [[abortion clinic]] in the Atlanta suburb of [[Sandy Springs, Georgia|Sandy Springs]] on January 16, 1997, 2) the Otherside Lounge of Atlanta, a [[gay bar|lesbian bar]], on February 21, 1997, injuring five,<ref>{{cite press release|url=http://www.justice.gov/opa/pr/2005/April/05_crm_176.htm|title=Eric Robert Rudolph To Plead Guilty To Serial Bombing Attacks In Atlanta And Birmingham; Will Receive Life Sentences|publisher=[[U.S. Justice Department]]|date=April 8, 2005|archive-url=https://web.archive.org/web/20130414003813/http://www.justice.gov/opa/pr/2005/April/05_crm_176.htm|archive-date=April 14, 2013}}</ref> and 3) an abortion clinic in [[Birmingham, Alabama]] on January 29, 1998, killing [[Birmingham Police Department|Birmingham police officer]] Robert Sanderson, who was off-duty but working as security in uniform, and critically injuring nurse [[Emily Lyons]]. Rudolph's [[nail bomb|bombs contained nails]], which acted as [[fragmentation (weaponry)|shrapnel]].

==Fugitive==
Rudolph was first identified as a suspect in the Alabama bombing by the [[United States Department of Justice|Department of Justice]] in February 1998, following tips from two witnesses, Jeffrey Tickal and Jermaine Hughes. Tickal and Hughes observed Rudolph departing the scene and noted his appearance and truck license plate.<ref>{{cite news|title=They Didn't Catch Rudolph, but They Stopped Him Cold|url=https://www.latimes.com/archives/la-xpm-2005-apr-15-na-witness15-story.html|access-date=April 16, 2013|newspaper=[[Los Angeles Times]]|date=April 15, 2005|first1=Ellen|last1=Barry|first2=Jenny|last2=Jarvie|archive-url=https://web.archive.org/web/20141227092743/http://articles.latimes.com/2005/apr/15/nation/na-witness15|archive-date=December 27, 2014|url-status=live}}</ref> He was named as a suspect in the other Atlanta and Alabama incidents in October.

On May 5, 1998, he became the 454th fugitive listed by the FBI on the [[FBI Ten Most Wanted Fugitives, 1990s|Ten Most Wanted]] list. The FBI considered him to be armed and extremely dangerous, and offered a $1 million reward for information leading directly to his arrest. He spent more than five years in the [[Appalachian Mountains|Appalachian]] wilderness as a fugitive, during which time federal and amateur search teams scoured the area without success.

Rudolph's family supported him and believed he was innocent of all charges.<ref>{{cite web|first=Henry|last=Schuster|website=[[CNN]]|url=http://www.cnn.com/2005/US/04/11/schuster.column/index.html|title=Why did Rudolph do it?|date=April 15, 2005|access-date=November 26, 2006|archive-url=https://web.archive.org/web/20060624092056/http://www.cnn.com/2005/US/04/11/schuster.column/index.html |archive-date=June 24, 2006 }}</ref> They were questioned and placed under [[surveillance]].<ref>{{cite web|first=Jeff|last=Stein|website=[[Salon.com]]|url=http://www.salon.com/1999/01/29/newsc_20/|title=A twisted tale of two brothers|date=January 29, 1999|archive-url=https://web.archive.org/web/20160911184055/http://www.salon.com/1999/01/29/newsc_20/ |archive-date=September 11, 2016 }}</ref> On March 7, 1998, Rudolph's older brother, Daniel, videotaped himself cutting off his left hand with a [[radial arm saw]] in order to, in his words, "send a message to the FBI and the media."<ref>{{cite news|title=Bombing suspect's brother cuts hand off with saw|url=http://www-cgi.cnn.com/US/9803/09/briefs.pm/rudolph.amputation/|archive-url=https://web.archive.org/web/20001011025711/http://www-cgi.cnn.com/US/9803/09/briefs.pm/rudolph.amputation/|author=|date=March 9, 1998|website=[[CNN]]|archive-date=October 11, 2000|access-date=November 26, 2006}}</ref> The hand was successfully reattached later by surgeons.<ref>{{cite news | url=https://www.nytimes.com/1998/03/11/us/bomb-suspect-s-brother-mutilates-himself.html | work=The New York Times | title=Bomb Suspect's Brother Mutilates Himself | date=March 11, 1998 | access-date=February 16, 2017 | archive-url=https://web.archive.org/web/20170204132742/http://www.nytimes.com/1998/03/11/us/bomb-suspect-s-brother-mutilates-himself.html | archive-date=February 4, 2017 | url-status=live }}</ref>
According to Rudolph's own writings, he survived during his years as a fugitive by camping in the [[Nantahala National Forest]] near Cherokee and Graham Counties, in North Carolina, by gathering [[acorn]]s and [[salamander]]s, pilfering vegetables from gardens, stealing grain from a grain silo, and raiding dumpsters in [[Murphy, North Carolina]].<ref>[http://www.armyofgod.com/EricRudolphLickFloor.html Lick the Floor] {{Webarchive|url=https://web.archive.org/web/20070403223943/http://www.armyofgod.com/EricRudolphLickFloor.html |date=April 3, 2007 }} January 27, 2004</ref><ref>{{cite web |url=http://www.armyofgod.com/EricRudolphTil.html |title=Lil |publisher=Armyofgod.com |access-date=January 4, 2012 |archive-url=https://web.archive.org/web/20050627010308/http://www.armyofgod.com/EricRudolphTil.html |archive-date=June 27, 2005 |url-status=live }}</ref>

==Arrest and guilty plea==

Rudolph was arrested in [[Murphy, North Carolina]], on May 31, 2003, by rookie police officer Jeffrey Scott Postell of the Murphy Police Department while Rudolph was looking through a dumpster behind a [[Save-A-Lot]] store at about 4:00{{spaces}}a.m. Postell, on routine patrol, had initially suspected a burglary in progress.<ref name="CNN captured">{{cite news |title=Atlanta Olympic bombing suspect arrested|website=[[CNN]]|url=http://www.cnn.com/2003/US/05/31/rudolph.main/|archive-url=https://web.archive.org/web/20030621163319/http://www.cnn.com/2003/US/05/31/rudolph.main/|archive-date=June 21, 2003|date=May 31, 2003}}</ref>

Rudolph was unarmed and did not resist arrest. When arrested, he was clean-shaven with a trimmed mustache, had dyed black hair and wore a camouflage jacket, work clothes, and new sneakers.<ref name="CNN captured"/><ref>{{cite news |title=Finally caught: 5 year hunt for Eric Rudolph ends|url=http://www.ajc.com/news/content/news/rudolph/0603/01main.html|work=[[Atlanta Journal-Constitution]]|access-date=December 7, 2013|first1=Bill|last1=Torpy|first2=Don|last2=Plummer|archive-url=https://web.archive.org/web/20030605155128/http://www.ajc.com/news/content/news/rudolph/0603/01main.html|archive-date=June 5, 2003|date=June 1, 2003}}</ref> Federal authorities charged him on October 14, 2003. Rudolph was initially defended by attorney [[Richard S. Jaffe]]. After Jaffe withdrew, he was represented by [[Judy Clarke]].

In April 2005, the Department of Justice announced that Rudolph had agreed to a [[plea bargain]] under which he would plead guilty to all charges he was accused of in exchange for avoiding the [[death penalty]]. The deal was confirmed after the FBI found {{convert|250|lb}} of [[dynamite]] he had hidden in the forests of [[North Carolina]]. His revealing the hiding places of the dynamite was a condition of his plea agreement.<ref>{{cite news|last=Eggen|first=Dan|title=Rudolph To Plead Guilty to Bombings|url=https://www.washingtonpost.com/wp-dyn/articles/A37544-2005Apr8.html|newspaper=[[The Washington Post]]|access-date=December 7, 2013|date=April 9, 2005|archive-url=https://web.archive.org/web/20140630002149/http://www.washingtonpost.com/wp-dyn/articles/A37544-2005Apr8.html|archive-date=June 30, 2014|url-status=live}}</ref> He made his pleas in person in Birmingham and Atlanta courts on April 13.<ref>{{cite news|last=Lohr|first=Kathy|title=Rudolph Pleads Guilty, Admits Little Remorse|url=https://www.npr.org/templates/story/story.php?storyId=4599872|website=[[NPR]]|access-date=December 7, 2013|date=April 13, 2005|archive-url=https://web.archive.org/web/20131213150211/http://www.npr.org/templates/story/story.php?storyId=4599872|archive-date=December 13, 2013|url-status=live}}</ref>

Rudolph released a statement explaining his actions; he rationalized the bombings as serving the cause of [[anti-abortion]] and [[anti-gay]] terrorism. In his statement, he claimed that he had "deprived the government of its goal of sentencing me to death," and that "the fact that I have entered an agreement with the government is purely a tactical choice on my part and in no way legitimates the [[moral authority]] of the government to judge this matter or impute my guilt."<ref>{{cite news |url=https://www.usatoday.com/news/nation/2005-04-13-rudolph-statement_x.htm |title=Excerpts from Eric Rudolph's statement |date=April 13, 2005 |newspaper=[[USA Today]] |access-date=August 27, 2017 |archive-url=https://web.archive.org/web/20090211085812/http://www.usatoday.com/news/nation/2005-04-13-rudolph-statement_x.htm |archive-date=February 11, 2009 |url-status=live }}</ref>

The terms of the plea agreement were that Rudolph would be sentenced to four consecutive life terms. He was sentenced July 18, 2005, to two consecutive [[life imprisonment|life term]]s without [[parole]] for the 1998 murder of a police officer.<ref>{{cite web|agency=[[Associated Press]]|url=http://www.foxnews.com/story/0,2933,162790,00.html|title=Eric Rudolph Gets Life Without Parole|website=[[Fox News]]|date=July 18, 2005|access-date=November 26, 2006|archive-url=https://web.archive.org/web/20060829015310/http://www.foxnews.com/story/0,2933,162790,00.html |archive-date=August 29, 2006 }}</ref> He was sentenced for his bombings in Atlanta on August 22, 2005, receiving two consecutive life terms. That same day, Rudolph was sent to the [[ADX Florence]] [[Supermax]] federal prison. Like other Supermax inmates, he spends 23 hours per day alone in his {{convert|80|sqft|m2|adj=on|abbr=off|sp=us}} concrete cell.<ref>{{cite news|first=R. Scott|last=Rappold|url=http://www.gazette.com/display.php?id=1310494&secid=1|title=Olympic bomber Rudolph calls Supermax home|archive-url=https://web.archive.org/web/20051023225608/http://www.gazette.com/display.php?id=1310494&secid=1 |archive-date=October 23, 2005|newspaper=[[The Gazette (Colorado Springs)|Colorado Springs Gazette]]|date=September 14, 2005|access-date=November 26, 2006}}</ref><ref>"[http://www.bop.gov/iloc2/InmateFinderServlet?Transaction=IDSearch&needingMoreList=false&IDType=IRN&IDNumber=18282-058&x=399&y=290 Eric Robert Rudolph] {{Webarchive|url=https://web.archive.org/web/20090614050524/http://www.bop.gov/iloc2/InmateFinderServlet?Transaction=IDSearch&needingMoreList=false&IDType=IRN&IDNumber=18282-058&x=399&y=290 |date=June 14, 2009 }}." [[Federal Bureau of Prisons]]. Retrieved on January 5, 2010.</ref>

Rudolph unsuccessfully tried to have part of his sentence vacated in 2021.<ref>{{Cite web |title=Rudolph v. United States, Civil Action 2:20-cv-08024-CLS {{!}} Casetext Search + Citator |url=https://casetext.com/case/rudolph-v-united-states-13 |access-date=2022-05-17 |website=casetext.com |archive-date=May 17, 2022 |archive-url=https://web.archive.org/web/20220517215518/https://casetext.com/case/rudolph-v-united-states-13 |url-status=live }}</ref>

==Motivations==
After Rudolph's arrest for the bombings, ''[[The Washington Post]]'' reported that the FBI considered Rudolph to have "had a long association with the [[Christian Identity]] movement, which asserts that [[Northern European]] whites are the direct descendants of the [[Ten Lost Tribes|lost tribes of Israel]], God's [[Jews as a chosen people|chosen people]]."<ref name="Cooperman">{{Cite news |last=Cooperman |first=Alan |date=June 2, 2003 |title=Is Terrorism Tied To Christian Sect? |language=en-US |newspaper=[[The Washington Post]] |url=https://www.washingtonpost.com/archive/politics/2003/06/02/is-terrorism-tied-to-christian-sect/7510f762-4ac6-43b5-9b15-479a8cef16d4/ |access-date=November 12, 2020 |issn=0190-8286 |archive-date=November 24, 2020 |archive-url=https://web.archive.org/web/20201124072527/https://www.washingtonpost.com/archive/politics/2003/06/02/is-terrorism-tied-to-christian-sect/7510f762-4ac6-43b5-9b15-479a8cef16d4/ |url-status=live }}</ref> Christian Identity is a [[white supremacism|white supremacist]] movement which holds the view that those who are not white Christians cannot be saved.<ref>{{cite book |last=Quarles |first=Chester L. |title=Christian Identity: The Aryan American Bloodline Religion |publisher=[[McFarland & Company]] |location=Jefferson, North Carolina|year=2004 |page=68 |url=https://books.google.com/books?id=r5BzY2eeyngC |isbn=978-0-7864-1892-3 |access-date=November 2, 2016 |archive-url=https://web.archive.org/web/20170424002004/https://books.google.com/books?id=r5BzY2eeyngC |archive-date=April 24, 2017 |url-status=live }}</ref> In the same article, the ''Post'' reported that some FBI investigators believe that Rudolph may have written letters in which he claimed responsibility for the nightclub and abortion clinic bombings on behalf of the [[Army of God (USA)|Army of God]], a group that sanctions the use of force to combat abortions and is associated with Christian Identity.<ref>{{cite web | title= The Second Defensive Action Statement | url= http://www.armyofgod.com/defense2.html | access-date= May 14, 2007| archive-url= https://web.archive.org/web/20081006164433/http://www.armyofgod.com/defense2.html | archive-date= October 6, 2008 | url-status= live | df= mdy-all }}</ref><ref name="Cooperman" />

In a statement released after he entered a guilty plea, Rudolph denied being a supporter of the Christian Identity movement, claiming that his involvement amounted to a brief association with the daughter of a Christian Identity adherent, later identified as [[Daniel Gayman]]. When asked about his religion he said "I was born a Catholic, and with forgiveness I hope to die one."<ref name="AP">{{cite news |title=Eric Rudolph, proud killer |url=http://legacy.decaturdaily.com/decaturdaily/news/050414/rudolph.shtml |access-date=December 11, 2006|agency=[[Associated Press]] |newspaper=[[The Decatur Daily]] |date=April 14, 2005 |first=Kristen|last=Wyatt |archive-url=https://web.archive.org/web/20080421100819/http://legacy.decaturdaily.com/decaturdaily/news/050414/rudolph.shtml |archive-date=April 21, 2008 |url-status=live }}</ref><ref>{{cite news|first=Blake|last=Morrison|url=https://www.usatoday.com/news/nation/2005-07-05-rudolph-cover-partone_x.htm?POE=NEWISVA|title=Special report: Eric Rudolph writes home|newspaper=[[USA Today]]|date=July 6, 2005|access-date=November 26, 2006|archive-url=https://web.archive.org/web/20110528052340/http://www.usatoday.com/news/nation/2005-07-05-rudolph-cover-partone_x.htm?POE=NEWISVA |archive-date=May 28, 2011 }}</ref> In other written statements, Rudolph has cited [[Bible|biblical]] passages and offered religious motives for his militant opposition to abortion.<ref name="fulltext"/>

Some books and media outlets, like the [[NPR]] radio program ''[[On Point]],'' have described Rudolph as a "Christian Identity extremist";<ref>{{cite web |title=Most Wanted Extremist, Eric Rudolph, Caught |url=http://www.onpointradio.org/shows/2003/06/20030603_b_main.asp |url-status=dead |archive-url=https://web.archive.org/web/20061002032921/http://www.onpointradio.org/shows/2003/06/20030603_b_main.asp |archive-date=October 2, 2006 |date=June 3, 2003}}</ref> ''[[Harper's Magazine]]'' referred to him as a "[[Christian terrorism|Christian terrorist]]."<ref name="harpersterror">{{cite news |url=http://harpers.org/blog/2005/04/weeklyreview2005-04-19/ |work=Harpers Magazine |title=Week In Review |access-date=January 8, 2016|archive-url=https://web.archive.org/web/20150929120620/http://harpers.org/blog/2005/04/weeklyreview2005-04-19/ |archive-date=September 29, 2015 |url-status=live}}</ref> The [[Voice of America]] reported that Rudolph could be seen as part of an "attempt to try to use a [[Christianity|Christian faith]] to try to forge a kind of racial and social purity."<ref>{{cite web |url=https://www.voanews.com/archive/arrest-accused-olympic-park-bomber-sparks-debate-christian-terrorism-2003-06-05 |url-status=dead |archive-url=https://web.archive.org/web/20210802230709/https://www.voanews.com/archive/arrest-accused-olympic-park-bomber-sparks-debate-christian-terrorism-2003-06-05 |archive-date=August 2, 2021 |title=Arrest of Accused Olympic Park Bomber Sparks Debate on 'Christian Terrorism' |date=June 5, 2003 |website=[[VOANews]] |access-date=July 3, 2024}}{{cbignore}}</ref> Writing in 2004, authors [[Michael Shermer]] and [[Dennis McFarland]] saw Rudolph's story as an example of "religious extremism in America," warning that the phenomenon he represented was "particularly potent when gathered together under the umbrella of [[American militia movement|militia groups]]".<ref>{{cite book |title=[[The Science of Good and Evil]] |last=Shermer |first=Michael |isbn=9781429996754 |date=December 9, 2004 |publisher=Henry Holt and Co}}</ref>

The [[Anti-Defamation League]] noted in 2003 that "extremist chatter on the Internet has praised Rudolph as 'a hero' and some followers of [[hate group]]s are calling for further acts of violence to be modeled after the bombings he is accused of committing."<ref>[[Anti-Defamation League]], "[http://www.adl.org/PresRele/Teror_92/4264_72.htm Extremist Chatter Praises Eric Rudolph as 'Hero.'] {{webarchive|url=https://web.archive.org/web/20061124235649/http://www.adl.org/PresRele/Teror_92/4264_72.htm |date=November 24, 2006 }}", June 3, 2003. Retrieved November 26, 2006.</ref>

In a letter to his mother from prison, Rudolph has written, "Many good people continue to send me money and books. Most of them have, of course, an agenda; mostly [[born again (Christianity)|born-again]] Christians looking to save my soul. I suppose the assumption is made that because I'm in here I must be a 'sinner' in need of [[salvation]], and they would be glad to sell me a ticket to heaven. I do appreciate their charity, but I could really do without the condescension. They have been so nice I would hate to break it to them that I really prefer [[Friedrich Nietzsche|Nietzsche]] to the [[Bible]]." His mother would state she saw this as evidence of his intellectual side, not as a denial of his Catholic faith.<ref>{{cite news|first=Blake|last=Lowenstein|url=https://www.usatoday.com/news/nation/2005-07-05-rudolph-cover-partone_x.htm?POE=NEWISVA|title=Special report: Eric Rudolph writes home|newspaper=[[USA Today]]|date=July 5, 2005|archive-url=https://web.archive.org/web/20110528052340/http://www.usatoday.com/news/nation/2005-07-05-rudolph-cover-partone_x.htm?POE=NEWISVA |archive-date=May 28, 2011 }}</ref>

Rudolph has said, "The truth is I am a [[Christianity|Christian]]".<ref>{{Cite web |url=https://www.armyofgod.com/EricRudolphRacism.html |title=Eric Rudolph Racism "The truth is I am a Christian" |access-date=December 18, 2017 |archive-url=https://web.archive.org/web/20171222050831/https://www.armyofgod.com/EricRudolphRacism.html |archive-date=December 22, 2017 |url-status=live }}</ref> Rudolph remained unremorseful for his actions and, in a statement before the court, called his acts against abortion providers a “moral duty.” “As I go to a prison cell for a lifetime, I know that ‘I have fought a good fight, I have finished my course, I have kept the faith,’” Rudolph said, quoting scripture.<ref>{{Cite web |first=Crystal|last=Bonvillian|url=https://www.ajc.com/news/national/serial-bomber-eric-rudolph-targeted-olympics-gay-clubs-abortion-clinics/2Pm8Uoj3XucMLRowsheGtL/ |title=Serial bomber Eric Rudolph targeted Olympics, gay club, abortion clinics |newspaper=[[Atlanta Journal-Constitution]]|date=March 19, 2018|access-date=April 13, 2018 |archive-url=https://web.archive.org/web/20180414012046/https://www.ajc.com/news/national/serial-bomber-eric-rudolph-targeted-olympics-gay-clubs-abortion-clinics/2Pm8Uoj3XucMLRowsheGtL/ |archive-date=April 14, 2018 |url-status=live }}</ref>
In his work "White Lies: Abortion, Eugenics, And Racism" Rudolph expresses his admirations towards Catholicism.

==Writings from prison==
[[File:Florence ADMAX.jpg|thumb|Rudolph is incarcerated at [[ADX Florence]]]]
Essays written by Rudolph that condone violence and militant action have been published on the Internet by an [[Army of God (United States)|Army of God]] anti-abortion activist.<ref>[http://www.armyofgod.com/EricRudolphHomepage.html Army of God's homepage for Rudolph] {{Webarchive|url=https://web.archive.org/web/20070516044053/http://www.armyofgod.com/EricRudolphHomepage.html |date=May 16, 2007 }} December 18, 2007</ref> Although victims maintain that Rudolph's messages are harassment and could incite violence, the prison can do little to restrict their publication, according to [[Alice Martin]], who was the [[United States District Court for the Northern District of Alabama|United States Attorney for the Northern District of Alabama]] when Rudolph was prosecuted for the [[Alabama]] bombing. "An inmate does not lose his [[Freedom of speech in the United States|freedom of speech]]," she said.<ref>{{cite web|first=Jay|last=Reeves|url=http://www.foxnews.com/wires/2007May14/0,4670,EricRudolph,00.html|title=Extremist Taunts His Victims From Prison|website=[[Fox News]]|date=May 14, 2007|archive-url=https://web.archive.org/web/20070516044017/http://www.foxnews.com/wires/2007May14/0,4670,EricRudolph,00.html |archive-date=May 16, 2007}}</ref>

As reported in an April 8, 2013, Alabama blog article,<ref>{{cite news|first=Kent|last=Faulk|url=http://blog.al.com/spotnews/2013/04/birmingham_abortion_clinic_bom.html|title=Birmingham abortion clinic bomber Eric Robert Rudolph fights to get profits from his book|newspaper=[[The Birmingham News]]|access-date=April 17, 2013|archive-url=https://web.archive.org/web/20130413061815/http://blog.al.com/spotnews/2013/04/birmingham_abortion_clinic_bom.html|archive-date=April 13, 2013|url-status=live|date=April 8, 2013}}</ref> Rudolph's book ''Between the Lines of Drift: The Memoirs of a Militant'' was published, with help from his brother, by [[Lulu (company)|Lulu.com]] in February 2013. In April 2013, the U.S. Attorney General seized $200 to help pay off the $1 million that Rudolph owes in restitution to the state of Alabama. The book has since been republished and has been made available through the [[Army of God (terrorist organization)|Army of God]] website.<ref>{{cite news|url=https://www.denverpost.com/2013/02/23/bomber-eric-rudolph-self-publishes-autobiography-from-prison/|title=Bomber Eric Rudolph self-publishes autobiography from prison|newspaper=[[Denver Post]]|date=February 23, 2013|accessdate=June 20, 2021|agency=[[Associated Press]]|archive-date=June 22, 2021|archive-url=https://web.archive.org/web/20210622114405/https://www.denverpost.com/2013/02/23/bomber-eric-rudolph-self-publishes-autobiography-from-prison/|url-status=live}}</ref>

==In popular culture==
In a fictionalized portrayal, Rudolph appears as the primary [[antagonist]] in season 2 of the drama anthology series [[Manhunt (2017 TV series)|''Manhunt'']], portrayed by [[Jack Huston]].<ref>{{Cite web |title=Jack Huston plays the 'ultimate sociopath' in 'Manhunt: Deadly Games' - UPI.com |url=https://www.upi.com/Entertainment_News/TV/2020/02/03/Jack-Huston-plays-the-ultimate-sociopath-in-Manhunt/1671580732955/ |access-date=2024-07-29 |website=UPI |language=en}}</ref><ref>{{Cite news
| last=Spencer
| first=Samuel
| title='Manhunt Deadly Games': The True Story Details the Netflix Show Changed
| magazine=[[Newsweek]]
| publication-date=December 14, 2020
| url=https://www.newsweek.com/manhunt-deadly-games-real-life-true-story-richard-jewell-eric-rudolph-changes-netflix-1554533?amp=1
| access-date=January 9, 2021
| archive-date=November 12, 2021
| archive-url=https://web.archive.org/web/20211112011115/https://www.newsweek.com/manhunt-deadly-games-real-life-true-story-richard-jewell-eric-rudolph-changes-netflix-1554533?amp=1
| url-status=live
}}</ref> He is portrayed by Eric Mendenhall in the 2019 film ''[[Richard Jewell (film)|Richard Jewell]]''.

==See also==
* [[Anti-abortion violence]]
* [[Army of God (United States)]]
* [[Christian terrorism]]
* [[Forensic linguistics]]
* [[Former FBI Ten Most Wanted Fugitives]]

==References==
{{reflist}}

==Further reading==
{{wikisource|works=or}}
* {{cite book|first=Maryanne|last=Vollers|author-link=Maryanne Vollers|year=2006|title=Lone Wolf: Eric Rudolph: Murder, Myth and the Pursuit of an American Outlaw|publisher=HarperCollins|isbn=9780060598624}}
* {{cite book|first1=Henry|last1=Schuster|first2=Charles|last2=Stone|title=Hunting Eric Rudolph|publisher=Berkley Books|year=2005|isbn=0-425-19936-3}}
* [https://www.npr.org/templates/story/story.php?storyId=4600480 Text of Rudolph's statement]
* [https://web.archive.org/web/20201209081043/https://www.foxnews.com/story/raw-data-timeline-in-eric-rudolph-case Timeline in Eric Rudolph Case] – May 31, 2003
* 'Run, Rudolph, Run' essay in ''[[Seek: Reports from the Edges of America & Beyond]]'' (2001) by [[Denis Johnson]]
* {{cite book |url=https://www.armyofgod.com/EricLinesOfDrift%201_18_15Opened.pdf |title=Between the Lines of Drift: The Memoirs of a Militant, Third Edition |date=2015 |author=Eric Rudolph}}
{{Authority control}}

{{DEFAULTSORT:Rudolph, Eric}}
[[Category:1966 births]]
[[Category:1996 murders in the United States]]
[[Category:1996 Summer Olympics]]
[[Category:1998 murders in the United States]]
[[Category:20th-century American criminals]]
[[Category:20th-century American memoirists]]
[[Category:Activists from North Carolina]]
[[Category:American anti-abortion activists]]
[[Category:American male criminals]]
[[Category:American people convicted of murder]]
[[Category:American people imprisoned on terrorism charges]]
[[Category:American prisoners sentenced to life imprisonment]]
[[Category:American white supremacists]]
[[Category:Army of God (United States)]]
[[Category:Bombers (people)]]
[[Category:Christian fundamentalists]]
[[Category:Christian Identity people]]
[[Category:Crime in Atlanta]]
[[Category:Criminals from Florida]]
[[Category:Inmates of ADX Florence]]
[[Category:Lesbophobic violence]]
[[Category:Living people]]
[[Category:Military personnel from Florida]]
[[Category:People convicted of murder by the United States federal government]]
[[Category:People convicted on terrorism charges]]
[[Category:People from Macon County, North Carolina]]
[[Category:People from Merritt Island, Florida]]
[[Category:Perpetrators of religiously motivated violence in the United States]]
[[Category:Prisoners sentenced to life imprisonment by the United States federal government]]
[[Category:Serial bombers]]
[[Category:United States Army soldiers]]
[[Category:Violence against LGBTQ people in the United States]]
[[Category:Right-wing terrorism]]

Srebrenica massacre

2025-06-20T12:12:19Z

173.79.19.248: Undid revision 1295839707 by Roger 8 Roger (talk) ditto. please do not use this tool if you are not manually checking that the output improves the situation; wrapping something in a cite template is not an improvement if you also destroy other information at the same time

{{Short description|1995 mass murder by the Bosnian Serb Army}}
{{Use dmy dates|date=August 2014}}
{{Infobox civilian attack
| title = Srebrenica massacre Srebrenica genocide
| partof = the [[Bosnian War]] and the [[Bosnian genocide]]
| image = Srebrenica massacre memorial gravestones 2009 1.jpg
| map = {{Location map|Bosnia and Herzegovina
| width = 220
| float = center
| label = Srebrenica
| marksize = 8
| lat_deg = 44.1
| lon_deg = 19.3
}}
| caption = Some of the gravestones for the nearly 7,000 identified victims buried at the [[Srebrenica Genocide Memorial|Srebrenica-Potočari Memorial and Cemetery for the Victims of the 1995 genocide]].<ref>{{Cite web|url=https://sarajevotimes.com/at-least-80-more-mass-graves-with-remains-of-srebrenica-genocide-victims/|title=At least 80 more Mass Graves with Remains of Srebrenica Genocide Victims|date=4 July 2023|website=Sarajevo Times}}</ref>
| location = [[Srebrenica]], Bosnia and Herzegovina
| native_name = Genocid u Srebrenici / Геноцид у Сребреници
| native_name_lang = sh
| coordinates = {{coord|44|06|N|19|18|E|type:event_dim:10km|display=inline}}
| target = [[Bosniaks|Bosniak]] men and boys
| date = {{start date|df=yes|1995|7|11}} – {{end date and age|df=yes|1995|7|31}}
| type = [[Genocide]], [[ethnic cleansing]], [[genocidal rape]], [[androcide]]
| fatalities = 8,372<ref name="PWM"/>
| motive = Anti-Bosniak sentiment [[Greater Serbia|Serbian irredentism]] [[Islamophobia]] [[Serbianisation]]
| perps = {{plainlist|
*[[Army of Republika Srpska]]<ref name="UN SecGen 10th anniv">[https://www.un.org/News/Press/docs/2005/sgsm9993.doc.htm UN Press Release SG/SM/9993UN, 11/07/2005 "Secretary-General Kofi Annan's message to the ceremony marking the tenth anniversary of the Srebrenica massacre in Potocari-Srebrenica"]. Retrieved 9 August 2010.</ref><ref>{{cite web|url=https://iwpr.net/global-voices/briefly-noted-95|work=Institute for War & Peace Reporting|id=TU No 398|date=18 March 2005|title=Tribunal Update: Briefly Noted|access-date=4 May 2017}}</ref>
*[[Scorpions (paramilitary)|Scorpions]] paramilitary group<ref>{{cite web|url=https://iwpr.net/global-voices/serbia-mladic-%E2%80%9Crecruited%E2%80%9D-infamous-scorpions|title=Serbia: Mladic "Recruited" Infamous Scorpions|first=Daniel|last=Sunter|publisher=Institute for War and Peace Reporting|date=5 August 2005|access-date=30 April 2017}}</ref><ref name="Williams"/><ref>{{cite book |last1=Totten |first1=Samuel |last2=Bartrop |first2=Paul R. |year=2008 |title=Dictionary of Genocide |publisher=Greenwood Publishing Group |location=[[Westport, Connecticut]] |isbn=978-0-31334-644-6 |url=https://books.google.com/books?id=_02aVgTaFDYC |page=390 }}</ref>}}
| dfens =
}}
{{Campaignbox Bosnian War}}
{{multiple image
| align = right
| direction = vertical
| width = 230
| image1 = SrebrenicaStone.jpg
| caption1 = [[Srebrenica Genocide Memorial]] Stone at [[Donji Potočari|Potočari]]
| image2 = Srebrenica 2005 burial.jpg
| caption2 = Burial of 610 identified Bosniaks in 2005
| image3 = Srebrenica2007.jpg
| caption3 = Burial of 465 identified Bosniaks in 2007
| image4 = 20114124.jpg
| caption4 = Burial of 775 identified Bosniaks in 2010
| image5 = Le mémorial de Srebrenica- Potocari.jpg
| caption5 = The Srebrenica-Potočari memorial, and the cemetery for the victims of the genocide.
}}
The '''Srebrenica massacre''',{{efn|{{lang-sh-Latn-Cyrl|Masakr u Srebrenici|Масакр у Сребреници}}}} also known as the '''Srebrenica genocide''',{{efn|{{lang-sh-Latn-Cyrl|Genocid u Srebrenici|Геноцид у Сребреници}}}}<ref>
* {{cite news |last1=Lederer |first1=Edith M. |title=UN approves resolution to commemorate the 1995 Srebrenica genocide annually over Serb opposition |url=https://apnews.com/article/un-commemorate-srebrenica-genocide-bosnia-muslims-serbs-ed58930f9e53639f22939a6efedd478b |work=The Associated Press |date=23 May 2024}}
* {{cite web |title=European Parliament resolution of 15 January 2009 on Srebrenica |url=http://www.europarl.europa.eu/sides/getDoc.do?pubRef=-//EP//TEXT+TA+P6-TA-2009-0028+0+DOC+XML+V0//EN&language=EN |access-date=10 August 2009 |publisher=European Parliament}}
* {{cite web |title=Office of the High Representative – "Decision Enacting the Law on the Center for the Srebrenica-Potocari Memorial and Cemetery for the Victims of the 1995 Genocide" |url=http://www.ohr.int/print/?content_id=40028 |url-status=dead |archive-url=https://web.archive.org/web/20110606075959/http://www.ohr.int/print/?content_id=40028 |archive-date=6 June 2011 |access-date=10 August 2009 |publisher=Office of the High Representative in Bosnia and Herzegovina}}
* {{cite web |title=Youth Initiative for Human Rights in Serbia letter to the Serbian President to commemorate the Srebrenica genocide |url=http://www.advocacynet.org/blogs/index.php/2009/02/11/serbian-ngos-press-for-commemoration-of-?blog=145 |url-status=dead |archive-url=https://web.archive.org/web/20110718144942/http://www.advocacynet.org/blogs/index.php/2009/02/11/serbian-ngos-press-for-commemoration-of-?blog=145 |archive-date=18 July 2011 |access-date=10 August 2009 |publisher=Youth Initiative for Human Rights in Serbia}}
* {{cite news |date=21 August 2006 |title=Mladic shadow hangs over Srebrenica trial |url=https://www.theguardian.com/world/2006/aug/21/warcrimes |access-date=1 November 2008 |work=The Guardian |location=London}}
* {{cite news |author=Mike Corder |date=20 August 2006 |title=Srebrenica Genocide Trial to Restart |url=https://www.washingtonpost.com/wp-dyn/content/article/2006/08/20/AR2006082000360.html |access-date=26 October 2010 |newspaper=The Washington Post}}</ref> was the July 1995 [[genocidal]] killing<ref>* {{cite web|title=International Criminal Tribunal for the former Yugoslavia (ICTY)|url=http://www.icty.org/x/cases/popovic/acjug/en/150130_summary.pdf|access-date=10 July 2015}}
* {{cite news|title=The New York Times|url=https://www.nytimes.com/2001/08/03/world/judge-s-words-cleansing-became-genocide.html?scp=6&sq=srebrenica%20genocide%20krstic%202001&st=cse|access-date=10 July 2015|date=3 August 2001}}
* {{cite web |title=The International Court of Justice |url=http://www.icj-cij.org/docket/files/91/13685.pdf |access-date=10 July 2015 |archive-date=1 March 2011 |archive-url=https://web.archive.org/web/20110301032417/http://www.icj-cij.org/docket/files/91/13685.pdf |url-status=dead }}</ref> of more than 8,000<ref>
* {{cite web |title=ICTY: The Conflicts |publisher=The [[International Criminal Tribunal for the former Yugoslavia]] |access-date=5 August 2013 |url=http://www.icty.org/sid/322}}
* {{cite book |url=https://books.google.com/books?id=2jrsEaRIzFkC&pg=PA81 |title=The United Nations |author=Kirsten Nakjavani Bookmiller |year=2008 |publisher=Infobase Publishing |access-date=4 August 2013|isbn=9781438102993 |page=81}}
* {{cite book |url=https://books.google.com/books?id=0J_JZbLElKkC&pg=PA25 |title=Victory Has a Thousand Fathers: Sources of Success in Counterinsurgency |author1=Christopher Paul |author2=Colin P. Clarke |author3=Beth Grill |year=2010 |publisher=Rand Corporation |access-date=4 August 2013|isbn=9780833050786 |page=25}}
* {{cite news |title=Mladic Arrives in The Hague |date=31 May 2011 |work=[[The New York Times]] |url=https://www.nytimes.com/2011/06/01/world/europe/01serbia.html |first=Marlise |last=Simons}}</ref> [[Bosniak]] Muslim men and boys in and around the town of [[Srebrenica]] during the [[Bosnian War]].<ref>{{cite news|url=https://www.bbc.co.uk/news/world-europe-53346759 |title=Bosnia's Srebrenica massacre 25 years on – in pictures |work=BBC News |date=10 July 2020 |access-date=11 July 2020}}</ref> It was mainly perpetrated by units of the Bosnian Serb [[Army of Republika Srpska]] under [[Ratko Mladić]], though the Serb paramilitary unit [[Scorpions (paramilitary)|Scorpions]] also participated.<ref name="Williams">{{cite news |url=https://www.washingtonpost.com/wp-dyn/content/article/2005/06/24/AR2005062401501_pf.html |title=Srebrenica Video Vindicates Long Pursuit by Serb Activist |newspaper=The Washington Post |access-date=26 May 2011 |first=Daniel |last=Williams}}</ref><ref name="ReferenceC">{{cite web |url=http://www.icty.org/x/cases/kordic_cerkez/tjug/en/kor-tj010226e.pdf |title=ICTY – Kordic and Cerkez Judgement – 3. After the Conflict |access-date=11 July 2012}}</ref> The massacre was the first legally recognised genocide in Europe since the end of [[World War II]].<ref>{{cite web |access-date=2024-07-12 |archive-date=2014-10-06 |archive-url=https://web.archive.org/web/20141006113644/http://iwpr.net/report-news/briefly-noted-67 |title=IMPR Report News |url=http://iwpr.net/report-news/briefly-noted-67|quote=The July 1995 events have become the first legally established case of genocide in Europe since the Second World War.}}</ref>

Before the massacre, the [[United Nations]] (UN) had declared the [[Siege of Srebrenica|besieged enclave of Srebrenica]] a "[[United Nations Safe Areas|safe area]]" under its protection.<ref>{{Cite web |last=Nations |first=United |title=International Day of Reflection and Commemoration of the 1995 Genocide in Srebrenica |url=https://www.un.org/en/observances/srebrenica-genocide-commemoration-day |access-date=2025-01-27 |website=United Nations |language=en}}</ref> A [[UN Protection Force]] contingent of 370<ref name="spiegel.de" /> lightly armed [[Dutchbat|Dutch]] soldiers failed to deter the town's capture and subsequent massacre.<ref>ICTY, ''[https://web.archive.org/web/20010804090142/http://www.un.org/icty/krstic/TrialC1/judgement/index.htm]'', {{cite web |date=2 August 2001 |title=Case No. IT-98-33, United Nations |url=https://www.un.org/icty/krstic/TrialC1/judgement/krs-tj010802e.pdf |url-status=dead |archive-url=https://web.archive.org/web/20060608000517/http://www.un.org/icty/krstic/TrialC1/judgement/krs-tj010802e.pdf |archive-date=8 June 2006 |access-date=8 June 2006}} {{small|685 KB}}, "Findings of Fact", paragraphs 18 and 26</ref><ref>{{cite news|url=http://news.bbc.co.uk/2/hi/europe/7461310.stm|title=UN Srebrenica immunity questioned|date=18 June 2008|publisher=BBC |access-date=1 November 2008}}</ref><ref name="www.vandiepen.com" /><ref>"Under The UN Flag; The International Community and the Srebrenica Genocide" by Hasan Nuhanović, pub. DES Sarajevo, 2007, {{ISBN|978-9958-728-87-7}} [http://www.cobiss.ba/scripts/cobiss?ukaz=DISP&id=2315525138791328&rec=3&sid=1] {{Webarchive|url=https://web.archive.org/web/20150923205224/http://www.cobiss.ba/scripts/cobiss?ukaz=DISP&id=2315525138791328&rec=3&sid=1|date=23 September 2015}}[http://www.cobiss.ba/scripts/cobiss?ukaz=DISP&id=2315525138791328&rec=4&sid=1] {{Webarchive|url=https://web.archive.org/web/20191024091305/https://plus.bh.cobiss.net/opac7/bib/search|date=24 October 2019}}</ref> A list of people missing or killed during the massacre contains 8,372 names.<ref name="PWM">{{cite web|url=http://war-memorial.net/Victims-of-the-Srebrenica-Massacre-4.171|title=Victims of the Srebrenica Massacre |publisher=The Polynational War Memorial|date=|access-date=2 February 2022}}</ref> {{as of|2012|July}}, 6,838 genocide victims had been identified through DNA analysis of body parts recovered from mass graves;<ref>{{cite news|title=Over 7,000 Srebrenica Victims have now been recovered|url=http://www.ic-mp.org/press-releases/over-7000-srebrenica-victims-recovered/|newspaper=ICMP|date=11 July 2012|access-date=11 July 2012|archive-date=25 December 2018|archive-url=https://web.archive.org/web/20181225160047/https://www.ic-mp.org/press-releases/over-7000-srebrenica-victims-recovered/|url-status=usurped}}</ref> Some Serbs have claimed the massacre was retaliation for civilian casualties inflicted on Bosnian Serbs by [[Army of the Republic of Bosnia and Herzegovina|Bosniak soldiers]] from Srebrenica under the command of [[Naser Orić]].<ref>{{cite news |last1=Niksic |first1=Sabina |title=Bosnian court acquits ex-Srebrenica commander of war crimes |url=https://apnews.com/d9562388dd57466a91c453b37ea86584/Bosnian-court-acquits-ex-Srebrenica-commander-of-war-crimes |publisher=Associated Press |quote=Serbs continue to claim the 1995 Srebrenica slaughter was an act of revenge by uncontrolled troops because they say that soldiers under Oric's command killed thousands of Serbs in the villages surrounding the eastern town|date=9 October 2017}}</ref><ref>{{cite book |last1=Nettelfield |first1=Lara J. |last2=Wagner |first2=Sarah |title=Srebrenica in the Aftermath of Genocide |date=2014 |publisher=Cambridge University Press |isbn=978-1-10700-046-9 |page=253 |url=https://books.google.com/books?id=mQdaAQAAQBAJ&pg=PA253}}</ref> These 'revenge' claims have been rejected and condemned by the [[International Criminal Tribunal for the former Yugoslavia]] (ICTY) and the UN.

In 2004, the Appeals Chamber of the ICTY ruled the massacre of the enclave's male inhabitants constituted genocide.<ref name="Krstic para. 37">{{cite web|url=http://www.icty.org/x/cases/krstic/acjug/en/krs-aj040419e.pdf#page=16|title=Prosecutor vs Krstic, Appeals Chamber Judgement|publisher=ICTY|date=19 April 2004|access-date=25 April 2017|at=para. 37}}</ref> The ruling was also upheld by the [[International Court of Justice]] in 2007.<ref name="ICJ 2007">{{cite web |author=ICJ |date=26 February 2007 |url=http://www.icj-cij.org/docket/files/91/13685.pdf#page=250 |title=Application of the Convention on the Prevention and Punishment of the Crime of Genocide (Bosnia and Herzegovina v. Serbia and Montenegro), Judgment |id=General List No. 91 |page=166 § 297 |quote=The Court concludes that the acts committed at Srebrenica falling within Article II ''(a)'' and ''(b)'' of the Convention were committed with the specific intent to destroy in part the group of the Muslims of Bosnia and Herzegovina as such; and accordingly that these were acts of genocide, committed by members of the VRS in and around Srebrenica from about 13 July 1995. |access-date=27 April 2017 |archive-date=1 March 2011 |archive-url=https://web.archive.org/web/20110301032417/http://www.icj-cij.org/docket/files/91/13685.pdf#page=250 |url-status=dead }}</ref> The forcible transfer and abuse of between 25,000 and 30,000 Bosniak Muslim women, children and elderly, when accompanied by the massacre of the men, was found to constitute genocide.<ref>{{citation|url=http://www.icty.org/x/cases/krstic/acjug/en/krs-aj040419e.pdf |title=Prosecutor vs Radislav Krstic, ICTY Appeals Chamber Judgement |ref =Case No. IT-98-33-A|date=19 April 2004|at=Para. 33|work=International Tribunal for the Prosecution of Persons Responsible for Serious Violations of International Humanitarian Law Committed in the Territory of the Former Yugoslavia since 1991 |access-date= 21 March 2011}}</ref><ref>{{citation|url=http://www.icty.org/x/cases/tolimir/acjug/en/150408_judgement.pdf |title=Prosecutor v. Zdravko Tolimir, ICTY Appeals Chamber Judgement|work=International Tribunal for the Prosecution of Persons Responsible for Serious Violations of International Humanitarian Law Committed in the Territory of the Former Yugoslavia since 1991|date=8 April 2015|ref =Case No. IT-05-88/2-A}}</ref> In 2002, the [[Second Kok cabinet|government of the Netherlands]] resigned, citing its inability to prevent the massacre. In 2013, 2014 and 2019, the Dutch state was found liable by its supreme court and the Hague district court, of failing to prevent more than 300 deaths.<ref name="Netherlands 2013">{{cite web |url=http://www.nuhanovicfoundation.org/user/file/nuhanovic_vs_netherlands_supreme_court_decision_2013_6_sept_.pdf |title=Judgement of the Supreme Court of the Netherlands, First Chamber, 12/03324 LZ/TT (English translation) |date=6 September 2013 |access-date=17 July 2015 |archive-date=27 February 2021 |archive-url=https://web.archive.org/web/20210227203829/http://www.nuhanovicfoundation.org/user/file/nuhanovic_vs_netherlands_supreme_court_decision_2013_6_sept_.pdf |url-status=dead }}</ref><ref name="Amnesty NSC">{{cite web |url=http://www.amnesty.or.jp/en/news/2013/0927_4204.html |title=Netherlands Supreme Court hands down historic judgment over Srebrenica genocide |publisher=Amnesty Int. |date=27 Sep 2013 |access-date=15 July 2015}}</ref><ref>{{cite magazine|url=http://content.time.com/time/world/article/0,8599,2081634,00.html|title=Court Says the Dutch Are to Blame for Srebrenica Deaths|first=Lauren|last=Comiteau|access-date=14 July 2015|magazine=Time|location=The Hague|archive-date=25 February 2021|archive-url=https://web.archive.org/web/20210225172428/http://content.time.com/time/world/article/0,8599,2081634,00.html|url-status=dead}}</ref><ref name=":2">{{Cite news | url=https://www.bbc.co.uk/news/world-europe-49042372 | title=Netherlands '10% liable' for 350 Srebrenica deaths| work=BBC News| date=19 July 2019}}</ref> In 2013, Serbian president [[Tomislav Nikolić]] apologised for "the crime" of Srebrenica but refused to call it genocide.<ref>{{cite news|title=Serbian president apologises for Srebrenica 'crime'|url=https://www.bbc.co.uk/news/world-europe-22297089|date=25 April 2013 |work=BBC News}}</ref>

In 2005, then UN Secretary-General [[Kofi Annan]] described the massacre as "a terrible crime – the worst on European soil since the Second World War",<ref name="UN10th" /> and in May 2024, the UN [[United Nations General Assembly Resolution 78/282|designated]] July 11 as the annual ''International Day of Reflection and Commemoration of the 1995 Genocide in Srebrenica''.<ref name="UNChiefwelcomes" /><ref name="UNapproves" />

== Background ==
=== Conflict in Eastern Bosnia and Herzegovina ===
{{See also|Yugoslav Wars|Bosnian War}}

The [[Socialist Republic of Bosnia and Herzegovina]] was inhabited by mainly Muslim [[Bosniaks]] (44%), [[Serbian Orthodox Church|Orthodox Serbs]] (31%) and Catholic [[Croats]] (17%). As the former [[Socialist Federal Republic of Yugoslavia|Yugoslavia]] began to disintegrate, the region declared national sovereignty in 1991 and held a [[History of Bosnia and Herzegovina|referendum for independence]] in February 1992. The result, which favoured independence, was opposed by Bosnian Serb political representatives, who boycotted the referendum. The [[Republic of Bosnia and Herzegovina]] was formally recognised by the [[European Community]] in April 1992 and the UN in May 1992.<ref name=":3">{{cite court |litigants= Prosecutor v. Naser Orić |pinpoint=81, 89, 93, 99, 103, 110 |court=International Tribunal for the
Prosecution of Persons Responsible for Serious Violations of International Humanitarian Law Committed in the Territory of
Former Yugoslavia since 1991 |date=30 June 2006 |url=https://www.icty.org/x/cases/oric/tjug/en/ori-jud060630e.pdf |access-date=29 November 2020}}</ref><ref>{{UN doc |docid=A-RES-46-237 |body=A |session=46 |type=R |resolution_number=237 |title=Admission of the Republic of Bosnia and Herzegovina to membership in the United Nations |date=20 July 1992 |access-date=29 November 2020}}</ref>

Following the declaration of independence, Bosnian Serb forces, supported by the [[Serbian government]] of [[Slobodan Milošević]] and the [[Yugoslav People's Army]] (JNA), attacked Bosnia and Herzegovina, to secure and unify the territory under Serb control, and create an ethnically homogenous Serb state of [[Republika Srpska]].<ref>{{citation|url=http://www.icty.org/x/cases/krstic/acjug/en/krs-aj040419e.pdf |title=Prosecutor vs Radislav Krstic, ICTY Appeals Chamber Judgement |ref =Case No. IT-98-33-A|date=19 April 2004|at=Para. 15|work=International Tribunal for the Prosecution of Persons Responsible for Serious Violations of International Humanitarian Law Committed in the Territory of the Former Yugoslavia since 1991 |access-date= 21 March 2011}}</ref> In the struggle for territorial control, the non-Serb populations from areas under Serbian control, especially the Bosniak population in East Bosnia, near the Serbian borders, were subject to ethnic cleansing.<ref name="ICTY: Kunarac, Kovač and Vuković judgement – Foča">{{cite web|url=https://www.un.org/icty/kunarac/trialc2/judgement/kun-tj010222e-5.htm#VC |title=ICTY: The attack against the civilian population and related requirements |url-status=dead |archive-url=https://web.archive.org/web/20090219012918/http://un.org/icty/kunarac/trialc2/judgement/kun-tj010222e-5.htm#VC |archive-date=19 February 2009 }}</ref>

=== Ethnic cleansing ===
{{See also|Ethnic cleansing in the Bosnian War}}
Srebrenica, and the surrounding Central Podrinje region, had immense strategic importance to the Bosnian Serb leadership. It was the bridge to disconnected parts of the envisioned ethnic state of [[Republika Srpska]].<ref name="ICTY Krstic">ICTY, Prosecutor vs. Krstic; Trial Chamber Judgement; United Nations; para. 15.</ref> Capturing Srebrenica and eliminating its Muslim population would also undermine the viability of the Bosnian Muslim state.<ref name="ICTY Krstic"/>

In 1991, 73% of the population in Srebrenica were Bosnian Muslims and 25% Bosnian Serbs. Tension between Muslims and Serbs intensified in the early 1990s, as the local Serb population were provided with weapons and military equipment distributed by Serb paramilitary groups, the [[Yugoslav People's Army]] (''Jugoslovenska'' ''Narodna Armija'', JNA) and the [[Serb Democratic Party (Bosnia and Herzegovina)|Serb Democratic Party]] (''Srpska Demokratska Stranka'', SDS).<ref name=":3" />

By April 1992, Srebrenica had become isolated by Serb forces. On 17 April, the Bosnian Muslim population was given a 24-hour ultimatum to surrender all weapons and leave town. Srebrenica was briefly captured by the Bosnian Serbs and retaken by Bosnian Muslims on 8 May 1992. Nonetheless, the Bosnian Muslims remained surrounded by Serb forces, and cut off from outlying areas. The [[Naser Orić]] trial judgment described the situation:<ref name=":3" />

{{blockquote|Between April 1992 and March 1993 ... Srebrenica and the villages in the area held by Bosnian Muslims were constantly subjected to Serb military assaults, including artillery attacks, sniper fire ... occasional bombing from aircraft. Each onslaught followed a similar pattern. Serb soldiers and paramilitaries surrounded a Bosnian Muslim village ... called upon the population to surrender their weapons, and then began with indiscriminate shelling and shooting ... they then entered the village ... expelled or killed the population, who offered no significant resistance, and destroyed their homes ... Srebrenica was subjected to indiscriminate shelling from all directions daily. Potočari, in particular, was a daily target ... because it was a sensitive point in the defence line around Srebrenica. Other Bosnian Muslim settlements were routinely attacked as well. All this resulted in a great number of refugees and casualties.}}

Between April and June 1992 Bosnian Serb forces, with support from the JNA, destroyed 296 predominantly Bosniak villages around Srebrenica, forcibly uprooted 70,000 Bosniaks from their homes and systematically massacred at least 3,166 Bosniaks, including women, children and elderly.<ref name="bosnia.org.uk-Toljaga-2010-Prelude-Srebrenica">{{cite web |author1=Daniel Toljaga |title=Bosnian Institute UK, the 26-page study: Prelude to the Srebrenica Genocide – mass murder and ethnic cleansing of Bosniaks in the Srebrenica region during the first three months of the Bosnian War (April–June 1992) |url=http://www.bosnia.org.uk/news/news_body.cfm?newsid=2771 |website=bosnia.org.uk |publisher=Bosnian Institute |access-date=14 February 2021 |archive-url=https://web.archive.org/web/20191022155535/http://www.bosnia.org.uk/news/news_body.cfm?newsid=2771 |archive-date=22 October 2019 |location=London |language=en |date=18 November 2010}}</ref> In neighbouring [[Bratunac]], Bosniaks were either killed or forced to flee to Srebrenica, resulting in 1,156 deaths.<ref name="Bratunac">{{cite web |url=http://bratunac.com/content.php?content.12 |trans-title=Truth about Bratunac |title=Istina o Bratuncu|year=1995|publisher=Bratunac Municipality Officials |access-date=2006-06-29 |df=dmy-all }}{{dead link|date=July 2018|bot=medic}}{{cbignore|bot=medic}}</ref> Thousands of Bosniaks were killed in [[Foča massacres|Foča]], [[Zvornik massacre|Zvornik]], [[Cerska]] and [[Snagovo massacre|Snagovo]].<ref name="IDC: Podrinje victim statistics">{{cite web|url=http://www.idc.org.ba/onama/izvjestaj_analize_po_centrima.html#podrinje |title=IDC: Podrinje victim statistics |url-status=dead |archive-url=https://web.archive.org/web/20070707071037/http://www.idc.org.ba/onama/izvjestaj_analize_po_centrima.html |archive-date=7 July 2007 }}</ref>

=== 1992–1993: Struggle for Srebrenica ===
Over the remainder of 1992, offensives by Bosnian government forces from Srebrenica increased the area under their control, and by January 1993 they had linked with Bosniak-held [[Žepa]] to the south and Cerska to the west. The Srebrenica enclave had reached its peak size of {{convert|900|km2|sqmi|abbr=off|round=50}}, though it was never linked to the main area of Bosnian-government controlled land in the west and remained "a vulnerable island amid Serb-controlled territory".<ref>{{cite web|url=https://www.un.org/icty/krstic/TrialC1/judgement/ |title=Prosecutor vs. Radislav Krstic Judgement |access-date=8 June 2006 |author=ICTY |publisher=United Nations |url-status=dead |archive-url=https://web.archive.org/web/20060205104015/http://www.un.org/icty/krstic/TrialC1/judgement/ |archive-date=5 February 2006 }}</ref> [[Army of the Republic of Bosnia and Herzegovina]] (ARBiH) forces under [[Naser Orić]] used Srebrenica as a staging ground to attack neighboring Serb villages inflicting many casualties.<ref name="OricIndict">{{cite web|url=https://www.un.org/icty/indictment/english/ori-2ai041004e.htm|title=Naser Orić: second amended indictment|publisher=The International Criminal Tribunal for the former Yugoslavia|access-date=23 July 2015|url-status=dead|archive-url=https://web.archive.org/web/20090303134851/http://www.un.org/icty/indictment/english/ori-2ai041004e.htm|archive-date=3 March 2009}}</ref><ref name="NIOD Pt2Ch2">{{cite web|url=http://213.222.3.5/srebrenica/toc/p2_c02_s004_b01.html|title=Dutchbat in the enclave|publisher=Netherlands Institute for War Documentation (NIOD)|access-date=23 July 2015|url-status=dead|archive-url=https://web.archive.org/web/20060721152501/http://213.222.3.5/srebrenica/toc/p2_c02_s004_b01.html|archive-date=21 July 2006}}</ref> In 1993, the militarized Serb village of [[Kravica attack (1993)|Kravica]] was attacked by ARBiH, which resulted in Serb civilian casualties. The resistance to the Serb [[siege of Srebrenica]] by the ARBiH, under Orić was seen as a catalyst for the massacre.

Serbs started persecuting Bosniaks in 1992. Serbian propaganda deemed Bosniak resistance to Serb attacks as a ground for revenge. According to French General [[Philippe Morillon]], Commander of the UN Protection Force (UNPROFOR), in testimony at the ICTY in 2004:

{{poemquote|JUDGE ROBINSON: Are you saying, then, General, that what happened in 1995 was a direct reaction to what Naser Oric did to the Serbs two years before?
THE WITNESS: [Interpretation] Yes. Yes, Your Honour. I am convinced of that. This doesn't mean to pardon or diminish the responsibility of the people who committed that crime, but I am convinced of that, yes.<ref>{{cite web|url=http://www.icty.org/x/cases/slobodan_milosevic/trans/en/040212ED.htm|title=040212ED|publisher=International Criminal Tribunal for the former Yugoslavia|access-date=18 November 2015}}</ref>}}

Over the next few months, the Serb military captured the villages of Konjević Polje and Cerska, severing the link between Srebrenica and Žepa, and reducing the Srebrenica enclave to 150 square kilometres. Bosniak residents of the outlying areas converged on Srebrenica and its population swelled to between 50,000 and 60,000, about ten times the pre-war population.<ref name="Bartrop Essential Reference">{{cite book |last1=Bartrop |first1=Paul R. |title=Bosnian genocide: the essential reference guide |date=2016 |publisher=ABC-CLIO |location=Santa Barbara |isbn=978-1-4408-3868-2}}</ref> General Morillon visited Srebrenica in March 1993. The town was overcrowded and [[siege]] conditions prevailed. There was almost no running water as the advancing Serb forces had destroyed water supplies; people relied on makeshift generators for electricity. Food, medicine and other essentials were scarce. The conditions rendered Srebrenica a slow death camp.<ref name="Bartrop Essential Reference" /> Morillon told panicked residents at a public gathering that the town was under the protection of the UN, and he would never abandon them. During March and April 1993 several thousand Bosniaks were evacuated, under the auspices of the [[UN High Commissioner for Refugees]] (UNHCR). The evacuations were opposed by the Bosnian government in [[Sarajevo]], as contributing to the ethnic cleansing of predominantly Bosniak territory.

The Serb authorities remained intent on capturing the enclave. On 13 April 1993, the Serbs told the UNHCR representatives that they would attack the town within two days unless the Bosniaks surrendered and agreed to be evacuated.<ref>ICTY, Prosecutor vs. Krstic; Trial Chamber Judgement; United Nations; para. 13–17.</ref>

====Starvation====
With the failure to demilitarize and the shortage of supplies getting in, Orić consolidated his power and controlled the black market. Orić's men began hoarding food, fuel, cigarettes and embezzled money sent by aid agencies to support Muslim orphans.<ref>{{Harvnb|Danner|2009|p=232}}</ref> Basic necessities were out of reach for many in Srebrenica due to Orić's actions. UN officials were beginning to lose patience with the ARBiH in Srebrenica and saw them as "criminal gang leaders, pimps and black marketeers".<ref>{{Harvnb|Danner|2009|p=233}}</ref>

A former Serb soldier of the "[[JSO (Special Operations Unit)#History|Red Berets]]" unit described the tactics used to starve and kill the besieged population:

{{blockquote|It was almost like a game, a cat-and-mouse hunt. But of course, we greatly outnumbered the Muslims, so in almost all cases, we were the hunters and they were the prey. We needed them to surrender, but how do you get someone to surrender in a war like this? You starve them to death. So very quickly we realised that it wasn't really weapons being smuggled into Srebrenica that we should worry about, but food. They were truly starving in there, so they would send people out to steal cattle or gather crops, and our job was to find and kill them ... No prisoners. Well, yes, if we thought they had useful information, we might keep them alive until we got it out of them, but in the end, no prisoners ... The local people became quite indignant, so sometimes we would keep someone alive to hand over to them [to kill] just to keep them happy.<ref name="nytimes.com">{{cite news |work=The New York Times Magazine |first=Scott |last=Anderson |date=28 May 2014 |url=https://www.nytimes.com/interactive/2014/05/29/magazine/srebrenica-life-in-the-valley-of-death.html |title=Life in the Valley of Death |access-date=18 April 2017}}</ref>}}

When British journalist Tony Birtley visited Srebrenica in March 1993, he took footage of civilians starving to death.<ref>{{Cite web|url=https://www.youtube.com/watch?v=Cz_dVScxOtg|title=Starvation in Pre-Genocide Srebrenica / Gladovanje u Srebrenici (1993)|date=15 August 2010|via=YouTube}}</ref> The Hague Tribunal in the case of Orić concluded:

{{blockquote|Bosnian Serb forces controlling the access roads were not allowing international humanitarian aid—most importantly, food and medicine—to reach Srebrenica. As a consequence, there was a constant and serious shortage of food causing starvation to peak in the winter of 1992/1993. Numerous people died or were in an extremely emaciated state due to malnutrition. Bosnian Muslim fighters and their families, however, were provided with food rations from existing storage facilities. The most disadvantaged group among the Bosnian Muslims was that of the refugees, who usually lived on the streets and without shelter, in freezing temperatures. Only in November and December 1992, did two UN convoys with humanitarian aid reach the enclave, and this despite Bosnian Serb obstruction.<ref name=":3" />

=== 1993–1995: Srebrenica "safe area" ===
==== UN Security Council declares Srebrenica a "safe area" ====
[[File:Bosnia areas of control Sep 94.jpg|thumb|Areas of control in eastern Bosnia and Herzegovina enclaves near the Serbian border, September 1994]]

On 16 April 1993, the [[United Nations Security Council]] passed [[United Nations Security Council Resolution 819|Resolution 819]], which demanded "all parties ... treat Srebrenica and its surroundings as a [[United Nations Safe Areas|safe area]] which should be free from any armed attack or ... hostile act".<ref name="UN2">{{cite web|url=https://undocs.org/S/RES/819(1993)|id=S/RES/819(1993)|title=Resolution 819|publisher=United Nations|date=16 April 1993|at=para. No. 1}}</ref> On 18 April, the first group of UNPROFOR troops arrived in Srebrenica. UNPROFOR deployed Canadian troops to protect it as one of five newly established UN "safe areas".<ref name="Bartrop Essential Reference"/> UNPROFOR's presence prevented an all-out assault, though skirmishes and mortar attacks continued.<ref name="Bartrop Essential Reference"/>

On 8 May 1993 agreement was reached for demilitarization of Srebrenica. According to a UN report, <blockquote>General [Sefer] Halilović and General Mladić agreed on measures covering the whole of the Srebrenica enclave and ... Žepa. ... Bosniac forces ... would hand over their weapons, ammunition and mines to UNPROFOR, after which Serb 'heavy weapons and units that constitute a menace to the demilitarised zones ... will be withdrawn.' Unlike the earlier agreement, it stated specifically that Srebrenica was to be considered a 'demilitarised zone', as referred to in the ... Geneva Conventions.<ref>[https://documents.un.org/doc/undoc/gen/n99/348/76/pdf/n9934876.pdf?token=RmMMrBi9JokrXUxBmr&fe=true Unknown Title]{{Dead link|date=December 2024 |bot=InternetArchiveBot |fix-attempted=yes }}</ref></blockquote> Both parties violated the agreement, though two years of relative stability followed the establishment of the enclave.<ref name="ICTY Appeal Krstic">{{cite court |litigants= Prosecutor v. Radislav Krstic |pinpoint=9 |court=International Tribunal for the
Prosecution of Persons Responsible for Serious Violations of International Humanitarian Law Committed in the Territory of
Former Yugoslavia since 1991 |date=19 April 2004 |url=https://www.icty.org/x/cases/krstic/acjug/en/krs-aj040419e.pdf |access-date=6 December 2020}}</ref> [[Lieutenant colonel]] [[Thom Karremans]] (the [[Dutchbat]] Commander) testified that his personnel were prevented from returning to the enclave by Serb forces, and that equipment and ammunition were prevented from getting in.<ref name="ICTY Trial Krstic">{{cite court |litigants= Prosecutor v. Radislav Krstic |pinpoint=22 |court=International Tribunal for the
Prosecution of Persons Responsible for Serious Violations of International Humanitarian Law Committed in the Territory of
Former Yugoslavia since 1991 |date=2 August 2001 |url=https://www.icty.org/x/cases/krstic/tjug/en/krs-tj010802e.pdf |access-date=6 December 2020}}</ref> Bosniaks in Srebrenica complained of attacks by Serb soldiers, while to the Serbs it appeared Bosnian forces were using the "safe area", as a convenient base to launch counter-offensives and UNPROFOR was failing to prevent it.<ref name="ICTY Trial Krstic" />{{rp|24}} General [[Sefer Halilović]] admitted ARBiH helicopters had flown in violation of the no-fly zone and he had dispatched eight helicopters with ammunition for the 28th Division.<ref name="ICTY Trial Krstic" />{{rp|24}}

Between 1,000 and 2,000 soldiers from the VRS Drina Corps [[Brigade]]s were deployed around the enclave, equipped with tanks, armoured vehicles, artillery and [[Mortar (weapon)|mortars]]. The 28th Mountain Division of the [[Army of the Republic of Bosnia and Herzegovina]] (ARBiH) in the enclave was neither well organised nor equipped, and lacked a firm command structure and communications system. Some of its soldiers carried old hunting rifles or no weapons, few had proper uniforms.

==== UN failure to demilitarise ====
A Security Council mission led by [[Diego Arria]] arrived on 25 April 1993 and, in their report to the UN, condemned the Serbs for perpetrating "a slow-motion process of genocide".<ref name="UN report">{{cite web|title=Report of the Secretary-General pursuant to General Assembly resolution 53/35 – The fall of Srebrenica|url=https://undocs.org/A/54/549|website=undocs.org|publisher=United Nations|access-date=15 March 2017|page=1-21, 55, 103-107|language=en}}</ref> The mission stated "Serb forces must withdraw to points from which they cannot attack, harass or terrorise the town." Specific instructions from UN Headquarters in New York stated UNPROFOR should not be too zealous in searching for Bosniak weapons and the Serbs should withdraw their heavy weapons before the Bosniaks disarmed, which the Serbs never did.<ref name="UN report" />

Attempts to demilitarise the ARBiH and force withdrawal of the VRS proved futile. The ARBiH hid most of their heavy weapons, modern equipment and ammunition in the surrounding forest and only handed over disused and old weaponry.<ref name="NIOD Pt2Ch2" /> The VRS refused to withdraw from the front lines due to intelligence they received regarding ARBiH's hidden weaponry.<ref name="auto1">{{cite web |date=17 January 2005 |title=Srebrenica: 'a safe area' Part 2 Dutchbat in the enclave |url=http://www.cnj.it/documentazione/Srebrenica/NIOD/NIOD%20Part%20II.pdf |access-date=5 May 2017 |work=Netherlands Institute of War Documentation}}</ref>

In March 1994, UNPROFOR sent 600 Dutch soldiers ("Dutchbat") to replace the Canadians. By March 1995, Serb forces controlled all territory surrounding Srebrenica, preventing even UN access to the supply road. Humanitarian aid decreased and living conditions quickly deteriorated.<ref name="Bartrop Essential Reference"/> UNPROFOR presence prevented all-out assault on the safe area, though skirmishes and mortar attacks continued.<ref name="Bartrop Essential Reference"/> The Dutchbat alerted UNPROFOR command to the dire conditions, but UNPROFOR declined to send humanitarian relief or military support.<ref name="Bartrop Essential Reference"/>

====Organisation of UNPROFOR and UNPF====
{{main|United Nations Protection Force}}

In April 1995, UNPROFOR became the name used for the Bosnia and Herzegovina regional command of the now-renamed United Nations Peace Forces (UNPF),<ref name="cnj.it">{{cite web |url=http://www.cnj.it/documentazione/Srebrenica/NIOD/NIOD%20Part%20II.pdf |title=Srebrenica – a 'safe' area |publisher=NIOD |date=2011 |access-date=21 July 2015 |page=9}}</ref> with "12,500 British, French and Dutch troops equipped with tanks and high calibre artillery to increase the effectiveness and the credibility of the peacekeeping operation".<ref name="cnj.it" /> The report states:

{{blockquote|In the UNPROFOR [[command hierarchy|chain of command]], [[Dutchbat]] occupied the fourth tier, with the sector commanders occupying the third tier. The fourth tier primarily had an operational task ... Dutchbat was expected to operate as an independent unit with its own logistic arrangements. Dutchbat was dependent on the UNPROFOR organization to some extent for crucial supplies such as fuel. For the rest, it was expected to obtain its supplies from the Netherlands. From an organizational point of view, the battalion had two lifelines: UNPROFOR and the Royal Netherlands Army. Dutchbat had been assigned responsibility for the Srebrenica Safe Area. Neither UNPROFOR nor Bosnia-Hercegovina paid much attention to Srebrenica, however. Srebrenica was situated in eastern Bosnia and Herzegovina, which was geographically and mentally far removed from Sarajevo and Zagreb. The rest of the world was focused on the fight for Sarajevo ... As a Safe Area, Srebrenica only occasionally managed to attract the attention of the world press or the UN Security Council. That is why the Dutch troops there remained of secondary importance, in operational and logistic terms, for so long; and why the importance of the enclave in the battle for domination between the Bosnian Serbs and Bosnian Muslims failed to be recognised for so long.<ref name="cnj.it2">{{cite web |url=http://www.cnj.it/documentazione/Srebrenica/NIOD/NIOD%20Part%20II.pdf |title=Srebrenica – a 'safe' area |publisher=NIOD |date=2011 |access-date=21 July 2015 |page=20}}</ref>}}

==== Situation deteriorates ====
By early 1995, fewer and fewer supply convoys were making it through to the enclave. The situation in Srebrenica and other enclaves had deteriorated into lawless violence as prostitution among young Muslim girls, theft and black marketeering proliferated.<ref>[[#refRamet2006|Ramet (2006)]], p. 443</ref> Already meager resources dwindled further, and even the UN forces started running dangerously low on food, medicine, ammunition and fuel, eventually being forced to start patrolling on foot. Dutch soldiers who left on leave were not allowed to return,<ref name="UN report" /> and their number dropped from 600 to 400 men. In March and April, the Dutch soldiers noticed a build-up of Serb forces.

In March 1995, [[Radovan Karadžić]], President of the Republika Srpska (RS), despite pressure from the international community to end the war and efforts to negotiate peace, issued a directive to the VRS concerning long-term strategy in the enclave. The directive, known as "Directive 7", specified the VRS was to:

{{blockquote|Complete the physical separation of Srebrenica from Žepa as soon as possible, preventing even communication between individuals in the two enclaves. By planned and well-thought-out combat operations, create an unbearable situation of total insecurity with no hope of further survival or life for the inhabitants of Srebrenica.<ref>{{cite web|url=http://www.icty.org/x/cases/krstic/acjug/en/krs-aj040419e.pdf#page=33|title=Prosecutor vs Krstic, Appeals Chamber Judgement|publisher=ICTY|date=19 April 2004|access-date=25 April 2017|page=29}}</ref>}}

By mid-1995, the humanitarian situation in the enclave was catastrophic. In May, following orders, Orić and his staff left the enclave, leaving senior officers in command of the 28th Division. In late June and early July, the 28th Division issued reports including urgent pleas for the [[humanitarian corridor]] to be reopened. When this failed, Bosniak civilians began dying from starvation. On 7 July the mayor reported eight residents had died.<ref>[http://www.hri.org/cgi-bin/brief?/news/misc/misc-news/1995/95-07-11_1.misc.txt BALKAN WATCH The Balkan Institute 10 July 1995 A Weekly Review of Current Events Volume 2.26 Week in Review 3–9 July 1995]</ref> On 4 June, UNPROFOR commander [[Bernard Janvier]], a Frenchman, secretly met with Mladić to obtain the release of hostages, many of whom were French. Mladić demanded of Janvier that there would be no more airstrikes.<ref>{{cite web |title=Remarks by the High Commissioner to the Security Council meeting on Bosnia and Herzegovina (Srebrenica) |url=https://www.ohchr.org/EN/NewsEvents/Pages/DisplayNews.aspx?NewsID=16218&LangID=E |website=ohchr}}</ref>

In the weeks leading up to the assault on Srebrenica by the VRS, ARBiH forces were ordered to carry out diversion and disruption attacks on the VRS by the high command.{{sfn|Ingrao|2012|p=218}} On one occasion on 25 June, ARBiH forces attacked VRS units on the [[Sarajevo]]–[[Zvornik]] road, inflicting high casualties and looting VRS stockpiles.{{sfn|Ingrao|2012|p=218}}

==== 6–11 July 1995: Serb takeover ====
{{main|Siege of Srebrenica}}

The Serb offensive against Srebrenica began in earnest on 6 July. The VRS, with 2,000 soldiers, were outnumbered by the defenders and did not expect the assault to be an easy victory.{{sfn|Ingrao|2012|p=218}} Five UNPROFOR observation posts in the south of the enclave fell in the face of the Bosnian Serb advance. Some Dutch soldiers retreated into the enclave after their posts were attacked, the crews of the other observation posts surrendered into Serb custody. The defending Bosnian forces numbering 6,000 came under fire and were pushed back towards the town. Once the southern perimeter began to collapse, about 4,000 Bosniak residents, who had been living in a Swedish housing complex for refugees nearby, fled north into Srebrenica. Dutch soldiers reported the advancing Serbs were "cleansing" the houses in the south of the enclave.<ref name="Krstic 2001 takeover">{{cite web |url=http://www.icty.org/x/cases/krstic/tjug/en/krs-tj010802e.pdf |title=Krstic Judgement Part II. Findings of Fact |pages=10-18, 27, 32-92 |date=2 August 2001 |publisher=ICTY }}</ref>

[[File:Pantserrupsvoertuig YPR-765.jpg|thumb|A Dutch YPR-765 of the type used at Srebrenica]]

On 8 July, a Dutch [[YPR-765]] armoured vehicle took fire from the Serbs and withdrew. A group of Bosniaks demanded the vehicle stay to defend them, and established a makeshift barricade to prevent its retreat. As the vehicle withdrew, a Bosniak farmer manning the barricade threw a grenade onto it and killed Dutch soldier Raviv van Renssen.<ref name="LeBor2008">{{cite book|last1=LeBor|first1=Adam|title="Complicity with Evil": The United Nations in the Age of Modern Genocide|date=1 October 2008|publisher=Yale University Press|isbn=978-0300135145|page=97}}</ref> On 9 July, emboldened by success, little resistance from the demilitarised Bosniaks and lack of reaction from the international community, President Karadžić issued a new order authorising the 1,500-strong<ref>{{cite news |publisher=CNN |url=http://www.cnn.com/WORLD/Bosnia/updates/july95/7-11/index4.html|title=1,500 Bosnian Serb troops overran the enclave of Srebrenica |date=11 July 1995 |access-date=7 October 2012}}</ref> VRS Drina Corps to capture Srebrenica.<ref name="Krstic 2001 takeover"/>

The following morning, 10 July, Lieutenant Colonel Karremans made urgent requests for air support from [[NATO]] to defend Srebrenica as crowds filled the streets, some of whom carried weapons. VRS tanks were approaching, and NATO airstrikes on these began on 11 July. NATO bombers attempted to attack VRS artillery locations outside the town, but poor visibility forced NATO to cancel this. Further air attacks were cancelled after VRS threats to bomb the UN's Potočari compound, kill Dutch and French military hostages and attack surrounding locations where 20,000 to 30,000 civilian refugees were situated.<ref name="Krstic 2001 takeover"/> 30 Dutchbat were taken hostage by Mladic's troops.<ref name="Bartrop Essential Reference"/>

Late in the afternoon of 11 July, General Mladić, accompanied by General Živanović (Commander of the Drina Corps), General Krstić (Deputy Commander of the Drina Corps) and other VRS officers, took a triumphant walk through the deserted streets of Srebrenica.<ref name="Krstic 2001 takeover"/> In the evening,<ref>{{cite news|url=http://www.thedailybeast.com/articles/2011/07/13/the-fatal-toast-before-the-srebrenica-massacre.html|title=Dutch Court Rules Netherlands Responsible for Three Deaths in Srebrenica Massacre in Bosnia |work=The Daily Beast|date=13 July 2011|last1=Visser|first1=Nadette De}}</ref> Lieutenant Colonel Karremans was filmed drinking a toast with Mladić during the bungled negotiations on the fate of the civilian population grouped in Potočari.<ref name="spiegel.de"/><ref name="Daruvalla">{{cite magazine |url=http://www.time.com/time/europe/magazine/article/0,13005,901020429-232505,00.html |archive-url=https://web.archive.org/web/20050316044527/http://www.time.com/time/europe/magazine/article/0,13005,901020429-232505,00.html |url-status=dead |archive-date=16 March 2005 |title=Anatomy of a Massacre |magazine=Time |last=Daruvalla |first=Abi |access-date=20 July 2006|date=21 April 2002}}</ref>

== Massacre ==
The two highest-ranking Serb politicians from Bosnia and Herzegovina, Karadžić and [[Momčilo Krajišnik]], both indicted for genocide, were warned by VRS commander Mladić (found guilty of genocide in 2017) that their plans could not be realized without committing genocide. Mladić said at a parliamentary session of 12 May 1992:

{{blockquote|People are not little stones or keys in someone's pocket, that can be moved from one place to another just like that. ... Therefore, we cannot precisely arrange for only Serbs to stay in one part of the country while removing others painlessly. I do not know how Mr. Krajišnik and Mr. Karadžić will explain that to the world. That is genocide.<ref>{{Cite web|url=http://www.bosnia.org.uk/news/news_body.cfm?newsid=2229|access-date=30 September 2007|title=Bosnia's 'Accidental' Genocide|date=30 September 2006|first=Edina|last=Becirevic|work=Bosnian Institute|location=United Kingdom|archive-url=https://web.archive.org/web/20070930185500/http://www.bosnia.org.uk/news/news_body.cfm?newsid=2229|archive-date=30 September 2007}}</ref>}}

=== Increasing concentration of refugees in Potočari ===
[[File:Dutchbat HQ 2009.jpg|thumb|Headquarters in [[Donji Potočari|Potočari]] for soldiers under [[United Nations]] command; "[[Dutchbat]]" had 370<ref name="spiegel.de">{{cite web|url=http://www.spiegel.de/international/dealing-with-genocide-a-dutch-peacekeeper-remembers-srebrenica-a-364902.html|title=Dealing With Genocide: A Dutch Peacekeeper Remembers Srebrenica|date=12 July 2005|work=Spiegel Online}}</ref> soldiers in Srebrenica during the massacre. The building was a disused battery factory.]]

By the evening of 11 July, approximately 20,000–25,000 Bosniak refugees from Srebrenica were gathered in Potočari, seeking protection within the [[UNPROFOR]] Dutchbat headquarters. Several thousand had pressed inside the compound, while the rest were spread throughout neighbouring factories and fields. Though most were women, children, elderly or disabled, 63 witnesses estimated there were at least 300 men inside the compound and between 600 and 900 in the crowd outside.<ref name="ICTY">{{Cite web|url=http://www.icty.org/x/cases/krstic/acjug/en/krs-aj040419e.pdf|title=Prosecutor vs. Kristic, Judgement}}</ref>

Conditions included "little food or water" and sweltering heat. A UNPROFOR Dutchbat officer described the scene:

{{blockquote|They were panicked, they were scared, and they were pressing each other against the soldiers, my soldiers, the UN soldiers that tried to calm them. People who fell were trampled on. It was a chaotic situation.<ref name="ICTY" />}}

On 12 July, the UN Security Council, in [[United Nations Security Council Resolution 1004|Resolution 1004]], expressed concern at the humanitarian situation in Potočari, condemned the offensive by Bosnian Serb forces and demanded immediate withdrawal. On 13 July, the Dutch forces expelled five Bosniak refugees from the compound despite knowing men outside were being killed.<ref name="NYT9613">{{cite news|title=Dutch Peacekeepers Are Found Responsible for Deaths|url=https://www.nytimes.com/2013/09/07/world/europe/dutch-peacekeepers-are-found-responsible-for-deaths.html|access-date=7 September 2013|newspaper=The New York Times|date=6 September 2013|first=Marlise |last=Simons|quote=Dutchbat soldiers knew that outside the compound men were being killed and abused, the court summary said, but the soldiers decided not to evacuate most refugees, including the three men, along with the battalion and instead sent them away on 13 July.}}</ref>

=== Crimes committed in Potočari ===
On 12 July the refugees in the compound could see VRS members setting houses and haystacks on fire. Throughout the afternoon, Serb soldiers mingled in the crowd and [[summary executions]] of men occurred.<ref name="ICTY" /> In the morning of 12 July, a witness saw a pile of 20–30 bodies heaped up behind the Transport Building, alongside a tractor-like machine. Another testified he saw a soldier slay a child with a knife, in the middle of a crowd of expellees. He said he saw Serb soldiers execute over 100 Bosniak Muslim men behind the Zinc Factory, then load their bodies onto a truck, though the number and nature of the murders contrasted with other evidence in the Trial Record, which indicated killings in Potočari were sporadic in nature. Soldiers were picking people out of the crowd and taking them away. A witness recounted how three brothers – one a child, the others in their teens – were taken out in the night. When the boys' mother went looking for them, she found them naked and with their throats slit.<ref name="ICTY" /><ref name="Krstic 2001 takeover" />

That night, a Dutchbat medical orderly witnessed two Serb soldiers raping a woman.<ref name="Krstic 2001 takeover" /> A survivor, Zarfa Turković, described the horrors: "Two [Serb soldiers] took her legs and raised them in the air, while the third began raping her. Four of them were taking turns on her. People were silent, and no one moved. She was screaming and yelling and begging them to stop. They put a rag into her mouth, and then we just heard silent sobs."<ref>{{cite news|url=https://apnews.com/article/736bd951f0ad2fe504a32d3476bde222|title=Srebrenicans Detail Serb Gang Rape, Other Atrocities|date=17 July 1995|work=Associated Press|first=Snjezana|last=Vukic}}</ref><ref>{{cite web|url=http://www.markdanner.com/articles/show/62|first=Mark|last=Danner|title=The US and the Yugoslav Catastrophe|date=20 November 1997|work=The New York Review of Books|access-date=2 June 2009|archive-date=28 December 2018|archive-url=https://web.archive.org/web/20181228034959/http://www.markdanner.com/articles/show/62|url-status=dead}}</ref>

==== Murder of Bosniak men and boys in Potočari ====
From the morning of July 12, Serb forces began gathering men and boys from the refugee population in Potočari and holding them in separate locations, and as the refugees began boarding the buses headed north towards Bosniak-held territory, Serb soldiers separated [[Military-age male|men of military age]] who were trying to clamber aboard. Occasionally, younger and older men were stopped as well (some as young as 14).<ref>{{cite web|url=http://www.icty.org/x/cases/krstic/trans/en/000726ed.htm |title=Separation of boys, Krstic ICTY Potocari |publisher=International Criminal Tribunal for the former Yugoslavia |date=26 July 2000 |access-date=26 May 2011}}</ref><ref>{{cite web|url=http://www.icty.org/x/cases/popovic/trans/en/070206ED.htm |title=Separation, Popovic ICTY |publisher=International Criminal Tribunal for the former Yugoslavia |access-date=26 May 2011}}</ref><ref>{{cite web|url=http://www.icty.org/x/cases/krstic/trans/en/000411ed.htm |title=Separation, Krstic ICTY |publisher=International Criminal Tribunal for the former Yugoslavia |date=11 July 1995 |access-date=26 May 2011}}</ref> These men were taken to a building referred to as the "White House". By the evening of 12 July, Major Franken of Dutchbat heard that no men were arriving with the women and children, at their destination in [[Kladanj]].<ref name="ICTY" /> UNHCR Director of Operations Peter Walsh was dispatched to Srebrenica by Chief of Mission, Damaso Feci, to evaluate what emergency aid could be provided rapidly. Walsh and his team arrived at Gostilj, just outside Srebrenica, in the afternoon only to be turned away by VRS forces. Despite claiming freedom of movement rights, the UNHCR team was not allowed to proceed and forced to head back north to Bijelina. Throughout, Walsh relayed reports back to UNHCR in Zagreb about the unfolding situation, including witnessing the enforced movement and abuse of Muslim men and boys, and the sound of executions taking place.{{citation needed|date=October 2023}}

On 13 July, Dutchbat troops witnessed definite signs Serb soldiers were murdering Bosniak men who had been separated. Corporal Vaasen saw two soldiers take a man behind the "White House", heard a shot and saw the two soldiers reappear alone. Another Dutchbat officer saw Serb soldiers murder an unarmed man with a gunshot to the head, and heard gunshots 20–40 times an hour throughout the afternoon. When the Dutchbat soldiers told Colonel Joseph Kingori, a [[United Nations Military Observer]] (UNMO) in the Srebrenica area, that men were being taken behind the "White House" and not coming back, Kingori went to investigate. He heard gunshots as he approached, but was stopped by Serb soldiers before he could find out what was going on.<ref name="ICTY" />

Some executions were carried out at night under arc lights, and bulldozers then pushed the bodies into mass graves.<ref name="CNN-2006-05-03">Graham Jones. [http://www.cnn.com/2006/WORLD/europe/02/22/warcrimes.srebrenica/ "Srebrenica: A Triumph of Evil"]. CNN, 3 May 2006</ref> According to evidence collected from Bosniaks by French policeman Jean-René Ruez, some were buried alive; he heard testimony describing Serb forces killing and torturing refugees, streets littered with corpses, people committing suicide to avoid having their noses, lips and ears chopped off, and adults being forced to watch soldiers kill their children.<ref name="CNN-2006-05-03" />

==== Rape and abuse of civilians ====
Thousands of women and girls suffered rape and sexual abuse and other forms of torture. According to the testimony of Zumra Šehomerovic:

{{blockquote|The Serbs began at a certain point to take girls and young women out of the group of refugees. They were raped. The rapes often took place under the eyes of others and sometimes even under the eyes of the children of the mother. A Dutch soldier stood by and he simply looked around with a [[Walkman]] on his head. He did not react at all to what was happening. It did not happen just before my eyes, for I saw that personally, but also before the eyes of us all. The Dutch soldiers walked around everywhere. It is impossible that they did not see it.

There was a woman with a small baby a few months old. A [[Chetniks#Derogatory usage|Chetnik]] told the mother that the child must stop crying. When the child did not stop crying, he snatched the child away and cut its throat. Then he laughed. There was a Dutch soldier there who was watching. He did not react at all.

I saw yet more frightful things. For example, there was a girl, who must have been about nine years old. At a certain moment, some Chetniks recommended to her brother that he rape the girl. He did not do it and I also think that he could not have done it for he was still just a child. Then they murdered that young boy. I have personally seen all that. I really want to emphasize that all this happened in the immediate vicinity of the base. In the same way, I also saw other people who were murdered. Some of them had their throats cut. Others were [[behead]]ed.<ref name="legal tools">{{cite web|url=http://www.legal-tools.org/en/doc/ca1e99/|website=ICC Legal Tools|author=Van Diepen Van der Kroef Advocaten|title=Writ of Summons: District Court, The Hague|date=4 June 2007|pages=101–108}}</ref>}}

Testimony of Ramiza Gurdić:

{{blockquote|I saw how a young boy of about ten was killed by Serbs in Dutch uniform. This happened in front of my own eyes. The mother sat on the ground and her young son sat beside her. The young boy was placed on his mother's lap. The young boy was killed. His head was cut off. The body remained on the lap of the mother. The Serbian soldier placed the head of the young boy on his knife and showed it to everyone. ... I saw how a pregnant woman was slaughtered. There were Serbs who stabbed her in the stomach, cut her open and took two small children out of her stomach and then beat them to death on the ground. I saw this with my own eyes.<ref name="legal tools" />}}

Testimony of Kada Hotić:

{{blockquote|There was a young woman with a baby on the way to the bus. The baby cried and a Serbian soldier told her that she had to make sure that the baby was quiet. Then the soldier took the child from the mother and cut its throat. I do not know whether Dutchbat soldiers saw that. ... There was a sort of fence on the left-hand side of the road to Potocari. I heard then a young woman screaming very close by (4 or 5 meters away). I then heard another woman beg: "Leave her, she is only nine years old." The screaming suddenly stopped. I was so in shock that I could scarcely move. ... The rumour later quickly circulated that a nine-year-old girl had been raped.<ref name="legal tools" />}}

That night, a Dutchbat medical orderly came across two Serb soldiers raping a young woman:

{{blockquote|[W]e saw two Serb soldiers, one of them was standing guard and the other one was lying on the girl, with his pants off. And we saw a girl lying on the ground, on some kind of mattress. There was blood on the mattress, even she was covered with blood. She had bruises on her legs. There was even blood coming down her legs. She was in total shock. She went totally crazy.

Bosnian Muslim refugees nearby could see the rape, but could do nothing about it because of Serb soldiers standing nearby. Other people heard women screaming, or saw women being dragged away. Several individuals were so terrified that they committed suicide by hanging themselves. Throughout the night and early the next morning, stories about the rapes and killings spread through the crowd and the terror in the camp escalated.

Screams, gunshots and other frightening noises were audible throughout the night and no one could sleep. Soldiers were picking people out of the crowd and taking them away: some returned; others did not.<ref>{{cite web|url=https://www.icty.org/x/cases/krstic/acjug/en/ |title=Radislav Krstic Trial Chambers Judgement |publisher=International Criminal Tribunal for the former Yugoslavia |date=5 March 2007 |access-date=26 May 2011}}</ref>}}

==== Deportation of women ====
As a result of exhaustive UN negotiations with Serb troops, around 25,000 Srebrenica women were forcibly transferred to Bosniak-controlled territory. Some buses apparently never reached safety. According to a witness account by Kadir Habibović, who hid himself on one of the first buses from the base in Potočari to Kladanj, he saw at least one vehicle full of Bosniak women being driven away from Bosnian government-held territory.<ref name="Rhode">{{citation|last1=Rohde |first1=David|title=Account of Women Taken|publisher=Columbia University|date=2 October 1995|url=http://www.columbia.edu/itc/journalism/nelson/rohde/p-1025.html|access-date=7 January 2018}}</ref>

=== Column of Bosniak men ===
[[File:Map 61 - Bosnia - Srebrenica & Zepa, July 1995.jpg|thumb|Map of military operations during the Srebrenica massacre. The green arrow marks the route of the Bosniak column.|upright=1.3]]

On the evening of 11 July, word spread that able-bodied men should take to the woods, form a [[column (formation)|column]] with the ARBiH's 28th Division and attempt a breakthrough towards Bosnian government-held territory in the north.<ref name="ICTY: Radislav Krstić verdict – The Column of Bosnian Muslim Men">{{cite web|url=https://www.un.org/icty/krstic/TrialC1/judgement/krs-tj010802e-1.htm#IIA7b |title=ICTY: Radislav Krstić verdict – The Column of Bosnian Muslim Men |url-status=dead |archive-url=https://web.archive.org/web/20080105091809/http://www.un.org/icty/krstic/TrialC1/judgement/krs-tj010802e-1.htm |archive-date=5 January 2008 }}</ref> They believed they stood a better chance of surviving by trying to escape, than if they fell into Serb hands.<ref>[http://www.icty.org/x/cases/blagojevic_jokic/tjug/en/bla-050117e.pdf Witness P-104 evidence to the Blagojevic trial], ICTY Case IT-06-60-T, Trial Chamber Judgment, 17 January 2005, Footnote 460, p. 52. Retrieved 9 April 2010.</ref> Around 10 pm on 11 July the Division command, with the municipal authorities, took the decision to form a column and attempt to reach government territory around Tuzla.<ref name="icty.org">{{Cite web|url=https://www.icty.org/x/cases/krstic/trans/en/010406ed.htm|title=Enver Hadzihasanovic evidence to the Krstic trial, 6 April 2001, ICTY transcript |page=9528|publisher=International Criminal Tribunal for the former Yugoslavia}}</ref> Dehydration, along with lack of sleep and exhaustion were further problems; there was little cohesion or common purpose.<ref name="auto3">{{Cite web|url=https://www.icty.org/x/cases/popovic/trans/en/081027ED.htm|title=Zoran Janković evidence to the Popović et al trial, 27 October 2008, ICTY transcript |page= 27369|website=International Criminal Tribunal for the former Yugoslavia}}</ref> Along the way, the column was shelled and ambushed. In severe mental distress, some refugees killed themselves. Others were induced to surrender. Survivors claimed they were attacked with a chemical agent that caused hallucinations, disorientation and strange behaviour.<ref name="Popovic 2006">{{Cite web|url=https://www.icty.org/x/cases/popovic/trans/en/061106ED.htm|title=Witness PW-139 evidence to the Popovic et al trial, 6 November 2006, ICTY transcript |pages=3665–66|website=International Criminal Tribunal for the former Yugoslavia}}</ref><ref name="commdocs.house.gov">{{cite web|url=http://commdocs.house.gov/committees/intlrel/hfa49268.000/hfa49268_0.htm#39|title=Testimony of Diane Paul to US House of Representatives Committee on International relations Subcommittee on International Operations and Human Rights hearing on the Betrayal of Srebrenica|page=39|access-date=24 July 2010}}</ref><ref name="news.bbc.co.uk">{{Cite web|url=http://news.bbc.co.uk/2/hi/europe/217753.stm|work=BBC News |title=Serbs accused of chemical attacks|date=19 November 1999}}</ref><ref name="sense-agency.com">{{Cite web|url=http://www.sense-agency.com/en/stream.php?sta=3&pid=8351&kat=3|archive-url=https://web.archive.org/web/20071008015959/http://www.sense-agency.com/en/stream.php?sta=3&pid=8351&kat=3|url-status=dead|title=SENSE Tribunal report |date=22 August 2006|archive-date=8 October 2007}}</ref> Infiltrators in civilian clothing confused, attacked and killed refugees.<ref name="Popovic 2006"/><ref>Testimony of I.N., J.C., J.T., G.I., N.T., Human Rights Watch Report [https://www.hrw.org/en/node/85087/section/12 "The Fall of Srebrenica and the Failure of UN Peacekeeping"], 15 October 1995 (12. Trek through Serbian-Controlled Territory). Retrieved 7 April 2010.</ref> Many taken prisoner were killed on the spot.<ref name="Popovic 2006"/> Others were collected and taken to remote locations, for execution.

The attacks broke the column into smaller segments. Only about one third succeeded in crossing the asphalt road between [[Konjević Polje]] and [[Nova Kasaba]]. This group reached Bosnian government territory on and after 16 July. A second, smaller group (700–800) attempted to escape into Serbia. It is not known how many were intercepted and killed. A third group headed for Žepa; estimates of how many vary between 300 and 850. Pockets of resistance apparently remained behind and engaged Serb forces.{{cn|date=June 2024}}

==== Tuzla column departs ====
Almost all the 28th Division, 5,500 to 6,000 soldiers, not all armed, gathered in [[Šušnjari]], in the hills north of Srebrenica, along with about 7,000 civilians. They included a few women.<ref name="icty.org"/> Others assembled in the nearby village of Jaglići.<ref>{{cite web|url=http://www.sudbih.gov.ba/bundles/websitenews/gallery/predmet/2544/Stevanovic_Trial_Verdict_290708.pdf|title=Court of Bosnia and Herzegovina – Prosecutor's office of Bosnia and Herzegovina v. Miladin Stevanović – First instance verdict|page=18|id=case X-KRŽ-05/24-2|date=29 July 2008 |access-date=30 April 2017|url-status=live |archive-url=https://web.archive.org/web/20170430161522/http://www.sudbih.gov.ba/bundles/websitenews/gallery/predmet/2544/Stevanovic_Trial_Verdict_290708.pdf |archive-date=30 April 2017 }}</ref>

Around midnight, the column started moving along the axis between Konjević Polje and Bratunac. It was preceded by four scouts, 5 km ahead.<ref name="Klip">{{cite book |last1=Klip |first1=André |title=The International Criminal Tribunal for the Former Yugoslavia 2001 |publisher=Intersentia nv |isbn=9789050953757 |page=611 |url=https://books.google.com/books?id=NlRD4yaHrEYC&q=on+11+July+1995,+the+column+started+moving+along+the+axis+between+Konjevi%C4%87+Polje+and+Bratunac.&pg=PA611|year=2005 }}</ref> Members walked one behind the other, following a trail a demining unit had marked with paper to guide them.<ref name="autogenerated1083">{{Cite web|url=https://www.icty.org/x/cases/popovic/trans/en/060831ED.htm|title=Mevludin Orić evidence to the Popovic et al trial, 31 August 2006, ICTY transcript |page= 1084|publisher=International Criminal Tribunal for the former Yugoslavia}}</ref>

The column was led by 50–100 of the best soldiers from each brigade, carrying the best equipment. Elements of the 284th Brigade were followed by the 280th Brigade. Civilians accompanied by other soldiers followed, and at the back was the independent battalion.<ref name="icty.org" /> The command and armed men were at the front, following the deminer unit.<ref name="autogenerated1083" /> Others included political leaders of the enclave, medical staff and families of prominent Srebrenicans. A few women, children and elderly travelled with the column in the woods.<ref name="ICTY: Radislav Krstić verdict – The Column of Bosnian Muslim Men" /><ref name="Neth-WarDocs">{{cite web|url=http://publications.niod.knaw.nl/publications/srebrenicareportniod_en.pdf|title=Srebrenica – Reconstruction, background, consequences and analyses of the fall of a 'safe' area|author=NIOD Institute for War-, Holocaust- and Genocide Studies|date=2002}}</ref> The column was 12-15km long, two and a half hours separating head from tail.<ref name="icty.org" />

The attempt to reach Tuzla surprised the VRS and caused confusion, as the VRS had expected the men to go to Potočari. Serb general [[Milan Gvero]], in a briefing, referred to the column as "hardened and violent criminals who will stop at nothing to prevent being taken prisoner".<ref>{{Cite web|url=https://www.icty.org/x/cases/popovic/trans/en/070911ED.htm|title=Col. Trkulja evidence concerning Gen. Gvero Order 03/4-1629, 13 July 1995, to the Popovic et al. trial, 11 September 2007, ICTY transcript |pages=15183, 15185|website=International Criminal Tribunal for the former Yugoslavia}}</ref> The Drina The VRS Main Staff ordered all available manpower to find any Muslim groups observed, prevent them crossing into Muslim territory, take them prisoner and hold them in buildings that could be secured by small forces.<ref>{{Cite web|url=https://www.icty.org/x/cases/blagojevic_jokic/trans/en/031113IT.htm|title=Richard Butler evidence to the Blagojević trial, 13 November 2003, ICTY transcript |pages=4470–4471|website=International Criminal Tribunal for the former Yugoslavia}}</ref>

==== Ambush at Kamenica Hill ====
During the night, poor visibility, fear of mines and panic induced by artillery fire split the column in two.<ref name="trial-ch.org">{{cite web|url=http://trial-ch.org/fileadmin/user_upload/documents/trialwatch/Srebrenica_Report2004.pdf|pages=1-6, 15-24|title=The Events in and Around Srebrenica Between 10th and 19th July 1995|archive-url=https://web.archive.org/web/20160303223001/http://trial-ch.org/fileadmin/user_upload/documents/trialwatch/Srebrenica_Report2004.pdf|archive-date=3 March 2016|date=11 June 2004}}</ref> On the afternoon of 12 July, the front section emerged from the woods and crossed the asphalt road from Konjević Polje and Nova Kasaba. Around 6pm, the VRS Army located the main part of the column around Kamenica. Around 8pm this part, led by the municipal authorities and wounded, started descending Kamenica Hill towards the road. After about 40 men had crossed, soldiers of the VRS arrived from the direction of [[Kravica]] in trucks and armoured vehicles, including a white vehicle with UNPROFOR symbols, calling over the loudspeaker, to surrender.<ref name="trial-ch.org" />

Yellow smoke was observed, followed by strange behaviour, including suicides, hallucinations and column members attacking one another.<ref name="Popovic 2006" /> Survivors claimed they were attacked with a chemical agent that caused hallucinations and disorientation.<ref name="commdocs.house.gov" /><ref name="news.bbc.co.uk" /> General Tolimir was an advocate of the use of chemical weapons against the ArBiH.<ref name="sense-agency.com" /><ref>{{cite web |url=https://fas.org/news/serbia/fbis-weu-1999-0426.htm |title=Yugoslavia Has Long-Standing Poison Gas Program |first=Karel |last=Knip |location=Rotterdam |work=NRC Handelsblad |date=24 April 1999 |pages=1, 5 |via=Federation of American Scientists |access-date=26 May 2011 |url-status=dead |archive-url=https://web.archive.org/web/20110807043057/http://www.fas.org/news/serbia/fbis-weu-1999-0426.htm |archive-date=7 August 2011 |df=dmy-all }}</ref> Shooting and shelling began, which continued into the night. Armed members of the column returned fire and all scattered. Survivors at least 1,000 engaged at close range by small arms. Hundreds appear to have been killed as they fled the open area and some were said to have killed themselves to escape capture.{{cn|date=June 2024}}

VRS and Ministry of Interior personnel persuaded column members to surrender, by promising them safe transportation towards Tuzla, under UNPROFOR and Red Cross supervision. Appropriated UN and Red Cross equipment was used to deceive them. Belongings were confiscated and some executed on the spot.<ref name="trial-ch.org" />

The rear of the column lost contact and panic broke out. Many remained in the Kamenica Hill area for days, with the escape route blocked by Serb forces. Thousands of Bosniaks surrendered or were captured. Some were ordered to summon friends and family from the woods. There were reports of Serb forces using megaphones to call on the marchers to surrender, telling them they would be exchanged for Serb soldiers. It was at Kamenica that VRS personnel in civilian dress were reported to have infiltrated the column. Men who survived described it as a manhunt.<ref name="ICTY: Radislav Krstić verdict – The Column of Bosnian Muslim Men" />

==== Sandići massacre ====
[[File:Exhumations in Srebrenica 1996.jpg|thumb|[[Exhumation]]s in Srebrenica, 1996]]

Close to Sandići, on the main road from Bratunac to Konjević Polje, a witness described the Serbs forcing a Bosniak man to call other Bosniaks down from the mountains. 200–300 men, including the witness' brother, descended to meet the VRS, presumably expecting an exchange of prisoners. The witness hid behind a tree and watched as the men were lined up in seven ranks, each 40 m long, with hands behind their heads; they were then mowed down by machine guns.<ref name="Popovic 2006"/>

Some women, children and elderly people who had been part of the column, were allowed to join buses evacuating women and children from Potočari.<ref>ABiH Tuzla. Tuzla (Intel Dept) to 2nd Corps, 25/07/95, (Tuzla no.) 11.6.-1-414/95 (2nd Corps no.) 06-712-24-30/95, Results of meeting with persons from Srebrenica. This report was signed by Sarajlic Osman.</ref>

==== Trek to Mount Udrč ====
The central section of the column managed to escape the shooting, reached Kamenica around 11am and waited for the wounded. [[Captain (land)|Captain]] Golić and the Independent Battalion turned back towards Hajdučko Groblje, to help the casualties. Survivors from the rear, crossed the asphalt roads to the north or the west, and joined the central section. The front third of the column, which had left Kamenica Hill by the time the ambush occurred, headed for Mount Udrč ({{Coord|44|16|59|N|19|3|6|E|type:mountain|name=Mount Udrc}}); crossing the main asphalt road. They reached the base of the mountain on Thursday 13 July and regrouped. At first, it was decided to send 300 ARBiH soldiers back to break through the blockades. When reports came that the central section had crossed the road at Konjević Polje, this plan was abandoned. Approximately 1,000 additional men managed to reach Udrč that night.<ref>ABiH Tuzla. ABiH 2nd Corps (unnumbered). Additional statement by Ramiz Bećirović, 16 April 1998, based on an earlier statement of 11 August 1995.</ref>

==== Snagovo ambush ====
From Udrč, the marchers moved toward the River Drinjača and Mount Velja Glava. Finding Serbs at Mount Velja Glava, on Friday, 14 July, the column skirted the mountain and waited on its slopes, before moving toward [[Liplje, Zvornik|Liplje]] and Marčići. Arriving at Marčići in the evening of 14 July, they were ambushed again near Snagovo by forces equipped with anti-aircraft guns, artillery, and tanks.<ref name="trial-ch.org" /> The column broke through and captured a VRS officer, providing them with a bargaining counter. This prompted an attempt at negotiating a ceasefire, but this failed.<ref name="auto3"/>

==== Approaching the frontline ====
The evening of 15 July saw the first radio contact between the 2nd Corps and the 28th Division. The Šabić brothers were able to identify each other as they stood on either side of the VRS lines. Early in the morning, the column crossed the road linking Zvornik with Caparde and headed towards Planinci, leaving 100–200 armed marchers behind to wait for stragglers.{{Citation needed|date=August 2008}} The column reached Križevići later that day, and remained while an attempt was made to negotiate with Serb forces, for safe passage. They were advised to stay where they were, and allow Serb forces to arrange safe passage. It became apparent, that the small Serb force was only trying to gain time to organise another attack. In the area of Marčići – Crni Vrh, VRS armed forces deployed 500 soldiers and policemen to stop the split part of the column, about 2,500 people, which was moving from Glodi towards Marčići.{{Citation needed|date=August 2008}} The column's leaders decided to form small groups of 100–200 and send these to reconnoitre ahead. The 2nd Corps and 28th Division of the ARBiH met each other in Potočani.

==== Breakthrough at Baljkovica ====
The hillside at Baljkovica ({{Coord|44|27|N|18|58|E|name=Baljkovica|type:city}}) formed the last VRS line separating the column from Bosnian-held territory. The VRS cordon consisted of two lines, the first of which presented a front on the Tuzla side, against the 2nd Corps and the other a front against the approaching 28th Division.{{Citation needed|date=April 2010}}

On the evening of 15 July a hailstorm caused Serb forces to take cover. The column's advance group took advantage to attack the Serb rear lines at Baljkovica. The main body of what remained of the column began to move from Krizevici. It reached the area of fighting around 3 am on Sunday, 16 July.{{Citation needed|date=April 2010}} At approximately 5am, the 2nd Corps made its first attempt to break through the VRS cordon. The objective was to breakthrough close to the hamlets of Parlog and Resnik. They were joined by Orić and some of his men.{{Citation needed|date=August 2008}} Around 8 am, parts of the 28th Division, with the 2nd Corps of the RBiH Army from Tuzla providing artillery support, attacked and breached VRS lines. There was fierce fighting across Baljkovica.<ref name="trial-ch.org" /> The column finally succeeded in breaking through to Bosnian government-controlled territory, between 1 and 2 pm.{{Citation needed|date=August 2008}}

==== Baljkovica corridor ====
Following radio negotiations between the 2nd Corps and Zvornik Brigade, Brigade Command agreed to open a corridor to allow "evacuation" of the column in return for the release of captured policemen and soldiers. The corridor was open 2–5pm.<ref name="trial-ch.org" /> After the corridor was closed between 5 and 6 pm, the Zvornik Brigade Command reported that around 5000 civilians, with probably "a certain number of soldiers" with them had been let through, but "all those who passed were unarmed".<ref name="trial-ch.org" />

[[File:Srebrenica 2008 1.jpg|thumb|upright=.8|Damaged building in Srebrenica after the war]]
By about 4 August, the ArBiH determined that 3,175 members of the 28th Division had managed to get through to Tuzla. 2,628 members of the Division, soldiers and officers, were considered certain to have been killed. Column members killed was between 8,300 and 9,722.<ref name="icty.org" />

==== After closure of the corridor ====
Once the corridor had closed, Serb forces recommenced hunting down parts of the column. Around 2,000 refugees were reported to be hiding in the woods in the area of Pobuđe.<ref name="trial-ch.org" /> On 17 July, four children aged between 8 and 14 captured by the Bratunac Brigade were taken to the military barracks in Bratunac.<ref name="trial-ch.org" /><ref name="B&J para.467">{{cite web|work=ICTY|title=Prosecutor v. Blagojević & Jokić Trial Chamber Judgment|id=Case No. IT-02-60|at=para. 467|url=http://www.icty.org/x/cases/blagojevic_jokic/tjug/en/bla-050117e.pdf#page=181|date=17 January 2005|access-date=5 May 2017}}</ref> Brigade Commander Blagojević suggested the Drina Corps' press unit record this testimony on video.<ref name="B&J para.467"/>

On 18 July, after a soldier was killed "trying to capture some persons during the search operation", the Zvornik Brigade Command issued an order to execute prisoners, to avoid any risks associated with their capture. The order was presumed to have remained effective until countermanded on 21 July.<ref name="trial-ch.org" />

====Impact on survivors====
According to a 1998 [[Qualitative research|qualitative study]] involving survivors, many column members exhibited symptoms of hallucinations to varying degrees.<ref name="auto2">{{cite journal|last=Hay|first=Alastair|title=Surviving the Impossible: The Long March from Srebrenica. An Investigation of the Possible Use of Chemical Warfare Agents|journal=Medicine, Conflict and Survival|volume=14|issue=2|pages=120–155|doi=10.1080/13623699808409383|date=1998|pmid=9633268}}</ref> Several times, Bosniak men attacked one another, in fear the other was a Serb soldier. Survivors reported seeing people speaking incoherently, running towards VRS lines in a rage and committing suicide using firearms and hand grenades. Although there was no evidence to suggest what exactly caused the behaviour, the study suggested fatigue and stress may have induced this.<ref name="auto2"/>

=== A plan to execute the men ===
Although Serb forces had long been blamed for the massacre, it was not until 2004—following the Srebrenica Commission's report—that Serb officials acknowledged their forces carried out the mass killing. Their report acknowledged the [[mass murder]] of the men and boys was planned, and more than 7,800 were killed.<ref name="Investigating Srebrenica" /><ref name=":0" /><ref name=":1" />

A concerted effort was made to capture all Bosniak men of military age.<ref name="Krstic 2001 takeover" /> In fact, those captured included many boys well below that age, and men years above that age, who remained in the enclave following the take-over of Srebrenica. These men and boys were targeted, regardless of whether they chose to flee to Potočari or join the column. The operation to capture and detain the men was well-organised and comprehensive. The buses which transported women and children, were systematically searched for men.<ref name="Krstic 2001 takeover" />

=== Mass executions ===
The amount of planning and high-level coordination invested in killing thousands in a few days, is apparent from the scale and methodical nature in which the executions were carried out.{{cn|date=June 2024}}

The [[Army of Republika Srpska]] took the largest number of prisoners on 13 July, along the Bratunac-Konjević Polje road. Witnesses describe the assembly points, such as the field at Sandići, agricultural [[warehouses]] in Kravica, the school in Konjević Polje, the [[football pitch]] in Nova Kasaba, Lolići and Luke school. Several thousand people were herded in the field near Sandići and on the Nova Kasaba pitch, where they were searched and put into smaller groups. In a video by journalist Zoran Petrović, a Serb soldier states that at least 3,000–4,000 men gave themselves up on the road. By the late afternoon of 13 July, the total had risen to 6,000 according to intercepted radio communication; the following day, Major Franken of Dutchbat was given the same figure by Colonel Radislav Janković of the Serb army. Many prisoners had been seen in the locations described, by passing convoys taking women and children to Kladanj by bus, while aerial photos provided evidence to confirm this.<ref name="Neth-WarDocs"/><ref name="Krstic 2001 takeover" />

One hour after the evacuation of women from Potočari was complete, the Drina Corps staff diverted the buses to the areas in which the men were being held. Colonel Krsmanović, who on 12 July had arranged the buses for the evacuation, ordered the 700 men in Sandići to be collected, and the soldiers guarding them, made them throw their possessions on a heap and hand over valuables. During the afternoon, the group in Sandići was visited by Mladić, who told them they would come to no harm, be treated as prisoners of war, exchanged for other prisoners, and that their families had been escorted to Tuzla in safety. Some men were placed on transports to Bratunac and other locations, while some were marched to warehouses in Kravica. The men gathered on the pitch at Nova Kasaba were forced to hand over belongings. They too received a visit from Mladić during the afternoon of 13 July; on this occasion, he announced that the Bosnian authorities in Tuzla did not want them and so they were to be taken elsewhere. The men in Nova Kasaba were loaded onto buses and trucks and taken to Bratunac, or other locations.<ref name="Krstic 2001 takeover" />

The Bosnian men who had been separated from the women, children and elderly in Potočari, numbering approximately 1,000, were transported to Bratunac and joined by Bosnian men captured from the column.<ref>[http://www.icty.org/x/cases/blagojevic_jokic/trans/en/030519IT.htm Jean-René Ruez evidence to the Blagojevic trial], 19 May 2003 ICTY transcript p. 480</ref> Almost without exception, the thousands of prisoners captured after the take-over were executed. Some were killed individually, or in small groups, by the soldiers who captured them. Most were killed in carefully orchestrated mass executions, commencing on 13 July, just north of Srebrenica.

The mass executions followed a well-established pattern. The men were taken to empty schools or warehouses. After being detained for hours, they were loaded onto buses or trucks and taken to another site, usually in an isolated location. They were unarmed and often steps were taken to minimise resistance, such as [[blindfold]]ing, binding their wrists behind their backs with ligatures, or removing their shoes. Once at the killing fields, the men were taken off the trucks in small groups, lined up and shot. Those who survived the initial shooting were shot with an extra round, though sometimes only after they had been left to suffer.<ref name="Krstic 2001 takeover" />

==== Morning of 13 July: Jadar River ====
Prior to midday on 13 July, seventeen men were transported by bus a short distance to a spot on the banks of the Jadar River where they were lined up and shot. One man, after being hit in the hip by a bullet, jumped into the river and managed to escape.<ref name="Krstic 2001 takeover" />

==== Early afternoon of 13 July: Cerska Valley ====
[[File:Srebrenica Massacre - Massacre Victim 2 - Potocari 2007.jpg|upright=.8|thumb|Skull of a victim, at an exhumed mass grave outside the village of [[Donji Potočari|Potočari]], July 2007.]]
The first mass executions began on 13 July in the valley of the River Cerska, to the west of Konjević Polje. One witness, hidden among trees, saw two or three trucks, followed by an armoured vehicle and earthmoving machine proceeding towards Cerska. He heard gunshots for half an hour and then saw the armoured vehicle going in the opposite direction, but not the earthmoving machine. Other witnesses report seeing a pool of blood alongside the road to Cerska. Muhamed Duraković, a UN translator, probably passed this execution site later that day. He reports seeing bodies tossed into a ditch alongside the road, with some men still alive.<ref name="Krstic 2001 takeover" /><ref name="Cerska Valley">{{cite web|url=http://publications.niod.knaw.nl/publications/srebrenicareportniod_en.pdf#page=1966 |title=Srebrenica – a 'safe' area – Part IV: The repercussion and the aftermath until the end of 1995 – Chapter 2: The executions – 6. The afternoon of 13 July 1995: executions in the Cerska valley |page=1966 }}</ref>

Aerial photos, and excavations, confirmed the presence of a mass grave near this location. Bullet cartridges, found at the scene, showed that the victims were first lined up on one side of the road, whereupon their executioners shot from the other. The 150 bodies were covered with earth where they lay. It was later established they had been killed by gunfire. All were men, aged 14–50, and all but three were wearing civilian clothes. Many had their hands tied behind their backs. Nine were later identified on the Srebrenica missing persons list.<ref name="Krstic 2001 takeover" />

==== Late afternoon of 13 July: Kravica ====
Later on 13 July executions were conducted in the largest of four farm sheds, owned by the Agricultural Cooperative in Kravica. Between 1,000 and 1,500 men had been captured in fields near Sandići and detained in Sandići Meadow. They were brought to Kravica, either by bus or on foot, the distance being approximately 1km. A witness recalls seeing around 200 men, stripped to the waist and with their hands in the air, being forced to run in the direction of Kravica. An aerial photo taken at 2pm shows two buses standing in front of the sheds.<ref name="Krstic 2001 takeover" />

At around 6pm, when the men were all held in the warehouse, VRS soldiers threw in [[hand grenades]] and fired weapons, including [[rocket propelled grenade]]s. This mass murder seemed "well organised and involved a substantial amount of planning, requiring the participation of the Drina Corps Command".<ref name="Krstic 2001 takeover" />

Supposedly, there was more killing in and around Kravica and Sandići. Even before the murders in the warehouse, some 200 or 300 men were formed up in ranks near Sandići, then executed en masse with concentrated machine gun fire. At Kravica, it was claimed some local men assisted the killings. Some victims were mutilated and killed with knives. The bodies were taken to Bratunac, or simply dumped in the river that runs alongside the road. One witness stated this all took place on 14 July. There were three survivors of the mass murder in the farm sheds at Kravica.<ref name="Krstic 2001 takeover" />

Armed guards shot at the men who tried to climb out the windows to escape the massacre. When the shooting stopped, the shed was full of bodies. Another survivor, who was only slightly wounded, reports:

{{blockquote|I was not even able to touch the floor, the concrete floor of the warehouse ... After the shooting, I felt a strange kind of heat, warmth, which was coming from the blood that covered the concrete floor and I was stepping on the dead people who were lying around. But there were even men (just men) who were still alive, who were only wounded and as soon as I would step on him, I would hear him cry, moan, because I was trying to move as fast as I could. I could tell that people had been completely disembodied and I could feel bones of the people that had been hit by those bursts of bullets or shells, I could feel their ribs crushing. Then I would get up again and continue.<ref name="ICTY" />}}

When this witness climbed out of a window, he was seen by a guard who shot at him. He pretended to be dead and managed to escape the following morning. The other witness quoted above spent the night under a heap of bodies; the next morning, he watched as the soldiers examined the corpses for signs of life. The few survivors were forced to sing Serbian songs and were then shot. Once the final victim had been killed, an excavator was driven in to shunt the bodies out of the shed; the asphalt outside was then hosed down with water. In September 1996, however, it was still possible to find the evidence.<ref name="Krstic 2001 takeover" />

Analyses of hair, blood and explosives residue collected at the Kravica Warehouse provide strong evidence of the killings. Experts determined the presence of bullet strikes, explosives residue, bullets and shell cases, as well as human blood, bones and tissue adhering to the walls and floors of the building. Forensic evidence presented by the ICTY Prosecutor established a link between the executions in Kravica and the 'primary' mass grave known as Glogova 2, in which the remains of 139 people were found. In the 'secondary' grave known as Zeleni Jadar 5, there were 145 bodies, several were charred. Pieces of brick and window frame found in the Glogova 1 grave that was opened later, also established a link with Kravica. Here, the remains of 191 victims were found.<ref name="Krstic 2001 takeover" />

==== 13–14 July: Tišća ====

As the buses crowded with Bosnian women, children and elderly made their way from Potočari to Kladanj, they were stopped at Tišća village, searched, and the Bosnian men and boys found on board were removed. The evidence reveals a well-organised operation in Tišća.<ref name="Krstic 2001 takeover" />

From the checkpoint, an officer directed the soldier escorting the witness towards a nearby school where many other prisoners were being held. At the school, a soldier on a field telephone appeared to be transmitting and receiving orders. Around midnight, the witness was loaded onto a truck with 22 other men with their hands tied behind their backs. At one point the truck stopped and a soldier said: "Not here. Take them up there, where they took people before." The truck reached another stopping point and the soldiers came to the back of the truck and started shooting the prisoners. The survivor escaped by running away from the truck and hiding in a forest.<ref name="Krstic 2001 takeover" />

==== 14 July: Grbavci and Orahovac ====
A large group of prisoners held overnight in Bratunac were bussed in a convoy of 30 vehicles to the Grbavci school in [[Orahovica (Srebrenica)|Orahovica]], early on 14 July. When they arrived, the gym was already half-full with prisoners and within a few hours, the building was full. Survivors estimated there were about 2,000 men, some very young, others elderly, although the ICTY Prosecution suggested this may be an overestimation, with the number closer to 1,000. Some prisoners were taken outside and killed. At some point, a witness recalled, General Mladić arrived and told the men: "Well, your government does not want you and I have to take care of you."<ref name="Krstic 2001 takeover" />

After being held in the gym for hours, the men were led out in small groups to the execution fields that afternoon. Each prisoner was blindfolded and given water as he left. The prisoners were taken in trucks to the fields less than 1km away. The men were lined up and shot in the back; those who survived were killed with an extra shot. Two adjacent meadows were used; once one was full of bodies, the executioners moved to the other. While the executions were in progress, the survivors said earth-moving equipment dug the graves. A witness reported that Mladić watched some of the executions.<ref name="Krstic 2001 takeover" />

The forensic evidence supports crucial aspects of the testimony. Aerial photos show the ground in Orahovac was disturbed between 5 and 27 July and between 7 and 27 September. Two primary mass graves were uncovered in the area and named Lazete 1 and Lazete 2 by investigators.<ref name="Krstic 2001 takeover" /> Lazete 1 was exhumed by the ICTY in 2000. All of the 130 individuals uncovered, for whom sex could be determined, were male; 138 blindfolds were found. Identification material for 23 persons, listed as missing following the fall of Srebrenica, was located during the exhumations. Lazete 2 was partly exhumed by a joint team, from the Office of the Prosecutor and Physicians for Human Rights, in 1996 and completed in 2000. All of the 243 victims associated with Lazete 2 were male, and experts determined most died of gunshot injuries. 147 blindfolds were located.<ref name="Krstic 2001 takeover" /> Forensic analysis of soil/pollen samples, blindfolds, ligatures, shell cases and aerial images of creation/disturbance dates, further revealed that bodies, from Lazete 1 and 2, were reburied at secondary graves named Hodžići Road 3, 4 and 5. Aerial images show these secondary gravesites were begun in early September 1995 and all were exhumed in 1998.<ref name="Krstic 2001 takeover" />

==== 14–15 July: Petkovići ====
[[File:Srebrenica Massacre - Mass Gravesite - Potocari 2007.jpg|thumb|Delegates of the International Association of Genocide Scholars (IAGS) examine an exhumed mass grave, outside the village of Potočari, July 2007.]]
On 14 and 15 July 1995, another group of prisoners numbering 1,500 to 2,000 were taken from Bratunac to the school in Petkovići. The conditions at the Petkovići school were even worse than Grbavci. It was hot, and overcrowded and there was no food or water. In the absence of anything else, some prisoners chose to drink their urine. Now and then, soldiers would enter the room and physically abuse prisoners or call them outside. A few contemplated an escape attempt, but others said it would be better to stay since the International Red Cross would be sure to monitor the situation and they could not all be killed.<ref name="Krstic 2001 Petkovci School">{{cite web |url=http://www.icty.org/x/cases/krstic/tjug/en/krs-tj010802e.pdf#page=86 |title=Krstic Judgement |at=Part II. Findings of Fact – B. The Role of the Drina Corps in the Srebrenica Crimes – 5. Involvement of the Drina Corps in the Mass Executions – (f) 14–15 July 1995: Petkovci School Detention Site and Petkovci Dam Execution Site, para. 226, p. 81 |date=2 August 2001 |publisher=ICTY }}</ref>

The men were called outside in small groups. They were ordered to strip to the waist and remove their shoes, whereupon their hands were tied behind their backs. During the night of 14 July, the men were taken by truck to the dam at Petkovići. Those who arrived later could see immediately what was happening. Bodies were strewn on the ground, hands tied behind their backs. Small groups of five to ten men were taken out of the trucks, lined up and shot. Some begged for water but their pleas were ignored.<ref name="Krstic 2001 Petkovci School" /> A survivor described his feelings of fear combined with thirst:

{{blockquote|I was really sorry that I would die thirsty, and I was trying to hide amongst the people as long as I could, like everybody else. I just wanted to live for another second or two. And when it was my turn, I jumped out with what I believe were four other people. I could feel the gravel beneath my feet. It hurt ... I was walking with my head bent down and I wasn't feeling anything. ... And then I thought that I would die very fast, that I would not suffer. And I just thought that my mother would never know where I had ended up. This is what I was thinking as I was getting out of the truck. [As the soldiers walked around to kill the survivors of the first round of shooting] I was still very thirsty. But I was sort of between life and death. I didn't know whether I wanted to live or die anymore. I decided not to call out for them to shoot and kill me, but I was sort of praying to God that they'd come and kill me.<ref name="ICTY" />}}

After the soldiers had left, two survivors helped each other to untie their hands and crawled over the bodies towards the woods, where they intended to hide. As dawn arrived, they could see the execution site where bulldozers were collecting the bodies. On the way to the execution site, one survivor peeked out from under his blindfold and saw Mladić on his way to the scene.<ref name="ICTY" />

Aerial photos confirmed the earth near the Petkovići dam had been disturbed and it was disturbed again in late September 1995. When the grave was opened in April 1998, there seemed to be many bodies missing. Their removal had been accomplished with mechanical apparatus, causing considerable disturbance. The grave contained the remains of no more than 43 persons. Other bodies had been removed to a secondary grave, Liplje 2, before 2 October. Here, the remains of at least 191 individuals were discovered.<ref name="ICTY" />

==== 14–16 July: Branjevo ====
On 14 July, more prisoners from Bratunac were bussed northward to a school in Pilica. As at other detention facilities, there was no food or water and several died from heat and dehydration. The men were held at the school for two nights. On 16 July, following a now familiar pattern, the men were called out and loaded onto buses with their hands tied behind their backs, driven to the Branjevo Military Farm, where groups of 10 were lined up and shot.<ref name="ICTY-233">ICTY, [https://www.un.org/icty/krstic/TrialC1/judgement/index.htm ''Prosecutor vs Krstic, Judgement''] {{webarchive |url=https://web.archive.org/web/20080517075111/http://www.un.org/icty/krstic/TrialC1/judgement/index.htm |date=17 May 2008 }}, II, B, 5 (g) [https://www.un.org/icty/krstic/TrialC1/judgement/krs-tj010802e-1.htm#IIB5g "14 – 16 July 1995: Pilica School Detention Site and Branjevo Military Farm Execution Site"] {{webarchive |url=https://web.archive.org/web/20090508011123/http://www.un.org/icty/krstic/TrialC1/judgement/krs-tj010802e-1.htm#IIB5g |date=8 May 2009 }}, par. 233.</ref>

[[Dražen Erdemović]]—who confessed to killing at least 70 Bosniaks—was a member of the VRS 10th [[Sabotage]] Detachment. Erdemović appeared as a prosecution witness and testified: "The men in front of us were ordered to turn their backs ... we shot at them. We were given orders to shoot."<ref name="ICTY-234">ICTY, [https://www.un.org/icty/krstic/TrialC1/judgement/index.htm ''Prosecutor vs Krstic, Judgement''] {{webarchive |url=https://web.archive.org/web/20080517075111/http://www.un.org/icty/krstic/TrialC1/judgement/index.htm |date=17 May 2008 }}, II, B, 5 (g) [https://www.un.org/icty/krstic/TrialC1/judgement/krs-tj010802e-1.htm#IIB5g "14 – 16 July 1995: Pilica School Detention Site and Branjevo Military Farm Execution Site"] {{webarchive |url=https://web.archive.org/web/20090508011123/http://www.un.org/icty/krstic/TrialC1/judgement/krs-tj010802e-1.htm#IIB5g |date=8 May 2009 }}, par. 234.</ref> On this point, a survivor recalls:
{{blockquote|When they shot, I threw myself on the ground ... one man fell on my head. I think that he was killed on the spot. I could feel the hot blood pouring over me ... I could hear one man crying for help. He was begging them to kill him. And they simply said "Let him suffer. We'll kill him later."|Witness Q<ref>ICTY, [https://www.un.org/icty/krstic/TrialC1/judgement/index.htm Prosecutor vs Krstic, Judgement] {{webarchive |url=https://web.archive.org/web/20080517075111/http://www.un.org/icty/krstic/TrialC1/judgement/index.htm |date=17 May 2008 }}, II, B, 5 (g) [https://www.un.org/icty/krstic/TrialC1/judgement/krs-tj010802e-1.htm#IIB5g "14 – 16 July 1995: Pilica School Detention Site and Branjevo Military Farm Execution Site"] {{webarchive |url=https://web.archive.org/web/20090508011123/http://www.un.org/icty/krstic/TrialC1/judgement/krs-tj010802e-1.htm#IIB5g |date=8 May 2009 }}, par. 235.</ref>}}
[[File:Mass Graves.jpg|thumb|Satellite photo of [[Nova Kasaba]] mass graves]]

Erdemović said nearly all the victims wore civilian clothes and, except for one person who tried to escape, offered no resistance. Sometimes the executioners were particularly cruel. When some soldiers recognised acquaintances, they beat and humiliated them, before killing them. Erdemović had to persuade fellow soldiers to stop using machine guns; while it mortally wounded the prisoners, it did not cause death immediately and prolonged their suffering.<ref name="ICTY-234" />
Between 1,000 and 1,200 men were killed in that day at this execution site.<ref>ICTY, [https://www.un.org/icty/krstic/TrialC1/judgement/index.htm ''Prosecutor vs Krstic, Judgement''] {{webarchive |url=https://web.archive.org/web/20080517075111/http://www.un.org/icty/krstic/TrialC1/judgement/index.htm |date=17 May 2008 }}, II, B, 5 (g) [https://www.un.org/icty/krstic/TrialC1/judgement/krs-tj010802e-1.htm#IIB5g "14 – 16 July 1995: Pilica School Detention Site and Branjevo Military Farm Execution Site"] {{webarchive |url=https://web.archive.org/web/20090508011123/http://www.un.org/icty/krstic/TrialC1/judgement/krs-tj010802e-1.htm#IIB5g |date=8 May 2009 }}", par. 236.</ref>

Aerial photos, taken on 17 July of an area around the Branjevo Military Farm, show many bodies lying in a field, as well as traces of the excavator that collected the bodies.<ref name="ICTY-237">ICTY, [https://www.un.org/icty/krstic/TrialC1/judgement/index.htm ''Prosecutor vs Krstic, Judgement''] {{webarchive |url=https://web.archive.org/web/20080517075111/http://www.un.org/icty/krstic/TrialC1/judgement/index.htm |date=17 May 2008 }}, II, B, 5 (g) [https://www.un.org/icty/krstic/TrialC1/judgement/krs-tj010802e-1.htm#IIB5g "14 – 16 July 1995: Pilica School Detention Site and Branjevo Military Farm Execution Site"] {{webarchive |url=https://web.archive.org/web/20090508011123/http://www.un.org/icty/krstic/TrialC1/judgement/krs-tj010802e-1.htm#IIB5g |date=8 May 2009 }}, par. 237.</ref> Erdemović testified that, at around 3pm on 16 July, after he and fellow soldiers from the 10th Sabotage Detachment had finished executing prisoners at the Farm, they were told there was a group of 500 Bosnian prisoners from Srebrenica, trying to break out of a Dom Kultura club. Erdemović and other members of his unit refused to carry out more killings. They were told to meet with a [[Lieutenant Colonel]] at a café in Pilica. Erdemović and his fellow soldiers travelled to the café and, as they waited, could hear shots and grenades being detonated. The sounds lasted 15–20 minutes after which a soldier entered the café to inform them "everything was over".<ref name="ICTY-244">ICTY, [https://www.un.org/icty/krstic/TrialC1/judgement/index.htm ''Prosecutor vs Krstic, Judgement''] {{webarchive |url=https://web.archive.org/web/20080517075111/http://www.un.org/icty/krstic/TrialC1/judgement/index.htm |date=17 May 2008 }}, II, B, 5 (h) [https://www.un.org/icty/krstic/TrialC1/judgement/krs-tj010802e-1.htm#IIB5h "16 July 1995: Pilica Cultural Dom"] {{webarchive |url=https://web.archive.org/web/20090508011123/http://www.un.org/icty/krstic/TrialC1/judgement/krs-tj010802e-1.htm#IIB5h |date=8 May 2009 }}, par. 244.</ref>

There were no survivors to explain exactly what happened in the Dom Kultura.<ref name="ICTY-244" /> The executions there were remarkable as this was not remote, but a town centre on the main road from Zvornik to Bijeljina.<ref>{{Cite web|url=https://www.icty.org/x/cases/blagojevic_jokic/trans/en/030519IT.htm|title=Jean-René Ruez evidence to the Blagojevic trial, 19 May 2003 ICTY transcript |pages=535–536|publisher=International Criminal Tribunal for the former Yugoslavia}}</ref> Over a year later, it was still possible to find physical evidence of this crime. As in Kravica, many traces of blood, hair and body tissue were found in the building, with cartridges and shells littered throughout the two storeys.<ref name="ICTY-245">ICTY, [https://www.un.org/icty/krstic/TrialC1/judgement/index.htm ''Prosecutor vs Krstic, Judgement''] {{webarchive |url=https://web.archive.org/web/20080517075111/http://www.un.org/icty/krstic/TrialC1/judgement/index.htm |date=17 May 2008 }}, II, B, 5 (h) [https://www.un.org/icty/krstic/TrialC1/judgement/krs-tj010802e-1.htm#IIB5h "16 July 1995: Pilica Cultural Dom"] {{webarchive |url=https://web.archive.org/web/20090508011123/http://www.un.org/icty/krstic/TrialC1/judgement/krs-tj010802e-1.htm#IIB5h |date=8 May 2009 }}, par. 245.</ref> It could be established that explosives and machine guns had been used. Human remains and personal possessions were found under the stage, where blood had dripped down through the floorboards.

Two of the three survivors of the executions at the Branjevo Military Farm, were arrested by Bosnian Serb police on 25 July and sent to the prisoner of war compound at Batkovici. One had been a member of the group separated from the women in Potočari on 13 July. The prisoners who were taken to Batkovici survived<ref>[http://www.icty.org/x/cases/krstic/trans/en/000719it.htm Richard Butler evidence to the Krstic trial], 19 July 2000, ICTY transcript p. 5431. Retrieved 7 April 2010.</ref> and testified before the Tribunal.<ref>[http://www.icty.org/x/cases/popovic/trans/en/061107ED.htm Witness PW-139 evidence to the Popovice et al.], 7 November 2006, ICTY transcript p. 3690.</ref>

Čančari Road 12 was the site of the reinterment of at least 174 bodies, moved from the mass grave at the Branjevo Military Farm.<ref name="ICTY-238">ICTY, [https://www.un.org/icty/krstic/TrialC1/judgement/index.htm ''Prosecutor vs Krstic, Judgement''] {{webarchive |url=https://web.archive.org/web/20080517075111/http://www.un.org/icty/krstic/TrialC1/judgement/index.htm |date=17 May 2008 }}, II, B, 5 (g) [https://www.un.org/icty/krstic/TrialC1/judgement/krs-tj010802e-1.htm#IIB5g "14 – 16 July 1995: Pilica School Detention Site and Branjevo Military Farm Execution Site"] {{webarchive |url=https://web.archive.org/web/20090508011123/http://www.un.org/icty/krstic/TrialC1/judgement/krs-tj010802e-1.htm#IIB5g |date=8 May 2009 }}, par. 238.</ref> Only 43 were complete sets of remains, most of which established that death was due to rifle fire. Of the 313 body parts found, 145 displayed gunshot wounds of a severity likely to prove fatal.<ref name="Manning-A4">{{cite web|author=Dean Manning|url=http://abunodisceomnes.wellcomecollection.org/wp-content/uploads/2015/01/Manning-D.-16-May-2000-Srebrenica-investigation-Summary-of-forensic-evidence-Execution-points-and-mass-graves-ICTY-Investigator.pdf#page=45|title=Srebrenica Investigation: Summary of Forensic Evidence – Execution Points and Mass Graves|date=16 May 2000|at=Annex A, 4 "ČANČARI ROAD 12 – (SECONDARY GRAVE)", pp. 18–21|access-date=25 April 2017|archive-url=https://web.archive.org/web/20170425192611/http://abunodisceomnes.wellcomecollection.org/wp-content/uploads/2015/01/Manning-D.-16-May-2000-Srebrenica-investigation-Summary-of-forensic-evidence-Execution-points-and-mass-graves-ICTY-Investigator.pdf#page=45|archive-date=25 April 2017|url-status=dead|df=dmy-all}}</ref>

==== 14–17 July: Kozluk ====
[[File:Exhumation Site in Čančari valley.jpg|thumb|Exhumation of the Srebrenica massacre victims]]
The exact date of the executions at Kozluk is unknown, though most probably 15–16 July, partly due to its location, between Petkovići Dam and the Branjevo Military Farm. It falls within the pattern of ever more northerly execution sites: Orahovac on 14 July, Petkovići Dam on 15 July, the Branjevo Military Farm and Pilica Dom Kultura on 16 July. Another indication is that a Zvornik Brigade excavator spent eight hours in Kozluk on 16 July and a truck belonging to the same brigade made two journeys between Orahovac and Kozluk that day. A bulldozer is known to have been active in Kozluk on 18 and 19 July.<ref name="ICTY-249-254">ICTY, [https://www.un.org/icty/krstic/TrialC1/judgement/index.htm ''Prosecutor vs Krstic, Judgement''] {{webarchive |url=https://web.archive.org/web/20080517075111/http://www.un.org/icty/krstic/TrialC1/judgement/index.htm |date=17 May 2008 }}, II, B, 5 (i) [https://www.un.org/icty/krstic/TrialC1/judgement/krs-tj010802e-1.htm#IIB5i "Kozluk"] {{webarchive |url=https://web.archive.org/web/20090508011123/http://www.un.org/icty/krstic/TrialC1/judgement/krs-tj010802e-1.htm#IIB5i |date=8 May 2009 }}, par. 249-254.</ref>

Among Bosnian refugees in Germany, there were rumours of executions in Kozluk, during which 500 or so prisoners were forced to sing Serbian songs as they were being transported to the execution site. Though no survivors have come forward, investigations in 1999 led to the discovery of a mass grave near Kozluk.<ref name="ICTY-249-254" /> This proved to be the location of execution as well, and lay alongside the Drina accessible only by driving through the barracks occupied by the [[Drina Wolves]], a police unit of Republika Srpska. The grave was not dug specifically for the purpose: it had previously been a quarry and landfill site. Investigators found many shards of glass which the nearby 'Vitinka' bottling plant had dumped there. This facilitated the process of establishing links with the secondary graves along Čančari Road.<ref name="Manning-A11">{{cite web|author=Dean Manning|url=http://abunodisceomnes.wellcomecollection.org/wp-content/uploads/2015/01/Manning-D.-16-May-2000-Srebrenica-investigation-Summary-of-forensic-evidence-Execution-points-and-mass-graves-ICTY-Investigator.pdf#page=69|title=Srebrenica Investigation: Summary of Forensic Evidence – Execution Points and Mass Graves|date=16 May 2000|at=Annex A, 11 "Kozluk – (Primary Grave)", pp. 42–44.|access-date=25 April 2017|archive-url=https://web.archive.org/web/20170425192611/http://abunodisceomnes.wellcomecollection.org/wp-content/uploads/2015/01/Manning-D.-16-May-2000-Srebrenica-investigation-Summary-of-forensic-evidence-Execution-points-and-mass-graves-ICTY-Investigator.pdf#page=69|archive-date=25 April 2017|url-status=dead|df=dmy-all}}</ref> The grave at Kozluk had been partly cleared before 27 September 1995, but no fewer than 340 bodies were found there.<ref name="ICTY-249-254" /> In 237 cases, it was clear they had died as the result of rifle fire: 83 by a single shot to the head, 76 by one shot through the torso region, 72 by multiple bullet wounds, five by wounds to the legs and one by bullet wounds to the arm. Their ages were between 8 and 85. Some had been physically disabled, occasionally as the result of amputation. Many had been tied and bound using strips of clothing or nylon thread.<ref name="Manning-A11"/>

Along the Čančari Road are twelve known mass graves, of which only two—Čančari Road 3 and 12—have been investigated in detail ({{as of|2000|lc=y}}).<ref name=Manning>{{cite web|author=Dean Manning|url=http://abunodisceomnes.wellcomecollection.org/wp-content/uploads/2015/01/Manning-D.-16-May-2000-Srebrenica-investigation-Summary-of-forensic-evidence-Execution-points-and-mass-graves-ICTY-Investigator.pdf|title=Srebrenica Investigation: Summary of Forensic Evidence – Execution Points and Mass Graves|date=16 May 2000|access-date=25 April 2017|archive-url=https://web.archive.org/web/20170425192611/http://abunodisceomnes.wellcomecollection.org/wp-content/uploads/2015/01/Manning-D.-16-May-2000-Srebrenica-investigation-Summary-of-forensic-evidence-Execution-points-and-mass-graves-ICTY-Investigator.pdf|archive-date=25 April 2017|url-status=dead|df=dmy-all}}</ref> Čančari Road 3 is known to have been a secondary grave linked to Kozluk, as shown by the glass fragments and labels from the Vitinka factory.<ref name="ICTY-249-254" /> The remains of 158 victims were found here, of which 35 bodies were more or less intact and indicated most had been killed by gunfire.<ref name="Manning-p9">{{cite web|author=Dean Manning|url=http://abunodisceomnes.wellcomecollection.org/wp-content/uploads/2015/01/Manning-D.-16-May-2000-Srebrenica-investigation-Summary-of-forensic-evidence-Execution-points-and-mass-graves-ICTY-Investigator.pdf#page=72|title=Srebrenica Investigation: Summary of Forensic Evidence – Execution Points and Mass Graves|date=16 May 2000|at=Annex A, 12 "Čančari Road 3 – (Secondary Grave), pp. 45–47.|access-date=25 April 2017|archive-url=https://web.archive.org/web/20170425192611/http://abunodisceomnes.wellcomecollection.org/wp-content/uploads/2015/01/Manning-D.-16-May-2000-Srebrenica-investigation-Summary-of-forensic-evidence-Execution-points-and-mass-graves-ICTY-Investigator.pdf#page=72|archive-date=25 April 2017|url-status=dead|df=dmy-all}}</ref>

==== 13–18 July: Bratunac-Konjević Polje road ====
On 13 July, near Konjević Polje, Serb soldiers summarily executed hundreds of Bosniaks, including women and children.<ref name="Karadzic Mladic Indictment">{{cite web|url=http://www.icty.org/x/cases/karadzic/ind/en/kar-ii951116e.pdf|title=Initial Indictment against Radovan Karadzic and Ratko Mladic|date=14 November 1995|publisher=ICTY|access-date=22 April 2017}}</ref> The men found attempting to escape by the Bratunac-Konjević Polje road were told the Geneva Convention would be observed if they gave themselves up.<ref name="ICTY-63">ICTY, [https://www.un.org/icty/krstic/TrialC1/judgement/index.htm ''Prosecutor vs Krstic, Judgement''] {{webarchive |url=https://web.archive.org/web/20080517075111/http://www.un.org/icty/krstic/TrialC1/judgement/index.htm |date=17 May 2008 }}, II, A, 7 (b) [https://www.un.org/icty/krstic/TrialC1/judgement/krs-tj010802e-1.htm#IIA7b "The Column of Bosnian Muslim Men"] {{webarchive |url=https://web.archive.org/web/20080105091809/http://www.un.org/icty/krstic/TrialC1/judgement/krs-tj010802e-1.htm#IIA7b |date=5 January 2008 }}, par. 63.</ref> In Bratunac, men were told there were Serbian personnel standing by to escort them to [[Zagreb]] for a prisoner exchange. The visible presence of UN uniforms and vehicles, stolen from Dutchbat, were intended to contribute to the feeling of reassurance. On 17 to 18 July, Serb soldiers captured about 150–200 Bosnians in the vicinity of Konjevic Polje and summarily executed about one-half.<ref name="Karadzic Mladic Indictment"/>

==== 18–19 July: Nezuk–Baljkovica frontline ====
After the closure of the corridor at Baljkovica, groups of stragglers nevertheless attempted to escape into Bosnian territory. Most were captured by VRS troops in the Nezuk–Baljkovica area and killed on the spot. In the vicinity of Nezuk, about 20 small groups surrendered to Bosnian Serb military forces. After the men surrendered, soldiers ordered them to line up and summarily executed them.<ref name="Popovic 2006"/>

On 19 July, for example, a group of approximately 11 men was killed at Nezuk itself by units of the 16th [[Republic of Serbian Krajina|Krajina]] Brigade, then operating under the direct command of the Zvornik Brigade. Reports reveal a further 13 men, all ARBiH soldiers, were killed at Nezuk on 19 July.<ref name="ICTY-IIB5j">ICTY, [https://www.un.org/icty/krstic/TrialC1/judgement/index.htm ''Prosecutor vs Krstic, Judgement''] {{webarchive |url=https://web.archive.org/web/20080517075111/http://www.un.org/icty/krstic/TrialC1/judgement/index.htm |date=17 May 2008 }}, II, B, 5 (j) [https://www.un.org/icty/krstic/TrialC1/judgement/krs-tj010802e-1.htm#IIB5j "Smaller Scale Executions following the Mass Executions"] {{webarchive |url=https://web.archive.org/web/20090508011123/http://www.un.org/icty/krstic/TrialC1/judgement/krs-tj010802e-1.htm#IIB5j |date=8 May 2009 }}.</ref> The report of the march to Tuzla includes the account of an ARBiH soldier who witnessed executions carried out by police. He survived because 30 ARBiH soldiers were needed for an exchange of prisoners following the ARBiH's capture of a VRS officer at Baljkovica. The soldier was exchanged in late 1995; at that time, there were still 229 men from Srebrenica in the Batkovici [[prisoner of war camp]], including two who had been taken prisoner in 1994.{{Citation needed|date=August 2008}}

RS Ministry of the Interior forces searching the terrain from Kamenica as far as Snagovo killed eight Bosniaks.<ref name="trial-ch.org" /> Around 200 Muslims armed with [[automatic rifle|automatic]] and [[hunting rifle]]s were reported to be hiding near the old road near Snagovo.<ref name="trial-ch.org" /> During the morning, about 50 Bosniaks attacked the Zvornik Brigade line in the area of Pandurica, attempting to break through to Bosnian government territory.<ref name="trial-ch.org" /> The Zvornik Public Security Centre planned to surround and destroy these two groups the following day using all available forces.<ref name="trial-ch.org" />

==== 20–22 July: Meces area ====
According to ICTY indictments of Karadžić and Mladić, on 20 to 21 July near Meces, VRS personnel, using megaphones, urged Bosniak men who had fled Srebrenica to surrender and assured them they would be safe. Approximately 350 men responded to these entreaties and surrendered. The soldiers then took approximately 150, instructed them to dig their graves and executed them.<ref>{{cite web|url=https://www.un.org/icty/indictment/english/kar-ii951116e.htm |title=paras 20.6 and 20.7 |publisher=United Nations |date=5 March 2007 |access-date=26 May 2011 |url-status=dead |archive-url=https://web.archive.org/web/20090829194758/http://www.un.org/icty/indictment/english/kar-ii951116e.htm |archive-date=29 August 2009 }}</ref>

=== After the massacre ===
[[File:ICMP-PIP.jpg|thumb|ICMP's [[Podrinje Identification Project]] (PIP) was formed to deal with the victims of the Srebrenica massacre. PIP includes a facility for storing, processing, and handling exhumed remains. Much are only fragments or commingled body fragments since they were recovered from secondary [[mass grave]]s. The photo depicts one section of the refrigerated [[mortuary]].]]

During the days following the massacre, US [[Reconnaissance aircraft|spy planes]] overflew Srebrenica and took photos showing the ground in vast areas around the town had been removed, a sign of mass burials.

On 22 July, the commanding officer of the [[Zvornik Brigade]], Lieutenant Colonel [[Vinko Pandurević]], requested the Drina Corps set up a committee to oversee the exchange of prisoners. He asked for instructions on where the prisoners of war his unit had already captured should be taken and to whom they should be handed over. Approximately 50 wounded captives were taken to the Bratunac hospital. Another group was taken to the Batkovići camp, and these were mostly exchanged later.<ref name="trial-ch.org" /> On 25 July, the Zvornik Brigade captured 25 more ARBiH soldiers who were taken directly to the camp at Batkovići, as were 34 ARBiH men captured the following day. Zvornik Brigade reports up until 31 July continue to describe the search for refugees and the capture of small groups of Bosniaks.<ref>{{cite web |title=In the Trial Chamber |url=http://www.icty.org/x/cases/krstic/tjug/en/krs-tj010802e-1.htm |website=International Criminal Tribunal for the former Yugoslavia |access-date=3 July 2019}}</ref>

Several Bosniaks managed to cross over the River Drina into Serbia at [[Ljubovija]] and Bajina Bašta. 38 were returned to RS. Some were taken to the Batkovići camp, where they were exchanged. The fate of the majority has not been established.<ref name="trial-ch.org" /> Some attempting to cross the Drina drowned.<ref name="trial-ch.org" />

By 17 July, 201 Bosniak soldiers had arrived in Žepa, exhausted and many with light wounds.<ref name="trial-ch.org" /> By 28 July another 500 had arrived in Žepa from Srebrenica.<ref name="trial-ch.org" /><ref>{{Cite web|url=https://www.icty.org/x/cases/popovic/trans/en/070206ED.htm|title=Esma Palic evidence to the Popovic et al trial, 6 February 2006, ICTY transcript |page=6939|publisher=International Criminal Tribunal for the former Yugoslavia}}</ref> After 19 July, small Bosniak groups were hiding in the woods for days and months, trying to reach Tuzla.<ref name="trial-ch.org" /> Numerous refugees found themselves cut off in the area around Mount Udrc.<ref name='A1'>{{cite news |last1=Milovanovic |first1=Selma |title=Srebrenica: A town still divided |url=https://www.aljazeera.com/indepth/features/2012/07/2012710133138811312.html |access-date=7 July 2019 |agency=aljazeera |publisher=aljazeera.com |date=12 July 2012}}</ref><ref name='b1'>{{cite book |last1=Nigel |first1=Cawthorne |last2=Cawthorne |first2=Nigel |title=The World's Ten Most Evil Men – From Twisted Dictators to Child Killers |date=4 May 2009 |publisher=John Blake |isbn=9781782191582 |pages=300 |url=https://books.google.com/books?id=oYg7AAAAQBAJ&q=the+bodies+from+around+Srebrenica,+%C5%BDepa,+Kamenica+and+Snagovo+collected+and+burned&pg=PT17 |access-date=7 July 2019}}</ref> They did not know what to do next or where to go; they managed to stay alive by eating vegetables and snails.<ref name='A1'/><ref name='b1'/> The MT Udrc had become a place for ambushing marchers, and the Bosnian Serbs swept through and, according to one survivor, killed many people there.<ref name='A1'/><ref name='b1'/>

Meanwhile, the VRS had commenced the process of clearing the bodies from around Srebrenica, Žepa, Kamenica and Snagovo. Work parties and municipal services were deployed to help.<ref name='b1'/><ref name='b2'>{{cite book |last1=Basílio Rissi |first1=Bruno |last2=de Lima |first2=Débora Hanna F. |last3=Pereira Campbell |first3=Mila |last4=Bennet Fagundes |first4=Raquel Fanny |last5=Santana Fernandes |first5=Wladimir |title=Long-lasting peaces: Overcoming the war-peace hiatus for a sustainable future |date=1 January 2015 |publisher=Art Letras |isbn=9788561326678 |pages=492 |url=https://books.google.com/books?id=-ETKCgAAQBAJ&q=%C5%BDepa,+Kamenica+and+Snagovo |access-date=8 July 2019}}</ref> In Srebrenica, the refuse that had littered the streets since the departure of the people was collected and burnt, the town disinfected and deloused.<ref name='b1'/><ref name='b2'/>

==== Wanderers ====
Many people in the part of the column which had not succeeded in passing Kamenica, did not wish to give themselves up and decided to turn back towards Žepa.<ref name='hrw'>{{cite web |title=The March of Death |url=https://www.hrw.org/legacy/reports98/bosniacw/Bosni98o-03.htm |publisher=Human Rights Watch |access-date=3 July 2019}}</ref> Others remained where they were, splitting up into smaller groups of no more than ten.<ref name="b3">{{cite book |last1=Leydesdorff |first1=Selma |title=Surviving the Bosnian Genocide: The Women of Srebrenica Speak |title-link=Surviving the Bosnian Genocide |publisher=Indiana University Press |year=2011 |isbn=9780253356697 |edition=illustrated |pages=242}}</ref> Some wandered around for months, either alone or groups of two, four or six men.<ref name='b3'/> Once Žepa had succumbed to the Serb pressure, they had to move on once more, either trying to reach Tuzla or crossing the River Drina into Serbia.<ref name='b4'>{{cite book |last1=Stewart |first1=Bob |title=Leadership Under Pressure: Tactics from the Front Line |publisher=Kogan Page Publishers |isbn=9780749458553 |pages=192 |url=https://books.google.com/books?id=_qMLy2cyTVIC&q=Some+of+the+Bosniak+trying+to+reach+Tuzla+or+crossing+the+River+Drina |access-date=9 July 2019|date=3 October 2009 }}</ref>

=====Zvornik 7=====

The most famous group of seven men wandered about in occupied territory for the entire winter. On 10 May 1996, after nine months on the run and over six months after the end of the war, they were discovered in a quarry by American [[IFOR]] soldiers. They immediately turned over to the patrol; they were searched and their weapons were confiscated. The men said they had been in hiding near Srebrenica since its fall. They did not look like soldiers and the Americans decided this was a matter for the police.<ref>{{cite web|author=Preston V. McMurry III |url=http://www.prestonm.com/military/bosnia/articles/zvornik7.html |title=The Officer Who First Took Charge of the Zvornik Seven |publisher=Prestonm.com |access-date=26 May 2011 |url-status=dead |archive-url=https://web.archive.org/web/20110715112511/http://www.prestonm.com/military/bosnia/articles/zvornik7.html |archive-date=15 July 2011 }}</ref> The operations officer of the American unit ordered that a Serb patrol should be escorted into the quarry whereupon the men would be handed over to the Serbs.

The prisoners said they were initially tortured after the transfer, but later treated relatively well. In April 1997 the local court in [[Republika Srpska]] convicted the group, known as the [[Zvornik 7]], for illegal possession of firearms and three of them for the murder of four Serbian woodsmen. When announcing the verdict the presenter of the TV of Republika Srpska described them as "the group of Muslim terrorists from Srebrenica who last year massacred Serb civilians".<ref>[http://archiv.medienhilfe.ch/Projekte/BiH/Monitoring/IWPR/1996/mon_v2_14.htm "Monitoring Report – Media trial of the Zvornik Seven"]. {{webarchive |url=https://web.archive.org/web/20071006140608/http://archiv.medienhilfe.ch/Projekte/BiH/Monitoring/IWPR/1996/mon_v2_14.htm |date=6 October 2007 }} {{lang|de|Medienhilfe Ex-Jugoslawien}}.</ref> The trial was condemned by the international community as "a flagrant miscarriage of justice",<ref>{{Cite web|url=http://ww1.unhchr.ch/html/menu2/5/ex_yug/yug_pr10.htm|archive-url=https://web.archive.org/web/20090627152318/http://www.unhchr.ch/html/menu2/5/ex_yug/yug_pr10.htm|url-status=dead|title=Resources and Information|publisher=United Nations High Commissioner for Human Rights|archive-date=27 June 2009}}</ref><ref>{{cite press release|url=http://www.ohr.int/?p=55215 |agency=Office of the High Representative |title=Zvornik 7 Verdict|date=15 December 1998|access-date=28 April 2017 |url-status=live |archive-url=https://web.archive.org/web/20170428094434/http://www.ohr.int/?p=55215 |archive-date=28 April 2017|author1=OHR |author2=OSCE |author3=UNMIBH |author4=UNHCHR}}</ref> and the conviction quashed for 'procedural reasons' following international pressure. In 1999, the three remaining defendants in the Zvornik 7 case were swapped for three Serbs serving 15 years each in a Bosnian prison.

==== Reburials in the secondary mass graves ====
[[File:HuseinovicSadik.jpg|upright=.8|thumb|Grave of a 13-year-old]]
From August to October 1995, there was organised effort to remove the bodies from primary gravesites and transport them to secondary and tertiary gravesites.<ref>{{cite web|work=ICTY|title=Prosecutor v. Blagojević & Jokić Trial Chamber Judgment|id=Case No. IT-02-60|at=Section II G. Reburial Operation 1 August – 1 November 1995|url=http://www.icty.org/x/cases/blagojevic_jokic/tjug/en/bla-050117e.pdf#page=150|date=17 January 2005|access-date=5 May 2017}}</ref> In the ICTY court case ''Prosecutor v. Blagojević and Jokić'', the trial chamber found that this reburial effort was an attempt to conceal evidence of the mass murders.<ref name="B&Jpara382+">{{cite web|work=ICTY|title=Prosecutor v. Blagojević & Jokić Trial Chamber Judgment|id=Case No. IT-02-60|at=paras. 382–383|url=http://www.icty.org/x/cases/blagojevic_jokic/tjug/en/bla-050117e.pdf#page=151|date=17 January 2005|access-date=5 May 2017}}</ref> The trial chamber found that the cover-up operation was ordered by the [[VRS Main Staff]] and carried out by members of the [[Bratunac]] and Zvornik Brigades.<ref name="B&Jpara382+"/>

The cover-up had a direct impact on the recovery and identification of the remains. The removal and reburial of the bodies caused them to become dismembered and co-mingled, making it difficult for forensic investigators to positively identify the remains.<ref name="Durnford-BBB">{{cite news|last=Durnford|first=Laura|url=http://www.radionetherlands.nl/features/science/050711rf|title=Bridges of Bone and Blood|date=11 July 2005|work=Radio Netherlands Worldwide|archive-url=https://web.archive.org/web/20070206153153/http://www.radionetherlands.nl/features/science/050711rf |archive-date=6 February 2007}}</ref> In one case, the remains of a single person were found in two locations, 30 km apart.<ref>{{cite news |title=Finding the Bodies To Fill Bosnia's Graves |newspaper=The Scotsman |url=http://news.scotsman.com/international.cfm?id=398822007 |url-status=dead|archive-url=https://web.archive.org/web/20160108061745/http://www.scotsman.com/news/world/finding-the-bodies-to-fill-the-bosnia-graves-1-690822|archive-date=2016-01-08}}</ref>{{verification failed|reason=Article does not contain this detail. Previously cited a "comment" by Boys; if that means a reader comment, that would be unverified|date=October 2024}} In addition to the ligatures and blindfolds found, the effort to hide the bodies has been seen as evidence of the organised nature of the massacres and the non-combatant status of the victims.<ref name="Durnford-BBB" /><ref>{{Cite news|url=http://news.bbc.co.uk/2/hi/science/nature/3640788.stm|title=Pollen helps war crime forensics|date=9 September 2004|work=BBC News}}</ref>

=== Greek Volunteers controversy ===
{{Main|Greek Volunteer Guard}}
10 Greek volunteers fought alongside the Serbs in the fall of Srebrenica.<ref>{{Cite web|url=http://www.hri.org/news/balkans/omri/1995/95-07-14.omri.html#07|title=OMRI Daily Digest II, No. 136, 14 July 1995|website=www.hri.org}}</ref><ref name="auto4">{{Cite web|url=https://balkaninsight.com/2023/07/10/enjoying-impunity-greek-pro-serb-fighters-still-brag-about-srebrenica/|title=Enjoying Impunity, Greek Pro-Serb Fighters Still Brag about Srebrenica |website=Balkan Insight|date=10 July 2023 }}</ref> They were members of the Greek Volunteer Guard, a contingent of paramilitaries requested by Mladić, as an integral part of the Drina Corps. The volunteers were motivated to support their "[[Eastern Orthodox Church|Orthodox]] brothers" in battle.<ref name="tiscali.co.uk">{{cite news |url=http://www.tiscali.co.uk/news/newswire.php/news/reuters/2005/06/27/world/greecestartsprobeintosrebrenicamassacre.html&template=/news/templates/newswire/news_story_reuters.html |title=Greece starts probe into Srebrenica massacre |last=Grohmann |first=Karolos |agency=Reuters |date=27 June 2006 |access-date=26 August 2008 |url-status=dead |archive-url=https://web.archive.org/web/20090104002056/http://www.tiscali.co.uk/news/newswire.php/news/reuters/2005/06/27/world/greecestartsprobeintosrebrenicamassacre.html%26template%3D/news/templates/newswire/news_story_reuters.html |archive-date=4 January 2009 |df=dmy-all }}</ref> They raised the Greek flag at Srebrenica, at Mladić's request, to honour "the brave Greeks fighting on our side"<ref>[[#refMichas2002|Michas 2002]], p. 22.</ref> and Karadžić decorated four.<ref>[[#refMichas2002|Michas 2002]], pp. 17–41.</ref><ref name="smith">{{cite news |url=http://observer.guardian.co.uk/milosevic/story/0,,868869,00.html |title=Greece faces shame of role in Serb massacre |last=Smith |first=Helena |date=5 January 2003 |work=The Guardian |location=UK |access-date=20 April 2010}}</ref><ref name="bosnia.org.uk-Koknar-2003-Russian-mercenaries">{{cite web |first=Ali M. |last=Koknar |title=The Kontraktniki: Russian mercenaries at war in the Balkans |url=http://www.bosnia.org.uk/news/news_body.cfm?newsid=1766 |publisher=Bosnian Institute |access-date=14 February 2021 |archive-url=https://web.archive.org/web/20160304083455/http://www.bosnia.org.uk/news/news_body.cfm?newsid=1766 |archive-date=4 March 2016 |location=London |language=en |date=14 July 2003}}</ref><ref name="rightswebpages">{{cite news|url=http://cm.greekhelsinki.gr/index.php?sec=194&cid=848|title=Greece's Balkan Closets|work=The Wall Street Journal Europe|publisher=The BALKAN Human Rights Web Pages|date=4 July 2005|archive-url=https://web.archive.org/web/20060304074611/http://cm.greekhelsinki.gr/index.php?sec=194&cid=881|archive-date=4 March 2006|url-status=dead|df=dmy-all}}</ref> In 2005, Greek deputy Andrianopoulos called for an investigation,<ref name="rightswebpages" /> Justice Minister [[Anastasios Papaligouras|Papaligouras]] commissioned an inquiry<ref>{{cite news |url=https://www.theguardian.com/world/2005/aug/01/warcrimes.comment |title=Helena Smith@Athens |last=Smith |first=Helena |date=1 August 2005 |work=The Guardian |location=UK |access-date=21 April 2010}}</ref> and in 2011, a judge said there was insufficient evidence to proceed.<ref name="auto4"/> In 2009, Stavros Vitalis announced the volunteers were suing [[Takis Michas]] for libel over allegations in his book ''Unholy Alliance'', which described Greece's support for the Serbs during the war. Insisting the volunteers had simply taken part in the "re-occupation" of Srebrenica, Vitalis was present with Serb officers in "all military operations".<ref name="balkaninsight.com">{{cite news|title=Greece: Suit Against Journalist For Srebrenica Claims to Go Forward |url=http://www.balkaninsight.com/en/article/greece-suit-against-journalist-for-srebrenica-claims-to-go-forward |newspaper=Balkan Investigative Reporting Network |date=21 June 2010 |access-date=19 April 2017}}</ref><ref>[http://www.bosniak.org/interview-greek-journalist-sued-for-writing-about-the-presence-of-greek-paramilitaries-in-bosnia/ Congress of North American Bosniaks, Interview, 5 August 2009] {{Webarchive|url=https://web.archive.org/web/20100726050535/http://www.bosniak.org/interview-greek-journalist-sued-for-writing-about-the-presence-of-greek-paramilitaries-in-bosnia/|date=26 July 2010}}. Retrieved 8 April 2010.</ref><ref>{{cite AV media |publisher=YouTube |url=https://www.youtube.com/watch?v=WGuHpzYPpMk | archive-url=https://ghostarchive.org/varchive/youtube/20211030/WGuHpzYPpMk| archive-date=2021-10-30|title=The Greek role in Bosnia's war |date=10 July 2010 |access-date=26 May 2011}}{{cbignore}}</ref>

== Post-war developments ==
=== 1995–2000: Indictments and UN Secretary-General's report ===
In November 1995, Karadžić and Mladić were indicted by the ICTY for their alleged direct responsibility for the war crimes committed against the Bosnian Muslim population of Srebrenica.<ref name="UN report" /> In 1999, UN Secretary-General Kofi Annan submitted his report on the Fall of Srebrenica. He acknowledged the international community as a whole had to accept its share of responsibility, for its response to the ethnic cleansing that culminated in the murder of 7,000 unarmed civilians from the town designated by the Security Council as a "safe area".<ref name="UN report" /><ref name=10thAnniversary>{{cite web|url=https://www.un.org/sg/en/content/sg/statement/2005-07-11/secretary-generals-message-ceremony-marking-10th-anniversary|title=Secretary-General's message to ceremony marking the 10th anniversary of the Srebrenica massacre (delivered by Mark Malloch Brown, Chef de Cabinet)|date=11 July 2005|publisher=United Nations|access-date=26 April 2017}}</ref>

=== 2002: Dutch government report ===
The failure of [[Dutchbat]] to protect the enclave became a [[national trauma]] in the Netherlands and led to long-running discussions.<ref name="vandeBilt">{{cite journal|url=http://www.bradford.ac.uk/social-sciences/peace-conflict-and-development/issue-21/Srebrenica---a-dutch-national-trauma.pdf#page=27|last=van de Bildt|first=Joyce|title=Srebrenica: A Dutch national trauma|journal=Journal of Peace, Conflict & Development|issue=21|date=March 2015|issn=1742-0601|access-date=30 December 2017|archive-date=31 December 2017|archive-url=https://web.archive.org/web/20171231155510/https://www.bradford.ac.uk/social-sciences/peace-conflict-and-development/issue-21/Srebrenica---a-dutch-national-trauma.pdf#page=27|url-status=dead}}</ref> In 1996, the Dutch government asked the [[NIOD Institute for War, Holocaust and Genocide Studies]] to research the events. The report was published in 2002—''Srebrenica: a 'safe' area''.<ref name="NIOD">J. C. H. Blom et al. (2002) Prologue [http://www.cnj.it/documentazione/Srebrenica/NIOD/NIOD%20Prologue.pdf NIOD Report: Srebrenica. Reconstruction, background, consequences and analyses of the fall of a Safe Area] </ref> It concluded the Dutchbat mission was not well considered and well-nigh impossible. The report is often cited, however, the [[Institute for War and Peace Reporting]] labelled it "controversial", as "the sheer abundance of information makes it possible for anyone to pluck from it whatever they need to make their point". One author claimed some sources were "unreliable", and only used to support another author's argument.<ref>{{cite news |author=Karen Meirik |url=http://iwpr.net/report-news/controversial-srebrenica-report-back-table |title=Controversial Srebrenica Report Back on Table |work=Institute for War & Peace Reporting |location=UK |access-date=20 November 2013 |date=9 November 2005 |archive-date=1 February 2014 |archive-url=https://web.archive.org/web/20140201174230/http://iwpr.net/report-news/controversial-srebrenica-report-back-table |url-status=dead }}</ref> Responding to the report, the Dutch government accepted partial political responsibility for the circumstances in which the massacre happened<ref>{{cite news |url=https://news.google.com/newspapers?nid=2710&dat=20020417&id=52g5AAAAIBAJ&pg=1520,4715887 |title=Dutch government resigns over Srebrenica report criticizing role in massacre |newspaper=The Sentinel-Review |location=Oxford County, Ontario |date=17 April 2002 |page=16 |access-date=5 February 2013 |via=Google News}}</ref> and the [[Second Kok cabinet]] resigned.<ref>[[Parlementair Documentatie Centrum]] [Parliamentary Documentation Centre] of [[Leiden University]], [http://www.parlement.com/9291000/modulesf/g87kjv8v "Parlementaire enquête Srebrenica (2002–2003)"] (in Dutch). Retrieved 17 February 2007.</ref><ref>{{Cite news|url=http://news.bbc.co.uk/2/hi/europe/1933144.stm|title=Dutch Government quits over Srebrenica|date=16 April 2002|work=BBC News}}</ref>

===2002: First Republika Srpska report===
In September 2002, the Republika Srpska Office of Relations with the ICTY issued the "[[Report about Case Srebrenica]]". The document, by Darko Trifunović, was endorsed by leading Bosnian Serb politicians. It concluded that 1,800 Bosnian Muslim soldiers died during fighting and a 100 more from exhaustion. "The number of Muslim soldiers killed by Bosnian Serbs out of personal revenge or lack of knowledge of international law is probably about 100 ... It is important to uncover the names of the perpetrators to accurately and unequivocally establish whether or not these were isolated instances." The report examined the [[mass graves]], claiming they were made for hygiene reasons, questioning the legitimacy of the missing person lists and undermining a key witness' mental health and military history.<ref>{{Cite web|url=http://www.slobodan-milosevic.org/documents/srebrenica.pdf|title=Report about Case Srebrenica – Banja Luka, 2002}}</ref> The [[International Crisis Group]] and UN condemned the manipulation of their statements.<ref>{{Cite magazine|url=http://www.time.com/time/health/article/0,8599,349957,00.html|title=Imaginary Massacres? |magazine=Time|date=11 May 2008|access-date=21 July 2021|archive-date=11 May 2008|archive-url=https://web.archive.org/web/20080511010329/http://www.time.com/time/health/article/0,8599,349957,00.html|url-status=dead}}</ref>

=== 2003: Srebrenica Genocide Memorial ===

[[File:Srebrenica massacre memorial wall of names 2009 2.jpg|thumb|Wall of names at the [[Srebrenica Genocide Memorial]]]]
In September 2003, former US President [[Bill Clinton]] officially opened the [[Srebrenica Genocide Memorial]] to honour the victims of the genocide. The total cost was around $6 million. "We must pay tribute to the innocent lives, many of them children who were snuffed out in what must be called genocidal madness", Clinton said.<ref>{{cite news|url=http://news.bbc.co.uk/2/hi/europe/3124642.stm |title=Clinton unveils Bosnia memorial |date=20 September 2003 |work=BBC News |access-date=20 April 2010}}</ref><ref>{{cite news|last=Wilkinson|first=Tracy|title=Clinton Helps Bosnians Mourn Their Men |url=https://www.latimes.com/archives/la-xpm-2003-sep-21-fg-memorial21-story.html|newspaper=Los Angeles Times|date=21 September 2003}}</ref>

=== 2004: Second Republika Srpska report and apology ===
In March 2003, the [[Human Rights Chamber for Bosnia and Herzegovina]] issued a decision which ordered the [[Republika Srpska]] (RS) to conduct a full investigation into the Srebrenica events, and disclose the results by September.<ref>{{cite web|url=http://www.hrc.ba/DATABASE/decisions/CH01-8365%20Selimovic%20Admissibility%20and%20Merits%20E.pdf#page=54|page=50|title=Decision on Admissibility and Merits – The Srebrenica Cases (49 applications) against The Republika Srpska|date=7 March 2003|work=Human Rights Chamber for Bosnia and Herzegovina|access-date=2 May 2017|archive-date=12 November 2017|archive-url=https://web.archive.org/web/20171112021635/http://www.hrc.ba/DATABASE/decisions/CH01-8365%20Selimovic%20Admissibility%20and%20Merits%20E.pdf#page=54|url-status=dead}}</ref> The Chamber had no coercive power to implement the decision, especially as dissolved in late 2003.<ref name="Investigating Srebrenica">{{Citation|last1=Picard|first1=Michèle|last2=Zinbo|first2=Asta|chapter=The Long Road to Admission: The Report of the Government of the Republika Srpska|pages=137–140|editor-last1=Delpla|editor-first1=Isabelle|editor-last2=Bougarel|editor-first2=Xavier|editor-last3=Fournel|editor-first3=Jean-Louis|title=Investigating Srebrenica: Institutions, Facts, Responsibilities|publisher=Berghahn Books|location=New York|date=2012|isbn=978-0-85745-472-0}}</ref> The RS then published reports, in September 2003, which the Human Rights Chamber concluded did not fulfil the RS' obligations.<ref name="Investigating Srebrenica"/> In October 2003, The [[High Representative in Bosnia and Herzegovina|High Representative]], [[Paddy Ashdown]], lamented that "getting the truth from the [Bosnian Serb] government is like extracting rotten teeth". He did, however, welcome a recommendation to form an independent commission to investigate Srebrenica and issue a report within six months.<ref name=RottenTeeth>{{cite web|url=http://www.tol.org/client/article/10827-pulling-rotten-teeth.html|work=[[Transitions Online]]|first=Anes|last=Alic|title=Pulling Rotten Teeth|date=20 October 2003|access-date=2 May 2017}}</ref>

The Srebrenica commission, officially titled the ''Commission for Investigation of the Events in and around Srebrenica between 10 and 19 July 1995'', was established in December 2003, and submitted its final report on 4 June 2004,<ref name="trial-ch.org" /> and then an addendum<ref>{{cite web|url=http://balkanwitness.glypx.com/srebr_final_e.pdf|date=15 October 2004|title=Addendum to the Report of 11 June 2004 on the Events in and around Srebrenica between 10 and 19 July 1995|archive-url=https://web.archive.org/web/20160429052959/http://balkanwitness.glypx.com/srebr_final_e.pdf|archive-date=29 April 2016|url-status=live}}</ref> in October 2004 after delayed information was supplied.<ref name="Investigating Srebrenica"/><ref name=":0">{{cite web|url=http://www.ohr.int/?p=44915|publisher=Office of the High Representative in Bosnia and Herzegovina |access-date=1 May 2017|title=Srebrenica Commission Report to be assessed by judges before public comments|date=18 October 2004}}</ref> The report acknowledged men and boys were killed by Bosnian Serbs, citing a provisional figure of 7,800.<ref>{{cite book |url=https://books.google.com/books?id=vaZAwBHxdUoC&pg=PA192|title=Peace and Justice |first=Rachel |last=Kerr|year=2007|publisher=Polity |access-date=5 August 2013|isbn=9780745634227 |page=192}}</ref> Because of "limited time" and to "maximize resources", the commission "accepted the historical background and the facts stated in the second-instance judgment 'Prosecutor vs. [[Radislav Krstić]]', when the ICTY convicted him for 'assisting and supporting genocide' in Srebrenica".<ref name="trial-ch.org" />

The findings remain disputed by Serb nationalists, who claim it was pressured by the High Representative, given the earlier RS government report which exonerated the Serbs was dismissed. Nevertheless, [[Dragan Čavić]], the president of Republika Srpska, acknowledged in a televised address that Serb forces killed several thousand civilians in violation of [[international law]], and asserted that Srebrenica was a dark chapter in Serb history.<ref>{{cite web |url=http://www.vreme.com/cms/view.php?id=384060 |title=Otvaranje najmračnije stranice |publisher=[[Vreme]] |author=Tanja Topić |date=1 July 2004 |language=sr}}</ref> On 10 November 2004, the government of Republika Srpska issued an official apology. The statement came after a government review of the report. "The report makes it clear that enormous crimes were committed in the area of Srebrenica in July 1995. The Bosnian Serb Government shares the pain of the families of the Srebrenica victims, is truly sorry and apologises for the tragedy", the Bosnian Serb government said.<ref name=":1">{{cite web |url=http://www.bosnia.org.uk/bosrep/report_format.cfm?articleid=1147&reportid=166 |title=Bosnian Serbs issue apology for massacre |agency=Associated Press |publisher=Bosnia and Herzegovina UK Network |date=11 November 2004 |access-date=26 May 2011 |archive-date=7 June 2011 |archive-url=https://web.archive.org/web/20110607144917/http://www.bosnia.org.uk/bosrep/report_format.cfm?articleid=1147&reportid=166 |url-status=dead }}</ref>

==== Republika Srpska Srebrenica Working Group ====
After a request by Ashdown, the RS established a working group to implement the recommendations of the report by the Srebrenica Commission. The group was to analyze the documentation in the report's confidential annexes and identify all possible perpetrators who were officials in RS institutions.<ref name="RS Working Group">{{Citation|last1=Picard|first1=Michèle|last2=Zinbo|first2=Asta|chapter=The Long Road to Admission: The Report of the Government of the Republika Srpska|page=141|editor-last1=Delpla|editor-first1=Isabelle|editor-last2=Bougarel|editor-first2=Xavier|editor-last3=Fournel|editor-first3=Jean-Louis|title=Investigating Srebrenica: Institutions, Facts, Responsibilities|publisher=Berghahn Books|location=New York|date=2012|isbn=978-0-85745-472-0}}</ref> A report on 1 April 2005 identified 892 such persons still employed by the RS, and the information was provided to the State Prosecutor of Bosnia and Herzegovina, with the understanding names would not be made public until official proceedings opened.<ref name="RS Working Group"/> On 4 October 2005, the working group said they had identified 25,083 people who were involved in the massacre, including 19,473 members of Bosnian Serb armed forces that actively gave orders or directly took part.<ref name="ISN">{{cite web|last=Alic|first=Anes|date=5 October 2005|url=http://www.isn.ethz.ch/news/sw/details.cfm?id=13046|title=25,000 participated in Srebrenica massacre|archive-url=https://web.archive.org/web/20070515114218/http://www.isn.ethz.ch/news/sw/details.cfm?id=13046 |archive-date=15 May 2007|work=ISN Security Watch}}</ref>

=== 2005: Release of Scorpions massacre video ===

On 1 June 2005, video evidence was introduced at the [[Slobodan Milošević trial]] to testify to the involvement of police from Serbia in the massacre.<ref name="Milošević transcript">{{cite web|url=http://www.icty.org/x/cases/slobodan_milosevic/trans/en/050601IT.htm|title=Milošević Case – Transcript|pages=40275ff|work=International Criminal Tribunal for the former Yugoslavia|date=1 June 2005|access-date=30 April 2017}}</ref> The video, the only undestroyed copy of 20 and previously available for rental in the Serbian town of Šid, was obtained and submitted to the ICTY by [[Nataša Kandić]], director of the Belgrade-based [[Humanitarian Law Center]].<ref name="publicinternationallaw.org">{{cite news|url=http://www.publicinternationallawandpolicygroup.org/wp-content/uploads/2011/04/wcpw_vol02issue18.html|publisher=Public International Law & Policy Group|work=War Crimes Prosecution Watch|volume=2|issue=18|title=Serb paramilitaries found guilty in war crimes trial|first1=Anes|last1=Alic|first2=Igor|last2=Jovanovic|date=30 April 2007|access-date=27 April 2017|url-status=dead|archive-url=https://web.archive.org/web/20170427195453/http://www.publicinternationallawandpolicygroup.org/wp-content/uploads/2011/04/wcpw_vol02issue18.html|archive-date=27 April 2017|df=dmy-all}}</ref>

The video shows an [[Eastern Orthodox Church|Orthodox]] priest blessing members of a Serbian unit known as the "Scorpions". Later these soldiers are shown physically abusing civilians. They were later identified as four minors as young as 16 and two men in their early twenties. The footage shows the execution of four of the civilians and them lying dead in a field. The cameraman expresses disappointment the battery is almost out.{{cn|date=June 2024}} The soldiers then ordered the two remaining captives to take the dead bodies into a nearby barn, where they were also killed upon completing this.<ref name="Milošević transcript"/><ref name="publicinternationallaw.org"/>

The video caused outrage in Serbia. Following its showing, the Serbian government arrested some former soldiers identified on it. The event was covered by the ''[[Danas (newspaper)|Danas]]'' newspaper, and [[radio station|radio]] and television station [[B92]]. Nura Alispahić, mother of 16-year-old Azmir Alispahić, saw her son's execution on television.<ref>{{cite news|url=http://www.janinedigiovanni.com/srebrenica-ten-years-on.html|author=Janine di Giovanni|work=International Herald Tribune|date=June 2005|title=Srebrenica – The Living Dead|url-status=dead |archive-url=https://web.archive.org/web/20100821092440/http://www.janinedigiovanni.com/srebrenica-ten-years-on.html |archive-date=21 August 2010 }}</ref> She said she was already aware of his death and had been told his body was burned following the execution; his remains were among those buried in Potočari in 2003.<ref>{{cite web|url=http://www.iol.co.za/news/world/heartbroken-mom-sees-son-shot-on-tv-242941|title=Heartbroken mom sees son shot on TV|first=Samir|last=Krilic|agency=Associated Press|work=Cape Argus|date=4 June 2005|access-date=25 April 2017}}</ref><ref>{{cite web |url=http://www.tuzilastvorz.org.rs/html_trz/OPTUZNICE/O_2008_10_01_ENG.pdf |title=Belgrade District Court War Crimes Chamber, amended indictment against Aleksandar Medic in Case K.V. 6/2005 |date=1 October 2008 |access-date=2010-04-07 |url-status=dead |archive-url=https://web.archive.org/web/20111006213607/http://www.tuzilastvorz.org.rs/html_trz/OPTUZNICE/O_2008_10_01_ENG.pdf |archive-date=6 October 2011 |df=dmy-all }}</ref> The executions took place on 16/17 July, in Trnovo, about 30 minutes from the Scorpions' base near Sarajevo.<ref name="publicinternationallaw.org"/>

On 10 April 2007, a special war crimes court in Belgrade convicted four former members of the ''Scorpions'' of war crimes, treating the killings as an isolated war crime unrelated to the Srebrenica genocide and ignoring allegations the Scorpions were acting under the authority of the Serbian Interior Ministry, MUP.<ref>{{cite web|url=http://www.crimesofwar.org/news-srebrenica3.html|title=Serbian Court Convicts Four over Killings in Scorpions Tape|first=Anthony |last=Dworkin|work=Crimes of War Project|url-status=dead |archive-url=https://web.archive.org/web/20100526184030/http://www.crimesofwar.org/news-srebrenica3.html |archive-date=26 May 2010 }}</ref>

===2005: 10th anniversary===
In June 2005, the [[United States House of Representatives]] passed a resolution commemorating the 10th anniversary, by 370 to 1 ([[Ron Paul]]).<ref>[http://projects.washingtonpost.com/congress/109/house/1/votes/322/ "Votes Database: Bill: H RES 199"]. ''The Washington Post'' (27 June 2005)</ref> It stated the "innocent people executed at Srebrenica ... should be solemnly remembered and honored; the policies ... implemented by Serb forces ... meet the terms ... in ... the Convention on the Prevention and Punishment of the Crime of Genocide."<ref>{{cite web|url=http://www.govtrack.us/congress/billtext.xpd?bill=hr109-199 |title=Read The Bill: H. Res. 199 109th |publisher=GovTrack.us |access-date=3 June 2011}}</ref> [[Missouri]] passed a resolution recognising the genocide<ref>{{Cite web|url=http://www.srebrenica95.com/index.php?option=com_content&view=article&id=98:rezolucija-missouri-kongresa&catid=4:vijesti&Itemid=86|title=Rezolucija Missouri Kongresa|date=29 March 2010|access-date=10 June 2024|archive-date=29 March 2010|archive-url=https://web.archive.org/web/20100329061607/http://www.srebrenica95.com/index.php?option=com_content&view=article&id=98:rezolucija-missouri-kongresa&catid=4:vijesti&Itemid=86|url-status=dead}}</ref> and [[St. Louis]] issued a proclamation declaring 11 July Srebrenica Remembrance Day.<ref>Association of the Srebrenica Genocide Survivors in St. Louis, [https://web.archive.org/web/20100329061311/http://srebrenica95.com/index.php?option=com_content&view=article&id=102%3Aproklamacija-grada-st-louis-a-o-srebrenickom-genocidu&catid=44%3Avijesti&Itemid=86 City of St. Louis Proclamation]</ref>

In his message to the commemoration, UN Secretary-General [[Kofi Annan]] paid tribute to the victims of "a terrible crime – the worst on European soil since the Second World War", on a date "marked as a grim reminder of man's inhumanity to man". He said the "first duty of the international community was to uncover and confront the full truth about what happened, a hard truth for those who serve the UN because great nations failed to respond adequately. There should have been stronger military forces in place, and a stronger will to use them".<ref name="UN10th">{{cite news |last1=Staff |title=UN officials commemorate 10th anniversary of Srebrenica massacre |url=https://news.un.org/en/story/2005/07/144502-un-officials-commemorate-10th-anniversary-srebrenica-massacre |access-date=2 July 2021 |work=UN News |date=11 July 2005 |language=en}}</ref> Annan added that the UN bore its share of responsibility, having made serious errors of judgement, "rooted in a philosophy of impartiality and non-violence which, however admirable, was unsuited to the conflict in Bosnia; because of that the tragedy of Srebrenica would haunt the UN's history forever".<ref name="UN10th"/> Bosnian Serb police found bombs at the memorial site, just days before the ceremony, when more than 50,000 people, including international politicians, were to attend. The bombs would have caused widespread loss of life.<ref>{{cite news|url=http://www.independent.co.uk/news/world/europe/bombs-found-at-memorial-for-srebrenica-genocide-497720.html |title=Bombs found at memorial for Srebrenica genocide |work=The Independent |location=UK |access-date=26 May 2011 |date=6 July 2005 |url-status=dead |archive-url=https://web.archive.org/web/20121111232220/http://www.independent.co.uk/news/world/europe/bombs-found-at-memorial-for-srebrenica-genocide-497720.html |archive-date=11 November 2012 }}</ref><ref>{{cite news|last=Wood |first=Nicholas |url=https://www.nytimes.com/2005/07/05/world/europe/05iht-bosnia.html |title=2 bombs found near Srebrenica |work=The New York Times |date=6 July 2005 |access-date=26 May 2011}}</ref>

=== 2006: Further mass graves and list of participants ===
[[File:Boy at 2006 Srebrenica funeral.jpg|thumb|A boy at a grave during the 2006 funeral of genocide victims]]
[[File:Srebrenica Massacre - Exhumed Grave of Victims - Potocari 2007.jpg|thumb|Exhumed grave, 2007]]
By 2006, 42 mass graves had been uncovered. 2,070 victims had been identified, while body parts in 7,000 bags awaited identification.<ref>Weinberg, Bill (11 July 2006). [http://ww4report.com/node/2187 "Srebrenica: 11 years later, still no justice"]. {{webarchive |url=https://web.archive.org/web/20060715210414/http://ww4report.com/node/2187 |date=15 July 2006 }}. ''World War 4 Report''. Retrieved 1 August 2008.</ref> In August 2006 over 1,000 body parts were exhumed from a mass grave in Kamenica.<ref>{{cite web|agency=Reuters|work=b92|date=11 August 2006|url=http://www.b92.net/eng/news/region.php?yyyy=2006&mm=08&dd=12&nav_id=36111 |title=Mass Grave Yields over 1,000 Body Parts |access-date=5 May 2017}}</ref>

In August 2006, Sarajevo newspaper [[Oslobođenje]] published a list of 892 Bosnian Serbs who had allegedly participated in the massacre and believed to still be employed by state institutions. They were listed among 28,000 Bosnian Serbs reported to have taken part by a Republika Srpska report. The list had been withheld from publication with the report, by the chief prosecutor of the Bosnian War Crimes Chamber, Marinko Jurčević who claimed "publishing this information might jeopardise the ongoing investigations".<ref>Avdić, Avdo (24 August 2006). [http://www.oslobodjenje.ba/index.php?option=com_content&task=view&id=48950&Itemid=44 {{"'}}Oslobođenje' objavljuje spisak za Srebrenicu"]. ''[[Oslobođenje]]''. Retrieved August 2008.</ref><ref>{{cite web|url=https://iwpr.net/global-voices/srebrenica-suspects-revealed|title=Srebrenica Suspects Revealed|author=IWPR|work=Institute for War & Peace Reporting|date=26 August 2006|id=TU No 465|access-date=3 May 2017}}</ref>

In December 2006, the Dutch government awarded the Dutch UN peacekeepers an insignia because they believed they "deserved recognition for their behaviour in difficult circumstances", noting the limited mandate and ill-equipped mission. However, survivors and relatives called it a "humiliating decision" and responded with protest rallies in The Hague, [[Assen]] and Sarajevo.<ref>{{cite news|title=Anger over Dutch Srebrenica medal |url=http://news.bbc.co.uk/2/hi/europe/6207254.stm|work=BBC News |date=4 December 2006 |access-date=8 January 2007}}</ref>

=== 2007–08: Arrests of Tolimir and Karadžić===
[[File:Srebrenica Massacre - Reinterment and Memorial Ceremony - July 2007 - Women and Monument.jpg|thumb|Women at the monument for victims, at the annual memorial ceremony in Potočari, 11 July 2007]]

In May 2007, former Bosnian Serb general [[Zdravko Tolimir]] was apprehended by police from Serbia and the Bosnian Serb republic. He was turned over to NATO forces at the Banja Luka airport where he was read the ICTY indictment and arrested. Mladić's deputy in charge of intelligence and security, and a key commander, Tolimir is believed to have been an organiser of the network protecting Mladić, helping him elude justice.<ref>{{Cite news|url=http://news.bbc.co.uk/2/hi/europe/6712903.stm|title=Net tightens on fugitive Mladic|date=1 June 2007|work=BBC News}}</ref><ref>{{cite web|url=http://www.balkaninsight.com/en/article/tolimir-genocide-trial-begins-at-icty |date=26 February 2010 |title=Tolimir Genocide Trial Begins at ICTY |access-date=19 April 2017|newspaper=Balkan Investigative Reporting Network}}</ref> Tolimir—"Chemical Zdravko"—is infamous for requesting the use of chemical weapons and proposing military strikes against refugees at Zepa.<ref>[http://www.sense-agency.com/en/stream.php?sta=3&pid=8351&kat=3 "Tolimir Requested Use of Chemical Weapons in Zepa"]. {{webarchive |url=https://web.archive.org/web/20071008015959/http://www.sense-agency.com/en/stream.php?sta=3&pid=8351&kat=3 |date=8 October 2007 }}. ''SENSE Tribunal'' (22 August 2006). Retrieved 31 July 2008.</ref> In June 2007, he was turned over to the ICTY. [[Radovan Karadžić]], with similar charges, was arrested in Belgrade in 2008, after 13 years on the run, and brought before [[Belgrade]]'s War Crimes Court.<ref name="serbia captures">{{cite news |title=Serbia captures fugitive Karadzic |work=BBC News |date=21 July 2008 |url=http://news.bbc.co.uk/2/hi/europe/7518543.stm |access-date=21 July 2008}}</ref><ref>{{cite web|url=http://www.icty.org/x/cases/karadzic/cis/en/cis_karadzic_en.pdf|author=ICTY|title=Karadzic case information sheet|access-date=26 April 2017}}</ref>

=== 2009: EU Parliament resolution ===
On 15 January 2009, the [[European Parliament]] voted in favour of a resolution calling for recognition of 11 July as a day for EU commemoration of the genocide.<ref>[http://www.b92.net/eng/news/crimes-article.php?yyyy=2009&mm=01&dd=15&nav_id=56447 "EP: 11 July to be Srebrenica remembrance day"]. {{webarchive |url=https://web.archive.org/web/20090124024906/http://www.b92.net/eng/news/crimes-article.php?yyyy=2009&mm=01&dd=15&nav_id=56447 |date=24 January 2009 }}. ''B92''. Retrieved 16 January 2008.</ref> Bosnian Serb politicians rejected it, stating such a commemoration is unacceptable to the Republika Srpska.<ref>[http://www.sarajevo-x.com/clanak/090116081 "Za RS neprihvatljivo obilježavanje 11. jula"]. {{webarchive |url=https://web.archive.org/web/20090118221112/http://www.sarajevo-x.com/clanak/090116081 |date=18 January 2009 }}. ''Sarajevo-x''. Retrieved 16 January 2008.</ref>

=== 2010 and 2013: Serbia's official apologies ===
In 2010, the [[National Assembly of Serbia|Serbian Parliament]] passed a resolution condemning the massacre, and apologizing for Serbia not doing more to prevent it. The motion was passed narrowly with 127 out of 250 [[Member of parliament|MPs]] voting in favour, with 173 legislators present. The [[Socialist Party of Serbia]], formerly under Slobodan Milošević and under new leadership, voted for. Opposition parties claimed the text was "shameful", either stating the wording was too strong or too weak.<ref>{{cite news
|url= http://news.bbc.co.uk/2/hi/europe/8594625.stm
|title= Serbian MPs offer apology for Srebrenica massacre
|work=BBC News
|date=31 March 2010
|access-date=31 March 2010
}}</ref> Some victims' relatives were unhappy with the apology, as it did not use the word 'genocide', but rather pointed at the [[Bosnian genocide case]] ruling.<ref>{{cite news
|url= https://www.cbc.ca/news/world/serbia-apologizes-for-srebrenica-massacre-1.932148
|title= Serbia apologizes for Srebrenica massacre
|work=CBC News
|date=31 March 2010
|access-date=31 March 2010
}}</ref> President [[Boris Tadić]] said the declaration is the highest expression of patriotism and it represents distancing from the crimes.<ref>{{cite web|url=http://www.b92.net/eng/news/politics-article.php?yyyy=2010&mm=03&dd=31&nav_id=66187 |title=Declaration represents distancing from crimes |work=B92 |access-date=26 May 2011 |url-status=dead |archive-url=https://web.archive.org/web/20110926150447/http://www.b92.net/eng/news/politics-article.php?yyyy=2010&mm=03&dd=31&nav_id=66187 |archive-date=26 September 2011 }}</ref> [[Sulejman Tihić]], former Bosniak member of the [[Presidency of Bosnia and Herzegovina]], stated that Bosnia and Herzegovina must adopt a similar resolution condemning crimes against Serbs and [[Croats]].<ref>{{cite web|url=http://otvoreno.ba/vijesti/bih/1545-tihi-sad-i-bih-mora-donijeti-deklaraciju-kojom-e-osuditi-zlocine-nad-srbima-i-hrvatima |title=Tihić: Sad i BiH mora donijeti deklaraciju kojom će osuditi zločine nad Srbima i Hrvatima |publisher=Otvoreno.ba |access-date=26 May 2011 |url-status=dead |archive-url=https://web.archive.org/web/20110706131528/http://otvoreno.ba/vijesti/bih/1545-tihi-sad-i-bih-mora-donijeti-deklaraciju-kojom-e-osuditi-zlocine-nad-srbima-i-hrvatima |archive-date=6 July 2011 }}</ref> In April 2013, President [[Tomislav Nikolić]] stated: "I kneel and ask for forgiveness for Serbia for the crime committed in Srebrenica. I apologise for the crimes committed by any individual in the name of our state and our people."<ref>{{cite web|url=http://www.aljazeera.com/news/europe/2013/04/2013425102523848273.html|title=Serbia president 'apologises' for massacre|work=Al Jazeera}}</ref>

=== 2010: Second Republika Srpska report revision ===
[[File:Srebrenica Massacre - Reinterment and Memorial Ceremony - July 2007 - Male Mourners.jpg|thumb|Bosniak mourners at the reburial ceremony for an exhumed victim of the Srebrenica massacre]]

On 21 April 2010, the government of [[Milorad Dodik]], the prime minister of Republika Srpska, initiated a revision of the 2004 report saying the numbers killed were exaggerated and the report was manipulated by a former peace envoy.<ref>{{cite news
|url= https://www.reuters.com/article/idUSLDE63K0E2
|title= Envoy slams Bosnia Serbs for questioning Srebrenica
|work=Reuters
|date= 21 April 2010
|access-date=21 April 2010
}}</ref> The [[Office of the High Representative]] responded by saying: "The Republika Srpska government should reconsider its conclusions and align itself with the facts and legal requirements and act accordingly, rather than inflicting emotional distress on the survivors, torture history and denigrate the public image of the country".<ref>{{cite news|url=http://www.ohr.int/ohr-dept/presso/pressr/default.asp?content_id=44835 |title=RS Government Special Session A Distasteful Attempt to Question Genocide |publisher=OHR |date=21 April 2010 |access-date=21 April 2010 |url-status=dead |archive-url=https://web.archive.org/web/20150318084354/http://www.ohr.int/ohr-dept/presso/pressr/default.asp?content_id=44835 |archive-date=18 March 2015 }}</ref> On 12 July 2010, at the 15th anniversary, Milorad Dodik said he acknowledged the killings, but did not regard what happened as genocide.<ref name="Srebrenica massacre 'not genocide'">[http://news.smh.com.au/breaking-news-world/srebrenica-massacre-not-genocide-20100713-1083q.html "Srebrenica massacre 'not genocide{{'"}}]. ''[[The Sydney Morning Herald]]''. [[Agence France-Presse]]. 13 July 2010</ref>

=== 2011: Arrest of Mladić ===
In May 2011, Mladić was arrested in [[Lazarevo]], Serbia after remaining at large for 16 years, sheltered by Serbian and Bosnian Serb security forces and family.{{cn|date=June 2024}} His capture was considered to be a pre-condition for [[Accession of Serbia to the European Union|Serbia obtaining candidate status for EU membership]]. See [[#Legal proceedings|Legal proceedings]] for his trial.

=== 2015: Russia vetoes UN resolution ===
In July 2015, Russia [[United Nations Security Council veto power|vetoed]] a [[United Nations Security Council resolution]] that would have condemned the massacre as a genocide. It was intended to mark the 20th anniversary. China, [[Nigeria]], [[Angola]] and [[Venezuela]] [[Abstention|abstained]] and the remaining 10 members voted in favour.<ref>{{cite news |url=https://www.nytimes.com/2015/07/09/world/europe/srebrenica-genocide-massacre.html |work=The New York Times |first1=Dan |last1=Bilefsky |first2=Somini |last2=Sengupta |title=Srebrenica Massacre, After 20 Years, Still Casts a Long Shadow in Bosnia |date=8 July 2015}}</ref> The veto was praised by Serbian President [[Tomislav Nikolić]] who said Russia had "prevented an attempt of smearing the entire Serbian nation as genocidal" and proven itself as a true and honest friend.<ref name="theguardian.com">{{cite web|date=8 July 2015|title=Russia threatens veto on UN vote calling Srebrenica 'a crime of genocide'|url=https://www.theguardian.com/world/2015/jul/08/russia-threatens-veto-on-un-vote-calling-srebrenica-a-of-genocide|work=The Guardian}}</ref><ref>{{cite news |url=https://www.reuters.com/article/us-bosnia-srebrenica-un-idUSKCN0PI1W620150708 |title=Russia blocks U.N. condemnation of Srebrenica as a genocide |work=Reuters |date=8 July 2015}}</ref>

=== 2024: International Day of Commemoration ===
In May 2024, July 11 was designated as the annual ''International Day of Reflection and Commemoration of the 1995 Genocide in Srebrenica'' by [[United Nations General Assembly Resolution 78/282]].<ref name="UNChiefwelcomes">{{Cite web |date=2024-05-23 |title=UN Human Rights Chief welcomes resolution to commemorate 1995 genocide in Srebrenica |url=https://www.ohchr.org/en/press-releases/2024/05/un-human-rights-chief-welcomes-resolution-commemorate-1995-genocide |access-date=2024-05-23|publisher=Office of the High Commissioner for Human Rights}}</ref><ref name="UNapproves">{{Cite web |date=2024-05-23 |title=UN approves annual commemoration of 1995 Srebrenica genocide |url=https://www.euronews.com/2024/05/23/un-approves-annual-commemoration-of-1995-srebrenica-genocide |access-date=2024-05-23|website=Euronews}}</ref>

The U.N. resolution, which was sponsored by Germany and Rwanda, was passed with 84 countries voting for the resolution, 68 abstaining, and 19 voting against.<ref>{{Cite web|url=https://news.un.org/en/story/2024/05/1150156|title=UN establishes International Day of reflection for Srebrenica genocide |website=UN News|date=May 23, 2024|access-date=June 9, 2024|archive-date=May 23, 2024|archive-url=https://web.archive.org/web/20240523211523/https://news.un.org/en/story/2024/05/1150156|url-status=live}}</ref> [[Politico]] described Serbia launching a ″full-blown diplomatic offensive″ to block the initiative, with Serbian leaders staging multiple press conferences and visiting the U.N. headquarters to meet with key stakeholders to try to sway the vote.<ref name="a886">{{cite web | last=Hajdari | first=Una | title=Serbia reels at UN resolution on Srebrenica massacre | website=POLITICO | date=22 May 2024 | url=https://www.politico.eu/article/serbia-un-resolution-srebrenica-genocide-massacre-aleksandar-vucic/ | access-date=18 November 2024}}</ref>

== Victims ==
''[[Research and Documentation Center in Sarajevo#The Bosnian Book of the Dead|The Bosnian Book of the Dead]]'' documented 8,331 victims killed in the massacre. The figure includes civilians and 1,416 soldiers who were taken as prisoners of war. Of the 8,331, 5,113 were from Srebrenica, 1,766 from Bratunac, 900 from Vlasenica, 437 from Zvornik and 115 from Rogatica/Žepa.<ref>{{Cite web | publisher=Research and Documentation Center | title=Human Losses in Bosnia-Herzegovina 1991–1995 part I | pages=173–174 | url=http://www.mnemos.ba/en/home/Download}}</ref>

{{as of|July 2020}} the [[International Commission on Missing Persons]] (ICMP) had identified 6,993 persons missing from the fall of Srebrenica, mostly through analysing [[DNA profiling|DNA profiles]] extracted from exhumed remains and matching them to profiles of relatives of the missing. The ICMP estimates total deaths was just over 8,000.<ref>{{cite web |title=ICMP and the Srebrenica Genocide |url=https://www.icmp.int/wp-content/uploads/2020/07/ICMP-srebrenica-leaflet-EN-web.pdf |website=[[International Commission on Missing Persons]] |access-date=30 October 2020 |date=July 2020 |archive-date=30 September 2020 |archive-url=https://web.archive.org/web/20200930161848/https://www.icmp.int/wp-content/uploads/2020/07/ICMP-srebrenica-leaflet-EN-web.pdf |url-status=dead }}</ref>

== Legal proceedings ==
{{See also|List of Bosnian genocide prosecutions}}

=== International Criminal Tribunal for the former Yugoslavia ===
[[File:Front view of the ICTY.jpg|thumb|[[International Criminal Tribunal for the former Yugoslavia]]]]

In 1993, the UN Security Council established the [[International Criminal Tribunal for the former Yugoslavia]] ('''ICTY''') to try those responsible for violations of international humanitarian law, including genocide.<ref>{{cite web |url=http://www.haguejusticeportal.net/eCache/DEF/9/564.html |title=Hague Justice Portal, "Srebrenica in summary – An overview of the legal proceedings relating to the 1995 genocide". Retrieved 31 July 2010 |publisher=Haguejusticeportal.net |access-date=26 May 2011 |archive-url=https://web.archive.org/web/20110925171659/http://www.haguejusticeportal.net/eCache/DEF/9/564.html |archive-date=25 September 2011 |url-status=dead }}</ref>
{{Quote box
|quote = The prosecution proved that genocide was committed in Srebrenica and that General Radislav Krstić, among others, was personally responsible for that.
|source =-- Olga Kavran
ICTY Outreach Programme<ref>{{cite web |url=http://www.icty.org/sid/10168 |title=Bridging the Gap in Srebrenica, Bosnia and Herzegovina |publisher=[[International Criminal Tribunal for the former Yugoslavia|ICTY]] |access-date=20 April 2010}}</ref>
|width = 30%
|align = right
}}
General [[Radislav Krstić]], who led the assault alongside Mladić, was convicted in 2001 of [[aiding and abetting]] genocide and received a sentence of 35 years. Colonel [[Vidoje Blagojević]] received 18 years for crimes against humanity. Krstić was the first European to be convicted of genocide since the [[Nuremberg trials]],<ref name="Newsweek13Aug2001">{{cite web |url=http://www.newsweek.com/id/76749 |title=Crimes Against Humanity |author=Rod Nordland |date=13 August 2001|work=Newsweek |access-date=20 April 2010}}</ref> and only the third person convicted under the 1948 [[Genocide Convention]]. The ICTY's final ruling against Krstić, judicially recognized the Srebrenica massacre, as an act of genocide:
{{blockquote|By seeking to eliminate a part of the Bosnian Muslims, the Bosnian Serb forces committed genocide. They targeted for extinction the 40,000 Bosnian Muslims living in Srebrenica, a group which was emblematic of the Bosnian Muslims in general. They stripped all the male Muslim prisoners, military and civilian, elderly and young, of their personal belongings and identification, and deliberately and methodically killed them solely on the basis of their identity.<ref name="Krstic para. 37"/>}}

Milošević was accused of genocide, or [[complicity in genocide]], including in Srebrenica,<ref>{{cite web|url=https://www.icty.org/x/cases/slobodan_milosevic/ind/en/mil-ai010629e.htm |title=ICTY: The prosecutor of the tribunal against Slobodan Milosevic – Amended Indictment |publisher=United Nations |date=5 March 2007 |access-date=26 May 2011 |archive-url=https://web.archive.org/web/20150419224309/https://www.icty.org/x/cases/slobodan_milosevic/ind/en/mil-ai010629e.htm |archive-date=19 April 2015 }}</ref> but died in 2006 during his trial. In June 2010, seven senior Serb military and police officers, [[Vujadin Popović]], [[Ljubiša Beara]], [[Drago Nikolić]], Ljubomir Borovčanin, Vinko Pandurević, Radivoje Miletić and Milan Gvero, were found guilty of various crimes, including genocide.<ref name="ICTYX2">{{cite web|title=Popovic et al. 'Srebrenica' Trial to Begin on 14 July 2006|url=http://www.icty.org/sid/8718|access-date=10 June 2010|publisher=ICTY|date=11 July 2006}}</ref><ref>{{cite news |url=https://edition.cnn.com/2010/WORLD/europe/06/10/hague.srebrenica.verdict/?hpt=T1 |title=Seven convicted over 1995 Srebrenica massacre |publisher=CNN |date=10 June 2010 |access-date=10 June 2010 |archive-date=7 August 2020 |archive-url=https://web.archive.org/web/20200807113020/http://edition.cnn.com/2010/WORLD/europe/06/10/hague.srebrenica.verdict/?hpt=T1 |url-status=live }}</ref><ref>{{cite news|last=Roumeliotis|first=Greg|title=Bosnian Serbs could face new Srebrenica war crimes trial|url=https://www.reuters.com/article/idAFTRE6893RQ20100910|work=Reuters |date=10 September 2010}}</ref> The former chief of the [[Staff (military)|general staff]] of the [[Armed Forces of Serbia and Montenegro|Yugoslav Army]], [[Momčilo Perišić]], was sentenced to 27 years for aiding and abetting murder, because he provided [[salary|salaries]], [[ammunition]], staff and [[fuel]] to the VRS officers.<ref name="Perisic Verdict">{{cite web|url=http://www.icty.org/x/cases/perisic/tjug/en/110906_summary.pdf|title=Summary of the Judgement in the Case of Prosecutor v. Momčilo Perišić|date=6 September 2011|publisher=International Criminal Tribunal for the former Yugoslavia|access-date=7 September 2011|location=The Hague}}</ref> However, the evidence proved Perišić's inability to impose binding orders on Mladić.<ref>{{cite web|url=http://www.sense-agency.com/icty/perisic-sentenced-to-27-years-for-crimes-in-bh-and-croatia.29.html?cat_id=1&news_id=13151|title=PERISIC SENTENCED TO 27 YEARS FOR CRIMES IN BH AND CROATIA|date=6 September 2011|publisher=Sense-Agency|access-date=7 September 2011|location=The Hague|archive-date=28 March 2019|archive-url=https://web.archive.org/web/20190328212556/http://www.sense-agency.com/icty/perisic-sentenced-to-27-years-for-crimes-in-bh-and-croatia.29.html%3Fcat_id%3D1%26news_id%3D13151|url-status=dead}}</ref> [[Zdravko Tolimir]], a former general in the Army of the Republika Srpska,<ref>{{cite web |url=http://www.icty.org/cases/party/791/4 |title=ICTY, "The Cases", Zdravko Tolimir. Retrieved 25 July 2010 |publisher=Icty.org |access-date=26 May 2011 |archive-date=6 September 2019 |archive-url=https://web.archive.org/web/20190906184216/http://www.icty.org/cases/party/791/4 |url-status=dead }}</ref> was accused of participating in the "criminal enterprise to remove the Muslim population" from Srebrenica and Zepa. He was convicted of genocide and sentenced to life imprisonment in 2012.<ref>{{cite news |title=Bosnian Serb Zdravko Tolimir convicted over Srebrenica |url=https://www.bbc.co.uk/news/world-europe-20700387 |work=BBC News |date=12 December 2012 |access-date=13 December 2012}}</ref>
{{Multiple image
| align = right
| direction = horizontal
| image1 = RadovanKaradzic.jpg
| width1 = 130
| image2 = Mladić_Trial_Judgement_(crop).jpg
| width2 = 173
| footer = [[Radovan Karadžić]] (left), former president of Republika Srpska, and [[Ratko Mladić]] (right), former Chief of Staff of the Army of the Republika Srpska. Both were found guilty of genocide, war crimes and crimes against humanity, and sentenced to life imprisonment.
}}
[[Radovan Karadžić]] and [[Ratko Mladić]] were indicted for genocide, and complicity in genocide, including in Srebrenica.<ref>{{cite news|url=https://www.bbc.co.uk/news/world-europe-13561407 |title=Ratko Mladic arrested: Bosnia war crimes suspect held |publisher=BBC |access-date=26 May 2011 |date=26 May 2011}}</ref> The [[trial of Radovan Karadžić]] began in 2010, and in 2016 he was convicted of genocide in Srebrenica and other crimes;<ref>{{cite news |url=http://news.bbc.co.uk/2/hi/europe/8320220.stm |title=Radovan Karadzic: The charges |date=23 October 2009 |work=BBC News |access-date=20 April 2010}}</ref><ref>{{cite web |url=http://www.icty.org/cases/party/703/4 |title=ICTY Cases, Radovan Karadžić. Retrieved 25 July 2010 |publisher=Icty.org |access-date=26 May 2011 |archive-date=5 September 2019 |archive-url=https://web.archive.org/web/20190905171133/http://www.icty.org/cases/party/703/4 |url-status=dead }}</ref> he was ultimately sentenced to life imprisonment.<ref name="nyt24-03-2016">{{cite news|url=https://www.nytimes.com/2016/03/25/world/europe/radovan-karadzic-verdict.html|title= Radovan Karadzic, a Bosnian Serb, Gets 40 Years Over Genocide and War Crimes|location=New York|newspaper=[[The New York Times]]|access-date=24 March 2016|date= 2016-03-24 |last1= Simons |first1= Marlise }}</ref><ref>{{Cite web|url=http://www.cnn.com/2016/03/24/europe/karadzic-war-crimes-verdict/index.html|title=Karadzic sentenced to 40 years for genocide|website=CNN|date=24 March 2016|access-date=2016-03-26}}</ref><ref name="life-sentance">{{cite web |last1=Borges |first1=Julian |title=Radovan Karadžić war crimes sentence increased to life in prison |url=https://www.theguardian.com/law/2019/mar/20/radovan-karadzic-faces-final-verdict-in-bosnia-war-crimes-case |website=the Guardian |access-date=20 September 2022 |language=en |date=20 March 2019}}</ref><ref>{{cite news|url=https://www.irmct.org/en/cases/mict-13-55| title=KARADŽIĆ, Radovan (MICT-13-55-ES)| date=20 March 2019| publisher=International Residual Mechanism for Criminal Tribunals| location=The Hague}}</ref> In 2017, the ICTY found Mladić guilty on 10 counts of war crimes, crimes against humanity and genocide, and sentenced him to life in prison.<ref>{{cite news|url=https://www.hrw.org/news/2017/11/22/icty/bosnia-life-sentence-ratko-mladic| title=ICTY/Bosnia: Life Sentence for Ratko Mladic| work=Human Rights Watch| date=22 November 2017}}</ref> As the top military officer with [[command responsibility]], Mladić was deemed responsible for the Srebrenica massacre.{{cn|date=June 2024}}

In 2023, the follow-up [[International Residual Mechanism for Criminal Tribunals]] sentenced [[State Security Service (FR Yugoslavia)|Serbian State Security]] officers [[Jovica Stanišić]] and [[Franko Simatović]] for aiding and abetting murder and persecution of six Bosniak men in Trnovo in 1995, through their control of Serb paramilitary, [[Scorpions (paramilitary)|Scorpions]], and sentenced each to 15 years.<ref>{{cite news|title=UN commends Criminal Tribunal for former Yugoslavia, as final judgement is delivered| work=UN News| date=31 May 2023| url=https://news.un.org/en/story/2023/05/1137222| access-date=17 August 2023}}</ref><ref>{{cite web| title=STANIŠIĆ and SIMATOVIĆ (MICT-15-96-A)| url=https://www.irmct.org/en/cases/mict-15-96| publisher=International Residual Mechanism for Criminal Tribunals| location=The Hague| date=31 May 2023| access-date=17 August 2023}}</ref> The Tribunal concluded:
{{blockquote|...in July 1995, Slobodan Medić (Boca) was ordered to transport Muslims, including six Muslim men and boys, to various locations, to be killed. The Trial Chamber found proven beyond reasonable doubt that the Scorpions, acting upon the orders of Slobodan Medić (Boca), killed the six Muslim men and boys in the rural area at Godinjske Bare.<ref>{{cite web| title=The Prosecutor vs. Jovica Stanišić & Franko Simatović — Judgement In the Appeals Chamber| url=https://www.irmct.org/sites/default/files/case_documents/IRMCT-Appeal-Judgement-Stanisic-Simatovic-ENG.pdf |page=219| publisher=International Residual Mechanism for Criminal Tribunals| location=The Hague| date=31 May 2023| access-date=17 August 2023}}</ref>}}

=== International Court of Justice ===
{{See also|Bosnian genocide case}}
The Srebrenica genocide was the core issue of the landmark [[Bosnian genocide]] case at the [[International Court of Justice]] through which Bosnia and Herzegovina accused [[Serbia and Montenegro]] of genocide. The ICJ presented its judgement in February 2007, which concurred with ICTY's recognition of the Srebrenica massacre as genocide.<ref name="ICJ 2007"/> It cleared Serbia of direct involvement,<ref>Hudson, Alexandra (26 February 2007). [https://web.archive.org/web/20070302091728/http://www.reuters.com/article/homepageCrisis/idUSL26638724._CH_.2400 "Serbia cleared of genocide, failed to stop killing"]</ref> but ruled that Belgrade breached international law by failing to prevent the genocide, and failing to try or transfer the persons accused to the ICTY, under its obligations in the Genocide Convention, particularly in respect of Mladić.<ref name="ICJ-2007-02-26">[http://www.icj-cij.org/presscom/index.php?pr=1897&pt=1&p1=6&p2=1 ICJ press release 2007/8] {{Webarchive|url=https://web.archive.org/web/20100213065153/http://www.icj-cij.org/presscom/index.php?pr=1897&pt=1&p1=6&p2=1 |date=13 February 2010 }} 26 February 2007, See points 7 and 8</ref><ref>{{cite web |url=http://www.icj-cij.org/docket/index.php?sum=667&code=bhy&p1=3&p2=2&case=91&k=f4&p3=5 |title=ICJ: Summary of the Judgment of 26 February 2007 – Bosnia v. Serbia |publisher=Icj-cij.org |date=11 July 1996 |access-date=26 May 2011 |archive-url=https://web.archive.org/web/20110605012548/http://www.icj-cij.org/docket/index.php?sum=667&code=bhy&p1=3&p2=2&case=91&k=f4&p3=5 |archive-date=5 June 2011 |url-status=dead |df=dmy-all }}</ref><ref name="NyTimes-BosniaGenocide">Simons, Marlise (27 February 2007). [https://www.nytimes.com/2007/02/27/world/europe/27hague.html?ref=world "Court Declares Bosnia Killings Were Genocide"], ''The New York Times''. Retrieved 31 July 2008.</ref> Citing national security, Serbia obtained permission from the ICTY to keep parts of its military archives out of the public eye during Milošević's trial. This may have decisively affected the ICJ's judgement in the lawsuit against Serbia, as the archives were not on the ICTY's public record – although the ICJ could have, but did not, [[subpoena]] the documents.<ref>Simons, Marlise (9 April 2007). [https://www.nytimes.com/2007/04/09/world/europe/09archives.html?_r=1&oref=slogin "Genocide Court Ruled for Serbia Without Seeing Full War Archive"], ''The New York Times''. Retrieved 31 July 2008.</ref> The Chief prosecutor's office, rejected allegations there was a deal with Belgrade to conceal documents from the ICJ case.<ref>{{cite web|url=https://iwpr.net/global-voices/del-ponte-denies-belgrade-deal-claims|title=Del Ponte Denies Belgrade Deal Claims|date=20 April 2007|first=Lisa|last=Clifford|work=Institute for War & Peace Reporting|access-date=4 May 2017}}</ref>

=== National courts ===

==== Serbia ====
{{See also|#2005: Release of Scorpions massacre video}}
On 10 April 2007, a Serbian war crimes court sentenced four members of the [[Scorpions (paramilitary)|Scorpions paramilitary group]] to a total of 58 years in prison for the execution of six Bosniaks during the Srebrenica massacre.<ref>{{cite news|last=Peric Zimonjic|first=Vesna|date=11 April 2007|url=https://www.independent.co.uk/news/world/europe/serb-scorpions-guilty-of-srebrenica-massacre-5332749.html |archive-url=https://ghostarchive.org/archive/20220524/https://www.independent.co.uk/news/world/europe/serb-scorpions-guilty-of-srebrenica-massacre-5332749.html |archive-date=24 May 2022 |url-access=subscription |url-status=live|title=Serb 'Scorpions' guilty of Srebrenica massacre|work=[[The Independent]]|access-date=4 May 2017}}</ref>

'''Guilty of war crimes'''
*Pera Petrasevic – sentenced to 13 years<ref name="auto5">{{Cite web|url=https://warcrimesmap.balkaninsight.com/verdicts/aleksandar-medic-branislav-medic-slobodan-medic-pera-petrasevic-and-aleksandar-vukov/|title=War Crimes Verdicts}}</ref>
*Branislav Medic – sentenced to 15 years<ref name="auto5"/>
*Aleksandar Medic – sentenced to five years<ref name="auto5"/>
'''Acquitted'''
*Aleksandar Vukov<ref name="auto5"/>

==== Bosnia and Herzegovina ====
The "[[Kravica]]" case was an important trial before the [[Court of Bosnia and Herzegovina]]; 11 men were accused of genocide.<ref name="Mitrovic -Kravice">The Court of Bosnia and Herzegovina – Mitrovic and others (Kravice) – Accused of the criminal offence of genocide in violation of Article 171 of the Criminal Code of Bosnia and Herzegovina ([http://www.sudbih.gov.ba/?opcija=predmeti&id=22&jezik=e X-KR-05/24 – Mitrovic and others (Kravice)]).</ref> In July 2008, after a two-year trial, the court found seven of them guilty of genocide for their role in Srebrenica, including the deaths of 1000 Bosniak men in a single day.<ref name="AP-2008-07-29"/><ref name="BBC-2008-07-29">{{cite news|url=http://news.bbc.co.uk/2/hi/europe/7531413.stm|title=Bosnian Serbs jailed for genocide |date=29 July 2008|work=BBC News |access-date=29 July 2008}}</ref> Men trying to escape were told they would be kept safe if they surrendered. Instead, they were transported to an agricultural [[cooperative]] in [[Kravica]], and executed.<ref name="AP-2008-07-29" /><ref name="BBC-2008-07-29" />

;Guilty of genocide
* Milenko Trifunović (commander of the 3rd "Skelani" Platoon, part of the 2nd Special Police Šekovići Squad)<ref name="Mitrovic -Kravice" /> – sentenced to 42 years.<ref name="AP-2008-07-29" /><ref name="BBC-2008-07-29" />
* Brano Džinić (special police force officer of the 2nd Special Police Šekovići Squad)<ref name="Mitrovic -Kravice" /> – sentenced to 42 years.<ref name="AP-2008-07-29" /><ref name="BBC-2008-07-29" />
* Slobodan Jakovljević (special police force member of the 3rd "Skelani" Platoon)<ref name="Mitrovic -Kravice" /> – sentenced to 40 years.<ref name="AP-2008-07-29" /><ref name="BBC-2008-07-29" />
* Branislav Medan (special police force member of the 3rd "Skelani" Platoon)<ref name="Mitrovic -Kravice" /> – sentenced to 40 years.<ref name="AP-2008-07-29" /><ref name="BBC-2008-07-29" />
* Petar Mitrović (special police force member of the 3rd "Skelani" Platoon)<ref name="Mitrovic -Kravice" /> – sentenced to 38 years.<ref name="AP-2008-07-29" /><ref name="BBC-2008-07-29" />
* Aleksandar Radovanović (special police force members of the 3rd "Skelani" Platoon)<ref name="Mitrovic -Kravice" /> – sentenced to 42 years.<ref name="AP-2008-07-29" /><ref name="BBC-2008-07-29" />
* [[Milorad Trbić]] (assistant commander for Security with the Zvornik Brigade of the Republika Srpska Army) found guilty on one count of genocide and sentenced to 30 years in jail.<ref>[http://www.bim.ba/en/188/10/22954/ Milorad Trbic Found Guilty of Genocide] {{webarchive |url=https://web.archive.org/web/20100329131528/http://www.bim.ba/en/188/10/22954/ |date=29 March 2010 }} 16 October 2009</ref><ref>{{cite news|url=http://www.voanews.com/english/2007-06-11-voa66.cfm|title=Hague Tribunal Transfers Trbic Case to Bosnian Court|archive-url=https://web.archive.org/web/20070613021322/http://www.voanews.com/english/2007-06-11-voa66.cfm |archive-date=13 June 2007|work=[[VOA News]]|date=11 June 2008}}</ref><ref>The Court of Bosnia and Herzegovina – Trbic case: Charged with genocide pursuant to Article 171 of the Criminal Code of Bosnia and Herzegovina (CC BiH) in conjunction with the killing members of the group, causing serious bodily or mental harm to members of the group, deliberately inflicting on the group conditions of life calculated to bring about its physical destruction in whole or in part, imposing measures intended to prevent births within the group ([http://www.sudbih.gov.ba/?opcija=predmeti&id=47&jezik=e X-KR-07/386 – Trbic Milorad] {{Webarchive|url=https://web.archive.org/web/20081227234155/http://www.sudbih.gov.ba/?opcija=predmeti&id=47&jezik=e |date=27 December 2008 }})</ref>
* Radomir Vuković (special police force officer of the 2nd Special Police Šekovići Squad) – sentenced to 31 years.<ref name="reuters 31 years">{{cite news|url=http://uk.reuters.com/article/uk-bosnia-warcrimes-sentence-idUKTRE63L23Z20100422|archive-url=https://web.archive.org/web/20170427202046/http://uk.reuters.com/article/uk-bosnia-warcrimes-sentence-idUKTRE63L23Z20100422|url-status=dead|archive-date=27 April 2017|first=Maja|last=Zuvela|location=Sarajevo|title=Two Bosnian Serbs Jailed For 31 Years Over Srebrenica |date=22 April 2010 |work=Reuters|access-date=27 April 2017}}</ref>
* Zoran Tomić (special police force officer of the 2nd Special Police Šekovići Squad) – sentenced to 31 years.<ref name="reuters 31 years"/>
* Marko Boškić (member of 10th Commando Squad of the Republika Srpska Army)<ref name="TBiH">{{cite news|url=http://www.tuzilastvobih.gov.ba/?id=673&jezik=e|title=Suspect Marko Boškić Extradited To BiH |date=29 April 2010|publisher=The Prosecutor's Office of Bosnia and Herzegovina|access-date=6 May 2010}}</ref> – pleaded guilty, sentenced to 10 years.<ref>{{cite web|title=Marko Boskic – TRIAL International|url=https://trialinternational.org/latest-post/marko-boskic/|website=trialinternational.org|access-date=5 May 2017|archive-date=19 April 2017|archive-url=https://web.archive.org/web/20170419045049/https://trialinternational.org/latest-post/marko-boskic/|url-status=dead}}</ref>

;Guilty of aiding and abetting genocide
* Duško Jević (deputy commander of the interior ministry special police brigade and commander of the Jahorina special police training center) – sentenced to 35 years.<ref name="Euronews 25 May 2012">{{cite news|title=Bosnian Serbs jailed for Srebrenica warehouse killings|url=https://www.reuters.com/article/oukwd-uk-bosnia-srebrenica-verdict-idAFBRE84O0QW20120525|first=Maja|last=Zuvela|work=Reuters|date=25 May 2012|location=Sarajevo|access-date=25 April 2017}}</ref>
* Mendeljev Đurić (commander of Jahorina special police training center's first company) – sentenced to 30 years.<ref name="Euronews 25 May 2012" />

;Guilty of crimes against humanity and war crimes
* Stanko Kojić (member of the 10th Sabotage Unit of the Republika Srpska Army) – sentenced to 43 years.<ref name="Balkan Insight 15 June 2012">{{cite news|title=Srebrenica: 142 Years of Prison For Branjevo Crimes|url=http://www.balkaninsight.com/en/article/srebrenica-trial-142-years-of-prison-for-branjevo-crimes|newspaper=Balkan Insight|date=15 June 2012}}</ref>
* Franc Kos (commander of the First Platoon of the 10th Sabotage Unit of the Republika Srpska Army) – sentenced to 40 years.<ref name="Balkan Insight 15 June 2012" />
* Zoran Goronja (member of the 10th Sabotage Unit of the Republika Srpska Army) – sentenced to 40 years.<ref name="Balkan Insight 15 June 2012" />
* Vlastimir Golijan (member of the 10th Sabotage Unit of the Republika Srpska Army) – plead guilty,<ref name="birn">{{cite news|title=Vlastimir Golijan Pleads Guilty to Genocide |url=http://www.bim.ba/en/235/10/30414/ |newspaper=Balkan Investigative Reporting Network |date=8 September 2010 |url-status=dead |archive-url=https://web.archive.org/web/20100915062431/http://www.bim.ba/en/235/10/30414/ |archive-date=15 September 2010 }}</ref> sentenced to 19 years.<ref name="Balkan Insight 15 June 2012" />
* Dragan Crnogorac (police officer) – sentenced to 13 years.<ref>{{cite news|title=Former Bosnian Serb police officer sentenced for Srebrenica war crimes|url=http://www.setimes.com/cocoon/setimes/xhtml/en_GB/newsbriefs/setimes/newsbriefs/2011/05/15/nb-06|newspaper=Southeast European Times|date=15 May 2011}}</ref>
* Božidar Kuvelja (Bosnian Serb police officer)- sentenced to 20 years.<ref>{{Cite web|url=https://warcrimesmap.balkaninsight.com/verdicts/bozidar-kuvelja/|title=War Crimes Verdicts}}</ref>
;Arrested
*On 12 September 2023, five Bosnian Serb ex-soldiers were arrested in [[Zvornik]], [[Šekovići]], [[Han Pijesak]], [[Vlasenica]], and [[Bileća]] for suspected involvement.<ref>{{Cite web |url=https://balkaninsight.com/2023/09/12/bosnia-arrests-five-serb-ex-soldiers-for-srebrenica-genocide/ |title=Bosnia Arrests Five Serb Ex-Soldiers for Srebrenica Genocide |website=Balkan Insight |last=Grebo |first=Lamija |date=12 September 2023 |access-date=13 September 2023}}</ref>

;On trial
* Aleksa Golijanin<ref name="B92 15 June 2012" />

;Acquitted
* Velibor Maksimović (special police force members of the 3rd "Skelani" Platoon)<ref name="Mitrovic -Kravice" /><ref name="AP-2008-07-29" /><ref name="BBC-2008-07-29" />
* Milovan Matić (member of the Republika Srpska Army)<ref name="Mitrovic -Kravice" /><ref name="AP-2008-07-29" /><ref name="BBC-2008-07-29" />
* Teodor Pavelvić (member of the Republika Srpska Army)<ref name="AP-2008-07-29" /><ref name="BBC-2008-07-29" />
* Miladin Stevanović (special police force members of the 3rd "Skelani" Platoon)<ref name="Mitrovic -Kravice" /><ref name="AP-2008-07-29" /><ref name="BBC-2008-07-29" />
* Dragiša Živanović (special police force members of the 3rd "Skelani" Platoon)<ref name="Mitrovic -Kravice" /><ref name="AP-2008-07-29" /><ref name="BBC-2008-07-29" />
* Miloš Stupar (commander of the 2nd Special Police Šekovići Squad)<ref name="Mitrovic -Kravice" /> – found guilty, sentenced to 40 years.,<ref name="AP-2008-07-29" /><ref name="BBC-2008-07-29" /> later acquitted.<ref>{{cite news|url=http://news.bbc.co.uk/2/hi/europe/8663764.stm|title=Bosnian Serb cleared of genocide |date=6 May 2010|publisher=BBC |access-date=6 May 2010}}</ref>
* Neđo Ikonić<ref name="SUDBIH 25 May 2012">{{cite news|title=Duško Jević et al. Verdict|url=http://www.sudbih.gov.ba/?opcija=predmeti&id=266&jezik=e|newspaper=Court of Bosnia and Herzegovina|date=25 May 2012}}</ref>
* Goran Marković<ref name="SUDBIH 25 May 2012" />
* Dejan Radojković<ref name="Fox 24 May 2012">{{cite news|title=US deports man sought for war crimes to Bosnia|url=http://www.foxnews.com/us/2012/05/24/apnewsbreak-us-deports-man-sought-for-war-crimes/|archive-url=https://web.archive.org/web/20120526040323/http://www.foxnews.com/us/2012/05/24/apnewsbreak-us-deports-man-sought-for-war-crimes/|url-status=dead|archive-date=26 May 2012|publisher=Fox News Channel|date=24 May 2012}}</ref><ref>{{Cite web|url=https://detektor.ba/case/case-radojkovic-dejan/?lang=en|title=Case: Radojkovic Dejan Archives|website=detektor.ba}}</ref>
* Aleksandar Cvetković (former member of the Tenth Reconnaissance Division of the Bosnian Serb Army).<ref>{{Cite web|url=https://balkaninsight.com/2015/07/02/ex-bosnian-serb-fighter-acquitted-of-srebrenica-genocide-charges/|title=Bosnian Serb Soldier Acquitted of Srebrenica Genocide | Balkan Insight|date=2 July 2015 }}</ref> He was accused of taking part in the executions of 800 people, initiating use of machine guns to speed up killing.<ref name="cnn18012011">{{cite news|title=Israel arrests suspect in Srebrenica massacre|url=http://edition.cnn.com/2011/WORLD/meast/01/18/israel.massacre.extradition/?hpt=Sbin|archive-url=https://web.archive.org/web/20120127034711/http://edition.cnn.com/2011/WORLD/meast/01/18/israel.massacre.extradition/?hpt=Sbin|url-status=dead|archive-date=27 January 2012|publisher=CNN|date=18 January 2011}}</ref><ref>{{cite news|first1=Josh|last1=Lederman|last2=Corder|first2=Mike|last3=Cerkez|first3=Aida|title=Israel arrests man wanted in Srebrenica massacre|url=http://www.sandiegouniontribune.com/sdut-israel-arrests-man-wanted-in-srebrenica-massacre-2011jan18-story.html|agency=Associated Press|work=The San Diego Union-Tribune|date=18 January 2011|access-date=26 April 2017|location=Jerusalem}}</ref>

;Indictment dismissed on medical grounds
* Nedeljko Milidragović<ref name="B92 15 June 2012">{{cite news|title=Two Serbian citizens indicted for Srebrenica crime |url=http://www.b92.net/eng/news/region-article.php?yyyy=2012&mm=06&dd=15&nav_id=80774 |newspaper=B82 |date=15 June 2012 |url-status=dead |archive-url=https://web.archive.org/web/20120617043010/http://www.b92.net/eng/news/region-article.php?yyyy=2012&mm=06&dd=15&nav_id=80774 |archive-date=17 June 2012 }}</ref>

==== Netherlands ====
Survivors and victims' relatives sought to establish the responsibility of the Netherlands and UN, in Dutch courts. In one case, 11 plaintiffs including "[[Mothers of Srebrenica]]",<ref name="www.vandiepen.com">{{cite web |url=http://www.vandiepen.com/en/about-us/corporate-social-responsibility/introduction-to-the-srebrenica-case.html |title=Srebrenica :: Introduction to the Case Srebrenica |website=Van Diepen Van der Kroef |access-date=26 May 2011 |archive-url=https://web.archive.org/web/20160306051405/http://www.vandiepen.com/en/about-us/corporate-social-responsibility/introduction-to-the-srebrenica-case.html |archive-date=6 March 2016 |url-status=dead |df=dmy-all }}</ref><ref>{{cite web|url=http://www.radionetherlands.nl/currentaffairs/080618-srebrenica-un-mc|title=Srebrenica genocide testcase for UN immunity|date=18 June 2008|first=Gottlied|last=Sebastian|work=Radio Netherlands Worldwide|archive-url=https://web.archive.org/web/20081220121549/http://www.radionetherlands.nl/currentaffairs/080618-srebrenica-un-mc |archive-date=20 December 2008 }}</ref> asked the court to rule that the Netherlands and UN breached their obligation to prevent genocide and hold them jointly liable to pay compensation.<ref>{{cite web|url=http://www.legal-tools.org/en/doc/ca1e99/|website=legal-tools.org|author=Van Diepen Van der Kroef Advocaten|title=Writ of Summons: District Court, The Hague|date=4 June 2007|page=198}}</ref> In July 2008, the court ruled it had no jurisdiction against the UN; the plaintiffs appealed this ruling in relation to UN immunity.<ref>[https://web.archive.org/web/20081226065315/http://www.dnevniavaz.ba/dogadjaji/aktuelno/advokatski-tim-podnio-zalbu "Advokatski tim podnio žalbu" F. V. (30 October 2008).]</ref>

Another action was brought by a former UN interpreter [[Hasan Nuhanović]] and the family of Rizo Mustafić, an electrician employed by the UN at Srebrenica. They claimed Dutch troops, responsible for security in the UN-protected zone, allowed VRS troops to kill Nuhanović's relatives<ref>[http://www.b92.net/eng/news/region-article.php?yyyy=2008&mm=06&dd=17&nav_id=51153 "Srebrenica lawsuit against Holland opens"]. {{webarchive |url=https://web.archive.org/web/20110607040529/http://www.b92.net/eng/news/region-article.php?yyyy=2008&mm=06&dd=17&nav_id=51153 |date=7 June 2011 }}. ''B92''. 17 June 2008.</ref> and Mustafić.<ref name="haguejusticeportal.net">[http://www.haguejusticeportal.net/eCache/DEF/9/321.TGFuZz1FTg.html "District Court hears Srebrenica cases"]. {{Webarchive|url=https://web.archive.org/web/20110716162855/http://www.haguejusticeportal.net/eCache/DEF/9/321.TGFuZz1FTg.html |date=16 July 2011 }}. ''The Hague Justice Portal''. 18 June 2008.</ref> They argued the Dutch Government had ''de facto'' operational command, in accordance with the [[Constitution of the Netherlands|Dutch Constitution]], which grants the government superior command over military forces.<ref name="haguejusticeportal.net" /> In September 2008, the district court dismissed these claims and held that the Netherlands could not be held responsible, because the Dutchbat peacekeepers were operating in Bosnia under a UN mandate and operational command had been transferred to the UN.<ref>[http://www.haguejusticeportal.net/eCache/DEF/9/758.html "Nuhanović vs The Netherlands"]. {{Webarchive|url=https://web.archive.org/web/20110716163233/http://www.haguejusticeportal.net/eCache/DEF/9/758.html |date=16 July 2011 }}. The Hague Justice Portal DomClic Project. Retrieved 5 August 2010.</ref> In July 2011, the Dutch court of appeal reversed this and held that the state was responsible for, and indeed actively coordinated the evacuation once Srebrenica fell, and therefore responsible for the decision to dismiss Nuhanović's brother and Mustafić from the compound. The court held that this decision was wrong, because the Dutch soldiers should have known they were in great danger of being tortured or killed. Both claimants were therefore eligible for compensation.<ref>{{cite web|url=http://www.rechtspraak.nl/Organisatie/Gerechtshoven/Den-Haag/Nieuws/Pages/NederlandseStaataansprakelijkvoordooddrieMoslimmannennavalSrebrenica.aspx |title=Nederlandse Staat aansprakelijk voor dood drie Moslimmannen na val Srebrenica |language=nl |work=De Rechtspraak |access-date=11 July 2012 |url-status=dead |archive-url=https://web.archive.org/web/20120318102557/http://www.rechtspraak.nl/Organisatie/Gerechtshoven/Den-Haag/Nieuws/Pages/NederlandseStaataansprakelijkvoordooddrieMoslimmannennavalSrebrenica.aspx |archive-date=18 March 2012 }}</ref> In September 2013, the [[Supreme Court of the Netherlands]] dismissed a government appeal,<ref>{{cite web|url=http://www.rechtspraak.nl/Organisatie/Hoge-Raad/Nieuws/Pages/StateresponsiblefordeathofthreeMuslimmeninSrebrenica.aspx |title=State responsible for death of three Muslim men in Srebrenica |date=6 September 2013 |website=De Rechtspraak |url-status=dead |archive-url=https://web.archive.org/web/20130911003011/http://www.rechtspraak.nl/Organisatie/Hoge-Raad/Nieuws/Pages/StateresponsiblefordeathofthreeMuslimmeninSrebrenica.aspx |archive-date=11 September 2013 }}</ref> a judgment the government accepted.<ref>{{cite news |url=http://www.nrc.nl/nieuws/2013/09/06/rutte-kabinet-zal-handelen-naar-uitspraak-hoge-raad-over-srebrenica/ |title=Kabinet zal handelen naar uitspraak Hoge Raad over Srebrenica |date=6 September 2013 |newspaper=NRC |last=Huiskamp |first=Frank |language=nl}}</ref> The court found it was the government which had "effective control" over its troops.<ref name="vandeBilt" /><ref>{{cite web|last=Boon|first=Kristen|title=Supreme Court Decision Rendered in Dutchbat Case: the Netherlands Responsible|date=6 September 2013|work=Opinio Juris|access-date=15 July 2015|url=http://opiniojuris.org/2013/09/06/supreme-court-decision-srebrenica-massacre-netherlands-responsible/}}</ref> The ruling meant relatives could pursue the government for compensation.<ref>{{cite news|title=Dutch state liable for three Srebrenica deaths – court|work=BBC News|date=6 September 2013|access-date=14 July 2015|url=https://www.bbc.co.uk/news/world-europe-23986063}}</ref>

On 16 July 2014, a Dutch court held the Netherlands liable for the killings of more than 300 Bosniaks, who had been expelled from the compound and the state was not liable for other deaths.<ref name="BBC liable for 300">{{cite web |url=https://www.bbc.com/news/world-europe-28313285 |title=Dutch state 'liable' for 300 Srebrenica deaths |publisher=BBC |date=16 July 2014 |access-date=17 July 2015}}</ref> The decision was upheld by The Hague appeals court in 2017.<ref>{{cite web|url=http://www.aljazeera.com/news/2017/06/court-dutch-partially-liable-srebrenica-victims-170627091809549.html|title=Court: Dutch partially liable for Srebrenica deaths|work=Al Jazeera|date=27 June 2017}}</ref><ref>{{cite news|url=https://www.telegraph.co.uk/news/2017/06/27/court-rules-dutch-state-partly-blame-death-350-bosnian-muslims/ |archive-url=https://ghostarchive.org/archive/20220111/https://www.telegraph.co.uk/news/2017/06/27/court-rules-dutch-state-partly-blame-death-350-bosnian-muslims/ |archive-date=11 January 2022 |url-access=subscription |url-status=live|title=Court rules Dutch state partly to blame for death of 350 Bosnian Muslims in Srebrenica|newspaper=The Telegraph|date=27 June 2017}}{{cbignore}}</ref> On 19 July 2019 the Supreme Court ruled the Dutch state was liable for 10%, for the 350 Bosnian men expelled from the compound. The 10% liability was the court's assessment of the likelihood the soldiers could have prevented the killings.<ref name="auto">{{Cite news | url=https://www.rechtspraak.nl/Organisatie-en-contact/Organisatie/Hoge-Raad-der-Nederlanden/Nieuws/Paginas/Nederlandse-Staat-in-zeer-beperkte-mate-aansprakelijk-in-zaak-Mothers-of-Srebrenica.aspx |title = Nederlandse Staat in zeer beperkte mate aansprakelijk in zaak 'Mothers of Srebrenica'|website=De Rechtspraak}}</ref><ref name=":2" />

== Analyses ==
=== Role of Bosnian forces ===
In response to the suggestion Bosniak forces in Srebrenica made no adequate attempt to defend the town, a report by the UN Secretary-General delivered to the [[United Nations General Assembly]] in 1999 states:

{{blockquote|
... military experts ... were largely in agreement that the Bosniaks could not have defended Srebrenica for long ... Many have accused the Bosniak forces of withdrawing from the enclave as the Serb forces advanced on the day of its fall. However ... the Dutchbat Commander urged the Bosniaks to withdraw from defensive positions south of Srebrenica town—the direction from which the Serbs were advancing ... because he believed that NATO aircraft would soon be launching widespread air strikes against the advancing Serbs.

A third accusation levelled at the Bosniak defenders of Srebrenica is that they provoked the Serb offensive by attacking out of that safe area ... there is no credible evidence to support it. Dutchbat personnel on the ground at the time assessed that the few "raids" the Bosniaks mounted out of Srebrenica were of little or no military significance. These raids were often organised in order to gather food, as the Serbs had refused access for humanitarian convoys into the enclave. Even Serb sources ... acknowledged that the Bosniak forces in Srebrenica posed no significant military threat to them. The biggest attack the Bosniaks launched out of Srebrenica ... appears to have been the raid on the village of Višnjica, on 26 June 1995, in which several houses were burned, up to four Serbs were killed and approximately 100 sheep were stolen. In contrast, the Serbs overran the enclave two weeks later, driving tens of thousands from their homes, and summarily executing thousands of men and boys. The Serbs repeatedly exaggerated the extent of the raids out of Srebrenica as a pretext for the prosecution of a central war aim: to create a geographically contiguous and ethnically pure territory along the Drina, while freeing their troops to fight in other parts of the country. The extent to which this pretext was accepted at face value by international actors and observers reflected the prism of "moral equivalency" through which the conflict in Bosnia was viewed by too many for too long.<ref name="UN report" />}}

=== Disputed Serb casualties ===
Serbs suffered casualties during military forays led by [[Naser Orić]]. The controversy over the nature and number of casualties came to a head in 2005.<ref name="hrw-oric">{{Cite web|url=https://www.hrw.org/news/2006/07/11/orics-two-years|title=Oric's Two Years | Human Rights Watch|date=11 July 2006}}</ref> According to [[Human Rights Watch]], the ultra-nationalist [[Serbian Radical Party]] "launched an aggressive campaign to prove that Muslims had committed crimes against thousands of Serbs in the area" which "was intended to diminish the significance of the July 1995 crime."<ref name="hrw-oric" /> A briefing by the ICTY Office of the Prosecutor (OTP) from July 2005 noted Serb deaths in the region alleged by Serbian authorities had increased from 1,400 to 3,500, a figure the OTP stated does, "not reflect the reality."<ref name="icty-july05">{{cite web|url=https://www.un.org/icty/briefing/2005/PB050706.htm |title=ICTY Weekly Press Briefing, July 2005 |publisher=United Nations |date=5 March 2007 |access-date=26 May 2011 |url-status=dead |archive-url=https://web.archive.org/web/20090510013827/http://www.un.org/icty/briefing/2005/PB050706.htm |archive-date=10 May 2009 }}</ref> The briefing cited previous accounts:
* The Republika Srpska's Commission for War Crimes gave the number of Serb victims as 995; 520 in Bratunac and 475 in Srebrenica.
* ''The Chronicle of Our Graves'' by Milivoje Ivanišević, president of the Belgrade Centre for Investigating Crimes Committed against the Serbs, estimated around 1,200.
* ''For the Honourable Cross and Golden Freedom'', a book published by the RS [[Ministry of Interior]], referred to 641 Serb victims

The accuracy of these numbers is challenged: the OTP noted that although Ivanišević's book estimated around 1,200 Serbs were killed, personal details were only available for 624.<ref name="icty-july05" /> The validity of labeling some casualties as "victims" is also challenged:<ref name="icty-july05" /> studies have found a significant majority of military, compared to civilian casualties.<ref name="RDC">[http://www.idc.org.ba/project/the_myth_of_bratunac.html RDC. "The Myth Of Bratunac: A Blatant Numbers Game"] {{webarchive |url=https://web.archive.org/web/20090508095038/http://www.idc.org.ba/project/the_myth_of_bratunac.html |date=8 May 2009 }}</ref> This is in line with the nature of the conflict—Serb casualties died in raids by Bosniak forces on outlying villages used as military outposts for attacks on Srebrenica.<ref>{{cite web|last=Sadović|first=Merdijana|date=4 November 2005|url=https://iwpr.net/global-voices/oric-3|title=Courtside: Oric|id=TU No 428|work=Institute for War and Peace Reporting|access-date=5 May 2017}}</ref><ref>{{cite web|url=http://www.hdmagazine.com/bosnia/census/cens-si.html |title=Bosnian Congress—census 1991—Northeast of Bosnia |publisher=Hdmagazine.com |access-date=26 May 2011 |url-status=dead |archive-url=https://web.archive.org/web/20110711154500/http://www.hdmagazine.com/bosnia/census/cens-si.html |archive-date=11 July 2011 }}</ref> For example, Kravica was attacked by Bosniak forces on [[Eastern Orthodox Church|Orthodox]] Christmas Day, 7 January 1993. Some Serb sources, such as Ivanišević, allege the village's 353 inhabitants were "virtually completely destroyed".<ref name="icty-july05" /> In fact, VRS' own records state 46 Serbs died,<ref>{{cite web |url=http://www.idc.org.ba/project/the_myth_of_bratunac.html |title=VRS, "Warpath of the Bratunac brigade", cited in: RDC. "The Myth Of Bratunac: A Blatant Numbers Game" |access-date=22 December 2010 |archive-url=https://web.archive.org/web/20080511233924/http://www.idc.org.ba/project/the_myth_of_bratunac.html  |archive-date=11 May 2008 }}</ref> while the OTP's investigation also found 43 people were killed.<ref>{{cite web |url=http://www.icty.org/en/press/icty-weekly-press-briefing-6th-jul-2005 |title=ICTY Weekly Press Briefing |date=6 July 2005 |author=Florence Hartmann, Spokesperson for the Office of the Prosecutor |access-date=22 April 2017}}</ref> Nevertheless, the event continues to be cited by Serb sources as the key example of crimes committed by Bosniak forces around Srebrenica.<ref name="hrw-oric" /> As for casualties in Kravica, [[Šiljkovići|Šiljković]], [[Bjelovac]], [[Fakovići]] and [[Sikirić]], the judgement states that the prosecution failed to present convincing evidence the Bosnian forces were responsible, because the Serb forces used artillery in the fighting in those villages. In the case of Bjelovac, Serbs even used warplanes.<ref>{{Cite web|url=http://www.sense-agency.com/en/stream.php?sta=3&pid=8198&kat=3|archive-url=https://web.archive.org/web/20090303224423/http://www.sense-agency.com/en/stream.php?sta=3&pid=8198&kat=3|url-status=dead|title=ICTY: Naser Orić verdict|archive-date=3 March 2009}}</ref> Another analysis was by the [[Research and Documentation Center in Sarajevo]], a non-partisan institution, whose data have been evaluated by international experts.<ref name="RDC" /><ref>Heil, Rebekah (23 June 2007). [http://iwpr.net/report-news/bosnias-book-dead "Bosnia's "Book of the Dead""] {{Webarchive|url=https://web.archive.org/web/20130710193617/http://iwpr.net/report-news/bosnias-book-dead |date=10 July 2013 }}, ''Institute for War & Peace Reporting''. Retrieved 31 July 2008.</ref><ref>[http://www.norveska.ba/press/rdc-bbd.htm RDC Norway] {{webarchive |url=https://web.archive.org/web/20070719175542/http://www.norveska.ba/press/rdc-bbd.htm |date=19 July 2007 }}—The Bosnian Book of Dead (short analysis)</ref> Its review found Serb casualties in the Bratunac municipality amounted to 119 civilians and 424 soldiers. It established that, although the 383 Serb victims buried in the Bratunac military cemetery are presented as casualties of ARBiH units from Srebrenica, 139, about a third, had fought and died elsewhere.<ref name="RDC" />

Serb sources maintain that casualties prior to the creation of the safe area gave rise to Serb demands for revenge against the Bosniaks based in Srebrenica. The ARBiH raids are presented as a key motivating factor for the genocide.<ref>Serbs accuse world of ignoring their suffering, AKI, 13 July 2006</ref> This view is echoed by international sources, including the 2002 report commissioned by the Netherlands.<ref name="NIOD-Appendix-IV">J.C.H. Blom et al. (2002) [http://www.cnj.it/documentazione/Srebrenica/NIOD/NIOD%20part%20IV.pdf NIOD Report: Srebrenica. Reconstruction, background, consequences and analyses of the fall of a Safe Area] {{webarchive|url=https://web.archive.org/web/20150923205116/http://www.cnj.it/documentazione/Srebrenica/NIOD/NIOD%20part%20IV.pdf |date=23 September 2015 }} (Appendix IV, History and Reminders in East Bosnia)</ref> Paul Mojzes notes much animosity towards the men of Srebrenica stems from May 1992 to January 1993, where forces under Orić's leadership attacked and destroyed scores of Serbian villages. Evidence indicated Serbs had been tortured, mutilated and others burned alive, when their houses were torched.<ref name="Mojzes">{{cite book |last1=Mojzes |first1=Paul |title=Balkan Genocides: Holocaust and Ethnic Cleansing in the Twentieth Century |date=2011 |publisher=Rowman & Littlefield |isbn=978-1-44220-665-6 |page=179 |url=https://books.google.com/books?id=m5bDUFZXQ5sC&pg=PA179}}</ref>

The efforts to explain the massacre as motivated by revenge have been dismissed as bad faith attempts to justify the genocide.<ref>{{Cite web|url=https://www.icty.org/sid/9762/en|title=ICTY Weekly Press Briefing – 14th Mar 2007 | International Criminal Tribunal for the former Yugoslavia|website=www.icty.org}}</ref> The ICTY Outreach Programme notes that the claim Bosnian Serb forces killed the prisoners, in revenge for crimes by Bosnian Muslims, provides no defence under law.

===Lack of military logic===
During [[Radislav Krstić]]'s trial, the prosecution's [[military advisor]], Richard Butler, pointed out that by carrying out a mass execution, the Serb Army deprived themselves of an extremely valuable bargaining counter. Butler suggested that they would have had far more to gain had they taken the men in Potočari as prisoners of war, under the supervision of the [[International Committee of the Red Cross]] and UN troops. It might then have been possible to enter into an exchange deal or they might have been able to force [[concession (politics)|political concessions]]. Based on this reasoning, the ensuing mass murder defied military explanation.<ref name="Krstic 2001 takeover" />

===Dutchbat===
{{See also|#Netherlands}}

[[Brigadier General]] [[Hagrup Haukland]] was [[UNPROFOR]]'s commander of<ref>{{cite web|title=The fall of Srebrenica|url=http://www.juliagorin.com/images/niod/p3_c09_s006_b01.html|publisher=[[Netherlands Institute for War Documentation]]|access-date=2013-04-10|page=Part III chapter 9.6}}</ref> the sector in which the killings started on 11 July, when he was on vacation.<ref name="vg.no">{{cite web|url=http://www.vg.no/nyheter/utenriks/artikkel.php?artid=106292|title=Norsk oberst får kritikk etter folkemord|author=Andreas Arnseth|work=VG|date=29 July 2005 }}</ref> His subordinate, Colonel Brantz, phoned Haukland twice on 9 July about the crisis.<ref name="Eraker">{{cite web|title=Forsøker å skjule sannheten |url=https://www.nytid.no/blank-118/ |work=[[Ny Tid]] |date=24 September 2005 |url-status=live |archive-url=https://web.archive.org/web/20140222142827/http://www.nytid.no/arkiv/artikler/20050921/blank/ |archive-date=22 February 2014 |access-date=25 April 2017 }}</ref> Confusion within Haukland's staff has been attributed in part, to his being slow<ref name="vg.no" /> to return to his place of work.<ref name="Eraker" /> The 2002 report ''Srebrenica: a 'safe' area'', and a military advisor,<ref name="Eraker" /> said "The cadres consisted of clans of [[Norwegians|Norwegian]], Pakistani and Dutch military that were incapable of adequate mutual cooperation."<ref>{{Cite web|url=http://www.juliagorin.com/images/niod/p3_c06_s003_b01.html|title=Srebrenica: Part 3, Chapter 6, Section 3|website=www.juliagorin.com}}</ref> The report did not assign any blame to Haukland for the massacre. In 2005 an unnamed officer on Haukland's staff, disputed the claim by Haukland and Norway's [[Chief of Defence of Norway|Chief of Defence]], [[Arne Solli]], that the attack was a surprise.<ref name=Eraker/> The officer said "We knew early on that the Serbs were amassing their forces around Srebrenica. At the end of June, Haukland informed the headquarters at Sarajevo again and again...".<ref name=Eraker/> In 2006 it was reported Haukland regularly informed Sollie about...Haukland's sector, and when Haukland departed Bosnia on his vacation to Norway, they travelled on the same plane.<ref>{{cite news|url=https://www.nytid.no/styrken_som_sviktet/|title=Styrken som sviktet|work=[[Ny Tid]]|date=13 July 2006|access-date=25 April 2017}}</ref>

In 2010, [[John J. Sheehan|John Sheehan]], NATO's [[Supreme Allied Commander Atlantic]] (1994–97), told the US Senate that the Dutch had "made a conscious effort to socialise their military...it includes open homosexuality", claiming gay soldiers could result in events like Srebrenica.<ref name=":4">{{cite news|author=Ian Traynor |url=https://www.theguardian.com/world/2010/mar/19/gay-dutch-soldiers-srebrenica?showCommentBox=true#post-area |title=US general: Gay Dutch soldiers caused Srebrenica massacre |work=The Guardian |location=UK |access-date=3 June 2011 |date=19 March 2010}}</ref> He claimed his opinion was shared by Dutch military leadership, mentioning "Hankman Berman", who Sheehan said had told him the presence of gay soldiers had contributed to the disaster.<ref>{{cite web|url=https://www.youtube.com/watch?v=0V7Or0kbqY0 | archive-url=https://ghostarchive.org/varchive/youtube/20211030/0V7Or0kbqY0| archive-date=2021-10-30|title=General Sheehan: gays responsible for Srebrenica massacre | date=18 March 2010|publisher=YouTube |access-date=26 May 2011}}{{cbignore}}</ref> General [[Henk van den Breemen]] denied saying this and called Sheehan's comments "total nonsense"; the Dutch authorities described them as "disgraceful" and "unworthy of a soldier".<ref>{{cite web |url=http://www.nrc.nl/international/article2507443.ece/Former_US_general__gays_make_Dutch_military_weak |title=Former US general: 'gays make Dutch military weak' |work=NRC Handelsblad |access-date=3 June 2011 |archive-url=https://web.archive.org/web/20100322230137/http://www.nrc.nl/international/article2507443.ece/Former_US_general__gays_make_Dutch_military_weak |archive-date=22 March 2010 |url-status=dead |df=dmy-all }}</ref><ref>{{cite news|author=Ian Traynor |url=https://www.theguardian.com/world/2010/mar/19/gay-dutch-soldiers-srebrenica |title=US general: Gay Dutch soldiers caused Srebrenica massacre |work=The Guardian |location=UK |access-date=26 May 2011 |date=19 March 2010}}</ref><ref>{{cite web |url=http://www.javno.com/en-world/pm-slams-disgraceful-srebrenica-gay-comments/298611 |title=PM slams 'disgraceful' Srebrenica gay comments |publisher=Javno.com |date=19 March 2010 |access-date=26 May 2011 |archive-url=https://web.archive.org/web/20121019212358/http://dalje.com/en-world/pm-slams-disgraceful-srebrenica-gay-comments/298611 |archive-date=19 October 2012 |url-status=dead |df=dmy-all }}</ref><ref>{{cite news|url=http://news.bbc.co.uk/2/hi/8575717.stm |title=Dutch fury at US general's gay theory over Srebrenica |work=BBC News |date=19 March 2010 |access-date=26 May 2011}}</ref><ref name=":4" /> Sheehan apologised to Dutch military officials and blamed instead "the rules of engagement...developed by a political system with conflicting priorities and an ambivalent understanding of how to use the military."<ref>{{Cite web|url=http://voices.washingtonpost.com/federal-eye/2010/03/ex-general_apologizes_for_dutc.html|archive-url=https://web.archive.org/web/20111130162409/http://voices.washingtonpost.com/federal-eye/2010/03/ex-general_apologizes_for_dutc.html|url-status=dead|archive-date=2011-11-30|title=Federal Eye – Ex-general apologizes for Dutch gay soldier remark|date=30 November 2011}}</ref>

===Criticism of the UN Special Representative===
The Dutch government report from 2002, ''Srebrenica: a 'safe' area'', criticised the choice of Stoltenberg as a mediator.<ref>{{cite web|url=http://publications.niod.knaw.nl/publications/srebrenicareportniod_en.pdf#page=753|title=Srebrenica – Reconstruction, background, consequences and analyses of the fall of a 'safe' area |page=753|author=NIOD Institute for War-, Holocaust- and Genocide Studies|date=2002}}</ref><ref name="aftenposten.no"/> In 2005, Professor [[Arne Johan Vetlesen]] said, "[[Thorvald Stoltenberg]]'s co-responsibility in Srebrenica boils down to the fact that, over three years as a top mediator, he helped to create a climate—diplomatically, politically and indirectly militarily—that was such that Mladic calculated correctly, when he figured he could do exactly as he wanted with Srebrenica's Muslim population".<ref name="aftenposten.no">{{cite news|url=https://www.aftenposten.no/verden/i/5B08W/stoltenberg-har-medansvar-for-srebrenica|title=Stoltenberg shares responsibility for Srebrenica|language=Norwegian|work=Aftenposten|date=5 August 2005|access-date=26 April 2017}}</ref>

== Denial ==
{{Main|Bosnian genocide denial}}
Scepticism has ranged from challenging judicial recognition of the killings as genocide, to the denial of a massacre having taken place. The finding of genocide by the ICJ and ICTY, has been disputed on evidential and theoretical grounds. The number of the dead has been questioned as has the nature of their deaths. It has been alleged that considerably fewer than 8,000 were killed and/or that most died in battle, rather than execution. It has been claimed the interpretation of "genocide" is refuted by the survival of the women and children.<ref>Examples
* [http://128.121.186.47/ISSA/reports/Balkan/Sep1903.htm Report of ''International Strategic Studies Association'' (ISSA)] {{webarchive|url=https://swap.stanford.edu/20090419003619/http://128.121.186.47/ISSA/reports/Balkan/Sep1903.htm |date=19 April 2009 }}, claimed that the "alleged casualty number of 7,000 victims [is] vastly inflated and unsupported by evidence"
* [http://www.mail-archive.com/serbian_way@antic.org/msg00008.html The real story behind Srebrenica] by the former UNPROFOR commander, Gen. Lewis MacKenzie, ''The Globe and Mail'', 14 July 2005.
* [http://www.slobodan-milosevic.org/news/smorg-sreb101604.htm "The Forbidden Srebrenica report"], report denying the Srebrenica massacre issued by the Republika Srpska Bureau for Cooperation with the ICTY in September 2002.
* [http://arquivo.pt/wayback/20091001191412/http://www.zmag.org/content/showarticle.cfm?ItemID=8244 The Politics of the Srebrenica Massacre] article argues that only some Bosniaks were executed, most died in battle, and some of the bodies in mass graves are actually Serbs, by Edward S. Herman, ''Znet'', 7 July 2005</ref>

During the war, Milošević had effective control of most Serbian media.<ref>{{cite web|url=https://www.icty.org/x/cases/slobodan_milosevic/ind/en/mil-ii011122e.htm |title=ICTY Indictment of Milosevic, clause 25, section g |publisher=United Nations |date=5 March 2007 |access-date=26 May 2011 |url-status=dead |archive-url=https://web.archive.org/web/20040307125137/http://www.un.org/icty/indictment/english/mil-ii011122e.htm |archive-date=7 March 2004 }}</ref><ref>{{cite web|last=Brosse|first=Renaud de la|title=Political Propaganda and the Plan to Create 'A State For All Serbs:' Consequences of using media for ultra-nationalist ends|url=http://hague.bard.edu/reports/de_la_brosse_pt4.pdf|page=paragraph 74}}</ref> Following its end, denial of Serbian responsibility for the killings was widespread among Serbians.<ref>{{cite news|url=https://www.theguardian.com/world/2005/jun/05/balkans.warcrimes|location=London|work=The Guardian|title=How video that put Serbia in dock was brought to light|author=Tim Judah and Daniel Sunter|date=4 June 2005|access-date=2 August 2012}}</ref><ref>{{Cite web|url=https://www.blic.rs/vesti/politika/istina-o-srebrenici-tek-treba-da-bude-utvrdena/eqsxexc|title=Istina o Srebrenici tek treba da bude utvrđena|website=Blic.rs|date=25 November 2015 }}</ref> [[Sonja Biserko]] and Edina Bečirević, have pointed to a culture of denial of the genocide in Serbian society.<ref>[http://www.bosnia.org.uk/news/news_body.cfm?newsid=2638 Denial of genocide – on the possibility of normalising relations in the region] {{Webarchive|url=https://web.archive.org/web/20160303231517/http://www.bosnia.org.uk/news/news_body.cfm?newsid=2638 |date=3 March 2016 }} by Sonja Biserko (the Helsinki Committee for Human Rights in Serbia) and Edina Bečirević (Faculty of Criminology and Security Studies of the University of Sarajevo).</ref>

== See also ==
{{colbegin|colwidth=20em}}
* [[12 April 1993 Srebrenica shelling]]
* ''[[A Cry from the Grave]]''
* [[Dubh (ar thitim Shrebenice, 11ú Iúil, 1995)]] poem
* ''[[The Enclave]]''
* "''[[Nož, žica, Srebrenica]]''"
* ''[[Overture (novel)|Overture]]''
* [[Persecution of Muslims]]
* [[Propaganda during the Yugoslav Wars]]
* ''[[Quo Vadis, Aida?]]''
* [[Serbia in the Yugoslav Wars#War crimes|Serbian war crimes in the Yugoslav Wars]]
* ''[[Silvertown (drama)|Silvertown]]''
* ''[[A Town Betrayed]]''
* [[Zaklopača massacre]]
{{colend}}

== Notes ==
{{notelist}}

== References ==
{{Reflist|refs=
<ref name="AP-2008-07-29">
{{cite news
| url = http://usatoday30.usatoday.com/news/world/2008-07-29-bosnia-warcrimes_n.htm
| title = 7 Bosnian Serbs guilty of genocide in Srebrenica
| work = [[USA Today]]
| agency = Associated Press
| location = Sarajevo
| author = Aida Cerkez-Robinson
| date = 29 July 2008
| archive-url = https://web.archive.org/web/20160217151144/http://usatoday30.usatoday.com/news/world/2008-07-29-bosnia-warcrimes_n.htm
| archive-date = 17 February 2016
| url-status = live
}}
</ref>
}}

=== Bibliography ===
* {{cite book |last1=Danner |first1=Mark |url=https://archive.org/details/strippingbarebod00dann |title=Stripping Bare the Body: Politics, Violence, War |date=1 October 2009 |publisher=Nation Books |isbn=9781568584133 |pages=[https://archive.org/details/strippingbarebod00dann/page/232 232]–[https://archive.org/details/strippingbarebod00dann/page/233 233] |url-access=registration}}
* {{cite book
| last = Ingrao
| first = Charles
| title = Confronting the Yugoslav Controversies: A Scholars' Initiative
| year = 2012
| publisher = Purdue University Press
| location = [[West Lafayette, Indiana]]
| isbn = 978-1-55753-617-4
| url = https://books.google.com/books?id=IDMhDgCJCe0C
}}
* {{cite book|last=Michas|first=Takis|title=Unholy Alliance: Greece and Milosevic's Serbia in the Nineties|year=2002|publisher=Texas A&M University Press|isbn=1-58544-183-X|ref=refMichas2002|url=https://archive.org/details/unholyalliance00taki}}
* {{cite book|last=Ramet|first=Sabrina P.|title=The Three Yugoslavias: State-Building And Legitimation, 1918–2005|year=2006|publisher=Indiana University Press|isbn=0-253-34656-8|ref=refRamet2006}}

== Further reading ==
{{Commons}}

'''National institutions'''
* Staff. [https://web.archive.org/web/20060807190123/http://www.ushmm.org/conscience/analysis/analysis_all.php?category=01-locations&topic=03-balkans Committee on Conscience Balkans Section]—[[U.S. Holocaust Memorial Museum]]
* [https://web.archive.org/web/20080511233924/http://www.idc.org.ba/project/the_myth_of_bratunac.html  Myth of Bratunac A blatant numbers game] [https://web.archive.org/web/20080501005423/http://www.idc.org.ba/aboutus.html Sarajevo-based Research & Documentation Centre]. Retrieved 16 March 2008.
* Srebrenica Genocide videos documented by the [[United States Holocaust Memorial Museum]], compiled by [http://www.srebrenicagenocide.org/ SrebrenicaGenocide.org] {{Webarchive|url=https://web.archive.org/web/20150220043833/http://www.srebrenicagenocide.org/ |date=20 February 2015 }}.

'''Academic articles'''
* Brunborg, H., Lyngstad, T.H. and Urdal, H. (2003): Accounting for genocide: How many were killed in Srebrenica? ''European Journal of Population'', 19(3):229–248. {{doi|10.1023/A:1024949307841}}
* Honig, Jan Willem. "Strategy and genocide: Srebrenica as an analytical challenge." Southeast European and Black Sea Studies 7.3 (2007): 399–416.
* David MacDonald, ([[University of Otago]]). [http://epress.lib.uts.edu.au/ojs/index.php/portal/article/viewFile/90/58 Globalizing the Holocaust: A Jewish ‘useable past’ in Serbian Nationalism (PDF)] {{Webarchive|url=https://web.archive.org/web/20220109204501/https://epress.lib.uts.edu.au/ojs/index.php/portal/article/viewFile/90/58 |date=9 January 2022 }}, PORTAL: Journal of Multidisciplinary International Studies Vol. 2, No. 2 July 2005 {{ISSN|1449-2490}}
* Miller, Paul B. "Contested memories: the Bosnian genocide in Serb and Muslim minds." Journal of Genocide Research 8.3 (2006): 311–324.
* Mulaj, Klejda. "Genocide and the ending of war: Meaning, remembrance and denial in Srebrenica, Bosnia." ''Crime, Law and Social Change'' 68.1–2 (2017): 123–143. [https://ore.exeter.ac.uk/repository/bitstream/handle/10871/27475/KM_Srebrenica_8_May_2017_Clean_Copy.pdf?sequence=2&isAllowed=y online]
* Jasmina Besirevic Regan, [https://web.archive.org/web/20070108090150/http://www.yale.edu/gsp/former_yugoslavia/index.html Genocide Studies Program: Former Yugoslavia] [[Yale University]]. Retrieved 16 March 2008
* van der Wilt, Harmen. "Srebrenica: on joint criminal enterprise, aiding and abetting and command responsibility." ''Netherlands International Law Review'' 62.2 (2015): 229–241. [https://link.springer.com/article/10.1007/s40802-015-0036-8 online]
* Ryngaert, Cedric, and Nico Schrijver. "Lessons learned from the Srebrenica massacre: from UN peacekeeping reform to legal responsibility." ''Netherlands international law review'' 62.2 (2015): 219–227. [https://link.springer.com/article/10.1007/s40802-015-0034-x online]
* Karčić, Hamza. "Remembering by resolution: the case of Srebrenica." ''Journal of Genocide Research'' 17.2 (2015): 201–210.
* Heynders, Odile. "Speaking the Self, Narratives on Srebrenica." ''European Journal of Life Writing'' 3 (2014): 1–22. [https://ejlw.eu/article/download/31423/28713 online]
* Gibbs, David N. "How the Srebrenica massacre redefined US foreign policy." ''Class, Race and Corporate Power'' 3.2 (2015): 5. [https://digitalcommons.fiu.edu/cgi/viewcontent.cgi?article=1061&context=classracecorporatepower online]

'''Books'''
* Allen, Beverly. ''Rape Warfare: The Hidden Genocide in Bosnia-Herzegovina and Croatia''. University of Minnesota Press, 1996.{{ISBN|0816628181}}.
* Bartrop. Paul R. ''Bosnian genocide: the essential reference guide''. ABC-CLIO LLC, 2016. {{ISBN|978-1-4408-3868-2}}
* Cigar, Norman. ''Genocide in Bosnia: The Policy of "Ethnic Cleansing"''. Texas A & M University Press, 1995.{{ISBN|0890966389}}.
* Cushman, Thomas and Stjepan G. Meštrović. ''This Time We Knew: Western Responses to Genocide in Bosnia''. New York University Press, 1996. {{ISBN|0814715346}}.
*Delpla, Isabelle, Xavier Bougarel, and Jean-Louis Fournel. ''Investigating Srebrenica: Institutions, Facts, Responsibilities''. Berghahn Books. {{ISBN|978-0-85745-472-0}}
*Drakulić, Slavenka. ''They Would Never Hurt a Fly: War Criminals on Trial in The Hague''. Penguin Books, 2003. {{ISBN|9780143035428}}.
* Gutman, Roy. ''[https://archive.org/details/witnesstogenocid0000gutm/mode/2up A Witness to Genocide]''. Lisa Drew Books, 1993. {{ISBN|9780020329954}}.
* Lebor, Adam. ''"Complicity with Evil": The United Nations in the Age of Modern Genocide''. Yale University Press, 2006. {{ISBN|0-300-11171-1}}.
*Leydesdorff, Selma (translated by Kay Richardson) ''Surviving the Bosnian Genocide: The Women of Srebrenica Speak''. Indiana University Press, 2011, {{ISBN|978-0253356697}}
* Nettelfield, Lara J. and Sarah E. Wagner. ''Srebrenica in the Aftermath of Genocide''. Cambridge University Press, 2013. {{ISBN|978-1-107-00046-9}}.
* [[David Rohde|Rohde, David]]. ''Endgame: The Betrayal and Fall of Srebrenica, Europe's Worst Massacre Since World War II''. Penguin Books, 2012. {{ISBN|9780143120315}}.
* Suljagić, Emir. [[Postcards from the Grave]], Saqi Books, 2005. {{ISBN|0-86356-519-0}}.

'''Reports'''
* [[Kofi Annan|Annan, Kofi]] [https://undocs.org/A/54/549 "Report of the Secretary-General pursuant to General Assembly resolution 53/35—The fall of Srebrenica", United Nations A/54/549 (15 November 1999)]
* Blom, J.C.H. et al. (2002) Prologue [http://www.cnj.it/documentazione/Srebrenica/NIOD/NIOD%20Prologue.pdf NIOD Report: Srebrenica. Reconstruction, background, consequences and analyses of the fall of a Safe Area] – The Dutch government's investigation of the massacre and of Dutch UN troops' role, April 2002 
* Centre for Interdisciplinary Postgraduate Studies /University of Sarajevo – [http://www.war-crimes-genocide-memories.org/books.shtml War Crimes, Genocide and Memories: The Roots of Evil: I Want to Understand]
* Ewa Tabeau, [http://www.helsinki.org.rs/doc/testimonies33.pdf Conflict in Numbers: Casualties of the 1990s Wars in the Former Yugoslavia (1991–1999)], published by the Helsinki Committee for Human Rights in Serbia
* ICTY Outreach Programme [http://www.icty.org/x/file/Outreach/view_from_hague/jit_srebrenica_en.pdf Facts About Srebrenica]

'''News media'''
* Slavenka Drakulic [https://web.archive.org/web/20160404194819/http://www.eurozine.com/articles/2004-02-12-drakulic-en.html Triumph of Evil], [[Eurozine]], 12 February 2004
* [https://www.docsonline.tv/topic/the-srebrenica-genocide/ SREBRENICA Genocide], DocsOnline, 2020
* Staff [http://www.balkaninsight.com/en/article/new-srebrenica-genocide-trial-begins-in-sarajevo New Srebrenica Genocide Trial Begins in Sarajevo] from [[Balkan Insight]], 17 October 2012
* Udo Ludwig and Ansgar Mertin. [http://www.spiegel.de/international/spiegel/0,1518,425024,00.html A Toast to the Dead: Srebrenica Widows Sue U.N., Dutch Government]—[[Der Spiegel]], 4 July 2006.
* Murat Karaali. [https://web.archive.org/web/20070930181029/http://www.turkishdailynews.com.tr/archives.php?id=5626 The secret killings grounds of Srebrenica], [[Turkish Daily News]], 10 January 1998.
* Marlise Simons. [https://web.archive.org/web/20080509125524/http://www.iht.com/articles/2007/04/08/news/serbia.php Serbia's darkest pages hidden from genocide court], [[International Herald Tribune]], 8 April 2007
* Staff. [http://news.bbc.co.uk/2/hi/europe/6138420.stm New Srebrenica mass grave found], BBC News, 11 November 2006
* Staff [https://web.archive.org/web/20170630235702/http://www.fabrika.com/en/?id=1&j=87&t=&m=444&media=wmv Tarik Samarah: Srebrenica] [https://web.archive.org/web/20170630235615/http://www.fabrika.com/en/?id=2&a=6 Fabrika Agency]. Retrieved 16 March 2008
* Staff [https://web.archive.org/web/20061016174356/http://www.sense-agency.com/en/stream.php?kat=3 Sense Tribunal], is a specialised project of [https://web.archive.org/web/20080106184127/http://www.sense-agency.com/en/stream.php?kat=20 Sense News Agency Sense] based in International War Crimes Tribunal for the former Yugoslavia in [[The Hague]]. The focus of this project is regular coverage of the work of the ICTY, and the activities of ICJ (International Court of Justice) and ICC (International Criminal Court). Retrieved 16 March 2008

'''NGOs'''
* Adam Jones. [https://web.archive.org/web/20000816221116/http://www.gendercide.org/case_srebrenica.html Case Study: The Srebrenica Massacre, July 1995] [[Gendercide Watch]], 1999–2002. Retrieved 16 March 2008
* Emir Suljagic [http://www.globalpolicy.org/intljustice/tribunals/yugo/2003/0619milosreb.htm Milosevic Linked to Srebrenica Massacre-Institute for War and Peace Reporting], from [http://www.iwpr.net Institute for War & Peace Reporting], republished by the [[Global Policy Forum]] 18 June 2003
* Staff. [https://web.archive.org/web/20080406211852/http://www.srebrenica.ba/index.en.php The Association Women of Srebrenica]. Retrieved 16 March 2008
* Staff. [https://web.archive.org/web/20060324190347/http://www.hrw.org/reports98/bosniacw/index.html Chemical Warfare in Bosnia? The Strange Experiences of the Srebrenica Survivors], [[Human Rights Watch]], Vol. 10, No.9 (D) November 1998
* [http://advocacynet.org/ The Advocacy Project], 2009 Peace Fellows [https://web.archive.org/web/20161019083817/http://advocacynet.org/wordpress-mu/alisonsluiter/ Alison Sluiter] and [https://web.archive.org/web/20160827235917/http://advocacynet.org/wordpress-mu/kbristow/ Kelsey Bristow] blogs, in partnership with [http://www.advocacynet.org/page/bosfam Bosnian Family (BOSFAM)]

'''Other'''
* Yves Billy, Gilles Hertzog [https://web.archive.org/web/20060602131147/http://www.documen.tv/asset/Srebrenica.html Documentary: Srebrenica, an Orchestrated Tragedy], [https://web.archive.org/web/20080404022048/http://www.documen.tv/web-html/html/html_index.htm www.documen.tv]
* Frankti. {{usurped|1=[https://archive.today/20121209103846/http://www.domovina.net/srebrenica/page_006.php Srebrenica:: Investigations, Reports, Books]}}, {{usurped|1=[https://archive.today/20130112073325/http://www.domovina.net/index.php Domovina Net]}}. Retrieved 16 March 2008.
* Merdijana Sadović. [https://iwpr.net/global-voices/srebrenica-status-question-wont-go-away Srebrenica Status Question Won't Go Away]. [http://www.iwpr.net Institute for War & Peace Reporting] TU No 503, 25 May 2007. —Bosniak returnees to Srebrenica now hope international community will back demand for town to be separated from Republika Srpska
* Staff. [https://web.archive.org/web/20060603005706/http://www.genocid.org/english Bosnian Genocide], [https://web.archive.org/web/20060603005706/http://www.genocid.org/english genocid.org]. Retrieved 16 March 2008.
* Staff. [https://archive.today/20121206003521/http://www.islam.co.ba/inferno/ Srebrenica's Inferno], Srebrenica in photographs. Retrieved 16 March 2008.
* [http://srebrenicamassacre1995.wordpress.com/2008/01/25/partial-list-of-child-victims-of-srebrenica-genocide/ Partial list of child victims of the Srebrenica massacre]

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Decimal floating point

2025-06-20T11:40:44Z

173.79.19.248: is not and has never been such an article

{{Short description|Decimal representation of real numbers in computing}}
{{use dmy dates|date=August 2019|cs1-dates=y}}
{{floating-point}}
{{Computer architecture bit widths}}
'''Decimal floating-point''' ('''DFP''') arithmetic refers to both a representation and operations on [[Decimal data type|decimal]] [[floating-point]] numbers. Working directly with decimal (base-10) fractions can avoid the rounding errors that otherwise typically occur when converting between decimal fractions (common in human-entered data, such as measurements or financial information) and binary (base-2) fractions.

The advantage of decimal floating-point representation over decimal [[Fixed-point arithmetic|fixed-point]] and [[Integer (computer science)|integer]] representation is that it supports a much wider range of values. For example, while a fixed-point representation that allocates 8 decimal digits and 2 decimal places can represent the numbers 123456.78, 8765.43, 123.00, and so on, a floating-point representation with 8 decimal digits could also represent 1.2345678, 1234567.8, 0.000012345678, 12345678000000000, and so on. This wider range can dramatically slow the accumulation of rounding errors during successive calculations; for example, the [[Kahan summation algorithm]] can be used in floating point to add many numbers with no asymptotic accumulation of rounding error.

== {{anchor|64-bit|96-bit}}Implementations ==
Early mechanical uses of decimal floating point are evident in the [[abacus]], [[slide rule]], the [[Smallwood calculator]], and some other [[calculator]]s that support entries in [[scientific notation]]. In the case of the mechanical calculators, the exponent is often treated as side information that is accounted for separately.

The [[IBM 650]] computer supported an 8-digit decimal floating-point format in 1953.<ref name="Beebe_2017"/> The otherwise binary [[Wang VS]] machine supported a 64-bit decimal floating-point format in 1977.<ref name="Savard_2018"/> The [[Motorola 68881]] supported a format with 17 digits of mantissa and 3 of exponent in 1984, with the floating-point support library for the [[Motorola 68040]] processor providing a compatible 96-bit decimal floating-point storage format in 1990.<ref name="Savard_2018"/>

Some [[computer language]]s have implementations of decimal floating-point arithmetic, including [[PL/I]], [[.NET]],<ref>{{cite web | url=https://learn.microsoft.com/en-us/dotnet/api/system.decimal | title=.NET API Documentation for System.Decimal | website=learn.microsoft.com | access-date=7 July 2024}}</ref> [[emacs]] with calc, and [[Python (programming language)|Python]]'s decimal module.<ref>{{cite web | url=https://docs.python.org/3/library/decimal.html | title=Python Documentation for decimal | access-date=7 July 2024 | website=docs.python.org}}</ref>
In 1987, the [[IEEE]] released [[IEEE 854]], a standard for computing with decimal floating point, which lacked a specification for how floating-point data should be encoded for interchange with other systems. This was subsequently addressed in [[IEEE 754-2008]], which standardized the encoding of decimal floating-point data, albeit with two different alternative methods.

IBM [[POWER6]] and newer POWER processors include DFP in hardware, as does the [[IBM System z9]]<ref>{{cite web |url=http://www-306.ibm.com/common/ssi/rep_ca/0/897/ENUS107-190/ENUS107190.PDF |title=IBM z9 EC and z9 BC — Delivering greater value for everyone |website=306.ibm.com |access-date=7 July 2018}}</ref> (and later zSeries machines). SilMinds offers SilAx, a configurable vector DFP [[coprocessor]].<ref>{{cite web |url=http://www.silminds.com/decimal-products/accelerator-cards/76 |title=Arithmetic IPs for Financial Applications - SilMinds |website=Silminds.com}}</ref> [[IEEE 754-2008]] defines this in more detail. [[Fujitsu]] also has 64-bit [[Sparc]] processors with DFP in hardware.<ref name="Fujitsu_2015"/><ref name="Savard_2018"/>

== IEEE 754-2008 encoding ==
The [[IEEE 754-2008]] standard defines 32-, 64- and 128-bit decimal floating-point representations. Like the binary floating-point formats, the number is divided into a sign, an exponent, and a [[significand]]. Unlike binary floating-point, numbers are not necessarily normalized; values with few [[significant digits]] have multiple possible representations: 1×102=0.1×103=0.01×104, etc. When the significand is zero, the exponent can be any value at all.

{|class=wikitable style="text-align:center"
|+IEEE 754-2008 decimal floating-point formats
! [[decimal32 floating-point format|decimal32]] !! [[decimal64 floating-point format|decimal64]] !! [[decimal128 floating-point format|decimal128]] !! decimal(32''k'') !!align=left| Format
|-
| 1 || 1 || 1 || 1
! Sign field (bits)
|-
| 5 || 5 || 5 || 5
! Combination field (bits)
|-
| 6 || 8 || 12 || ''w'' = 2×''k'' + 4
! Exponent continuation field (bits)
|-
| 20 || 50 || 110 || ''t'' = 30×''k''−10
! Coefficient continuation field (bits)
|-
| 32 || 64 || 128 || 32×''k''
! Total size (bits)
|-
| 7 || 16 || 34 || ''p'' = 3×''t''/10+1 = 9×''k''−2
! Coefficient size (decimal digits)
|-
| 192 || 768 || 12288 || 3×2''w'' = 48×4''k''
! Exponent range
|-
| 96 || 384 || 6144 || Emax = 3×2''w''−1
! Largest value is 9.99...×10Emax
|-
| −95 || −383 || −6143 || Emin = 1−Emax
! Smallest normalized value is 1.00...×10Emin
|-
| −101 || −398 || −6176 || Etiny = 2−p−Emax
! Smallest non-zero value is 1×10Etiny
|}

The exponent ranges were chosen so that the range available to normalized values is approximately symmetrical. Since this cannot be done exactly with an even number of possible exponent values, the extra value was given to Emax.

Two different representations are defined:
* One with a '''binary integer significand field''' encodes the significand as a large binary integer between 0 and 10''p''−1. This is expected to be more convenient for software implementations using a binary [[Arithmetic logic unit|ALU]].
* Another with a '''densely packed decimal significand field''' encodes decimal digits more directly. This makes conversion to and from binary floating-point form faster, but requires specialized hardware to manipulate efficiently. This is expected to be more convenient for hardware implementations.

Both alternatives provide exactly the same range of representable values.

The most significant two bits of the exponent are limited to the range of 0−2, and the most significant 4 bits of the significand are limited to the range of 0−9. The 30 possible combinations are encoded in a 5-bit field, along with special forms for infinity and [[NaN]].

If the most significant 4 bits of the significand are between 0 and 7, the encoded value begins as follows:

s 00mmm xxx Exponent begins with 00, significand with 0mmm
s 01mmm xxx Exponent begins with 01, significand with 0mmm
s 10mmm xxx Exponent begins with 10, significand with 0mmm

If the leading 4 bits of the significand are binary 1000 or 1001 (decimal 8 or 9), the number begins as follows:

s 1100m xxx Exponent begins with 00, significand with 100m
s 1101m xxx Exponent begins with 01, significand with 100m
s 1110m xxx Exponent begins with 10, significand with 100m

The leading bit (s in the above) is a sign bit, and the following bits (xxx in the above) encode the additional exponent bits and the remainder of the most significant digit, but the details vary depending on the encoding alternative used.

The final combinations are used for infinities and NaNs, and are the same for both alternative encodings:

s 11110 x ±Infinity (see [[Extended real number line]])
s 11111 0 quiet NaN (sign bit ignored)
s 11111 1 signaling NaN (sign bit ignored)

In the latter cases, all other bits of the encoding are ignored. Thus, it is possible to initialize an array to NaNs by filling it with a single byte value.

=== Binary integer significand field ===
This format uses a binary significand from 0 to 10p−1. For example, the Decimal32 significand can be up to 107−1 = {{gaps|9|999|999}} = 98967F16 = {{gaps|1001|1000100101|10011111112}}. While the encoding can represent larger significands, they are illegal and the standard requires implementations to treat them as 0, if encountered on input.

As described above, the encoding varies depending on whether the most significant 4 bits of the significand are in the range 0 to 7 (00002 to 01112), or higher (10002 or 10012).

If the 2 bits after the sign bit are "00", "01", or "10", then the exponent field consists of the 8 bits following the sign bit (the 2 bits mentioned plus 6 bits of "exponent continuation field"), and the significand is the remaining 23 bits, with an implicit leading 0 bit, shown here in parentheses:

<pre>
s 00eeeeee (0)ttt tttttttttt tttttttttt
s 01eeeeee (0)ttt tttttttttt tttttttttt
s 10eeeeee (0)ttt tttttttttt tttttttttt
</pre>

This includes [[subnormal numbers]] where the leading significand digit is 0.

If the 2 bits after the sign bit are "11", then the 8-bit exponent field is shifted 2 bits to the right (after both the sign bit and the "11" bits thereafter), and the represented significand is in the remaining 21 bits. In this case there is an implicit (that is, not stored) leading 3-bit sequence "100" in the true significand:

<pre>
s 1100eeeeee (100)t tttttttttt tttttttttt
s 1101eeeeee (100)t tttttttttt tttttttttt
s 1110eeeeee (100)t tttttttttt tttttttttt
</pre>

The "11" 2-bit sequence after the sign bit indicates that there is an ''implicit'' "100" 3-bit prefix to the significand.

Note that the leading bits of the significand field do ''not'' encode the most significant decimal digit; they are simply part of a larger pure-binary number. For example, a significand of {{gaps|8|000|000}} is encoded as binary {{gaps|0111|1010000100|1000000000}}, with the leading 4 bits encoding 7; the first significand which requires a 24th bit (and thus the second encoding form) is 223 = {{gaps|8|388|608}}.

In the above cases, the value represented is:

: (−1)sign × 10exponent−101 × significand 

Decimal64 and Decimal128 operate analogously, but with larger exponent continuation and significand fields. For Decimal128, the second encoding form is actually never used; the largest valid significand of 1034−1 = 1ED09BEAD87C0378D8E63FFFFFFFF16 can be represented in 113 bits.

=== Densely packed decimal significand field ===
In this version, the significand is stored as a series of decimal digits. The leading digit is between 0 and 9 (3 or 4 binary bits), and the rest of the significand uses the [[densely packed decimal]] (DPD) encoding.

The leading 2 bits of the exponent and the leading digit (3 or 4 bits) of the significand are combined into the five bits that follow the sign bit. This is followed by a fixed-offset exponent continuation field.

Finally, the significand continuation field made of 2, 5, or 11 10-bit ''[[declet (computing)|declet]]s'', each encoding 3 decimal digits.<ref name="Muller_2010">{{cite book |author-last1=Muller |author-first1=Jean-Michel |author-last2=Brisebarre |author-first2=Nicolas |author-last3=de Dinechin |author-first3=Florent |author-last4=Jeannerod |author-first4=Claude-Pierre |author-last5=Lefèvre |author-first5=Vincent |author-last6=Melquiond |author-first6=Guillaume |author-last7=Revol |author-first7=Nathalie |author7-link=Nathalie Revol|author-last8=Stehlé |author-first8=Damien |author-last9=Torres |author-first9=Serge |title=Handbook of Floating-Point Arithmetic |date=2010 |publisher=[[Birkhäuser]] |edition=1 |isbn=978-0-8176-4704-9 |doi=10.1007/978-0-8176-4705-6 |lccn=2009939668 |url=https://cds.cern.ch/record/1315760}}</ref>

If the first two bits after the sign bit are "00", "01", or "10", then those are the leading bits of the exponent, and the three bits after that are interpreted as the leading decimal digit (0 to 7):<ref>[http://speleotrove.com/decimal/dbspec.html Decimal Encoding Specification, version 1.00], from IBM</ref>
<pre>
Comb. Exponent Significand
s 00 TTT (00)eeeeee (0TTT)[tttttttttt][tttttttttt]
s 01 TTT (01)eeeeee (0TTT)[tttttttttt][tttttttttt]
s 10 TTT (10)eeeeee (0TTT)[tttttttttt][tttttttttt]
</pre>
If the first two bits after the sign bit are "11", then the second two bits are the leading bits of the exponent, and the last bit is prefixed with "100" to form the leading decimal digit (8 or 9):

<pre>
Comb. Exponent Significand
s 1100 T (00)eeeeee (100T)[tttttttttt][tttttttttt]
s 1101 T (01)eeeeee (100T)[tttttttttt][tttttttttt]
s 1110 T (10)eeeeee (100T)[tttttttttt][tttttttttt]
</pre>
The remaining two combinations (11110 and 11111) of the 5-bit field are used to represent ±infinity and NaNs, respectively.

== Floating-point arithmetic operations ==
The usual rule for performing floating-point arithmetic is that the exact mathematical value is calculated,<ref>Computer hardware doesn't necessarily compute the exact value; it simply has to produce the equivalent rounded result as though it had computed the infinitely precise result.</ref> and the result is then rounded to the nearest representable value in the specified precision. This is in fact the behavior mandated for IEEE-compliant computer hardware, under normal rounding behavior and in the absence of exceptional conditions.

For ease of presentation and understanding, 7-digit precision will be used in the examples. The fundamental principles are the same in any precision.

=== Addition ===
A simple method to add floating-point numbers is to first represent them with the same exponent. In the example below, the second number is shifted right by 3 digits. We proceed with the usual addition method:

The following example is decimal, which simply means the base is 10.

123456.7 = 1.234567 × 105
101.7654 = 1.017654 × 102 = 0.001017654 × 105
Hence:
123456.7 + 101.7654 = (1.234567 × 105) + (1.017654 × 102)
= (1.234567 × 105) + (0.001017654 × 105)
= 105 × (1.234567 + 0.001017654)
= 105 × 1.235584654
This is nothing other than converting to [[scientific notation]].
In detail:

e=5; s=1.234567 (123456.7)
+ e=2; s=1.017654 (101.7654)

e=5; s=1.234567
+ e=5; s=0.001017654 (after shifting)
--------------------
e=5; s=1.235584654 (true sum: 123558.4654)

This is the true result, the exact sum of the operands. It will be rounded to 7 digits and then normalized if necessary. The final result is:
e=5; s=1.235585 (final sum: 123558.5)

Note that the low 3 digits of the second operand (654) are essentially lost. This is [[round-off error]]. In extreme cases, the sum of two non-zero numbers may be equal to one of them:

e=5; s=1.234567
+ e=−3; s=9.876543

e=5; s=1.234567
+ e=5; s=0.00000009876543 (after shifting)
----------------------
e=5; s=1.23456709876543 (true sum)
e=5; s=1.234567 (after rounding/normalization)

Another problem of loss of significance occurs when ''approximations'' to two nearly equal numbers are subtracted. In the following example ''e'' = 5; ''s'' = 1.234571 and ''e'' = 5; ''s'' = 1.234567 are approximations to the rationals 123457.1467 and 123456.659.

e=5; s=1.234571
− e=5; s=1.234567
----------------
e=5; s=0.000004
e=−1; s=4.000000 (after rounding and normalization)

The floating-point difference is computed exactly because the numbers are close—the [[Sterbenz lemma]] guarantees this, even in case of underflow when [[gradual underflow]] is supported. Despite this, the difference of the original numbers is ''e'' = −1; ''s'' = 4.877000, which differs more than 20% from the difference ''e'' = −1; ''s'' = 4.000000 of the approximations. In extreme cases, all significant digits of precision can be lost.<ref name="Goldberg_1991"/><ref name="Sierra_1962"/> This ''[[Catastrophic cancellation|cancellation]]'' illustrates the danger in assuming that all of the digits of a computed result are meaningful. Dealing with the consequences of these errors is a topic in [[numerical analysis]]; see also [[#Accuracy problems|Accuracy problems]].

=== Multiplication ===
To multiply, the significands are multiplied, while the exponents are added, and the result is rounded and normalized.

e=3; s=4.734612
× e=5; s=5.417242
-----------------------
e=8; s=25.648538980104 (true product)
e=8; s=25.64854 (after rounding)
e=9; s=2.564854 (after normalization)

Division is done similarly, but that is more complicated.

There are no cancellation or absorption problems with multiplication or division, though small errors may accumulate as operations are performed repeatedly. In practice, the way these operations are carried out in digital logic can be quite complex.

{{further|Booth's multiplication algorithm|Division algorithm}}

== See also ==
* [[Binary-coded decimal]] (BCD)

== References ==
{{Reflist|refs=
<ref name="Savard_2018">{{cite web |title=The Decimal Floating-Point Standard |author-first=John J. G. |author-last=Savard |date=2018 |orig-year=2007 |work=quadibloc |url=http://www.quadibloc.com/comp/cp020302.htm |access-date=2018-07-16 |url-status=live |archive-url=https://web.archive.org/web/20180703002322/http://www.quadibloc.com/comp/cp020302.htm |archive-date=2018-07-03}}</ref>
<ref name="Beebe_2017">{{cite book |author-first=Nelson H. F. |author-last=Beebe |title=The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library |chapter=Chapter H. Historical floating-point architectures |date=2017-08-22 |location=Salt Lake City, UT, USA |publisher=[[Springer International Publishing AG]] |edition=1 |lccn=2017947446 |isbn=978-3-319-64109-6 |doi=10.1007/978-3-319-64110-2 |page=948|s2cid=30244721 }}</ref>
<ref name="Fujitsu_2015">{{cite book |title=Sparc64 X/X+ Specification |chapter=Chapter 4. Data Formats |date=January 2015 |location=Nakahara-ku, Kawasaki, Japan |page=13}}</ref>
<ref name="Goldberg_1991">{{cite journal |author-first=David |author-last=Goldberg |title=What Every Computer Scientist Should Know About Floating-Point Arithmetic |journal=[[ACM Computing Surveys]] |date=March 1991 |volume=23 |issue=1 |pages=5–48 |doi=10.1145/103162.103163 |s2cid=222008826 |url=http://perso.ens-lyon.fr/jean-michel.muller/goldberg.pdf |access-date=2016-01-20}} ([http://www.validlab.com/goldberg/paper.pdf], [http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html], [http://www.cse.msu.edu/~cse320/Documents/FloatingPoint.pdf])</ref>
<ref name="Sierra_1962">{{US patent reference |number=3037701A |issue-date=1962-06-05 |inventor=Huberto M Sierra |title=Floating decimal point arithmetic control means for calculator}}</ref>
}}

== Further reading ==
* [http://speleotrove.com/decimal/IEEE-cowlishaw-arith16.pdf Decimal Floating-Point: Algorism for Computers], Proceedings of the [[16th IEEE Symposium on Computer Arithmetic]] ([[Mike Cowlishaw|Cowlishaw, Mike F.]], 2003)

{{DEFAULTSORT:Decimal Floating Point}}
[[Category:Computer arithmetic]]
[[Category:Floating point]]
[[Category:10 (number)]]

Round-off error

2025-06-20T11:40:24Z

173.79.19.248: is not and has never been such an article

{{short description|Computational error due to rounding numbers}}
{{For|the acrobatic movement, roundoff|Roundoff}}
{{Use dmy dates|date=August 2019|cs1-dates=y}}

In [[computing]], a '''roundoff error''',<ref>{{citation |title=Introduction to Numerical Analysis Using MATLAB |author-first=Rizwan |author-last=Butt |publisher=Jones & Bartlett Learning |date=2009 |isbn=978-0-76377376-2 |pages=11–18 |url=https://books.google.com/books?id=QWub-UVGxqkC&pg=PA11}}</ref> also called '''rounding error''',<ref>{{citation |title=Numerical Computation 1: Methods, Software, and Analysis |author-first=Christoph W. |author-last=Ueberhuber |publisher=Springer |date=1997 |isbn=978-3-54062058-7 |url=https://books.google.com/books?id=JH9I7EJh3JUC&pg=PA139 |pages=139–146}}</ref> is the difference between the result produced by a given [[algorithm]] using exact [[arithmetic]] and the result produced by the same algorithm using finite-precision, [[Rounding|rounded]] arithmetic.<ref name="Forrester_2018">{{cite book |title= Math/Comp241 Numerical Methods (lecture notes) |author-first=Dick |author-last=Forrester |publisher=[[Dickinson College]] |date=2018}}</ref> Rounding errors are due to inexactness in the representation of [[real number]]s and the arithmetic operations done with them. This is a form of [[quantization error]].<ref>{{citation |title=Information Technology in Theory |author-first1=Pelin |author-last1=Aksoy |author-first2=Laura |author-last2=DeNardis |publisher=Cengage Learning |date=2007 |isbn=978-1-42390140-2 |page=134 |url=https://books.google.com/books?id=KGS5IcixljwC&pg=PA134}}</ref> When using approximation [[equation]]s or algorithms, especially when using finitely many digits to represent real numbers (which in theory have infinitely many digits), one of the goals of [[numerical analysis]] is to [[error analysis (mathematics)|estimate]] computation errors.<ref>{{citation |title=A First Course in Numerical Analysis |edition=2nd |series=Dover Books on Mathematics |author-first1=Anthony |author-last1=Ralston |author-first2=Philip |author-last2=Rabinowitz |publisher=Courier Dover Publications |date=2012 |isbn=978-0-48614029-2 |url=https://books.google.com/books?id=TVq8AQAAQBAJ&pg=PA2 |pages=2–4}}</ref> Computation errors, also called [[numerical error]]s, include both [[truncation error]]s and roundoff errors.

When a sequence of calculations with an input involving any roundoff error are made, errors may accumulate, sometimes dominating the calculation. In [[ill-conditioned]] problems, significant error may accumulate.<ref>{{citation |title=MATLAB Programming with Applications for Engineers |author-first=Stephen |author-last=Chapman |publisher=Cengage Learning |date=2012 |isbn=978-1-28540279-6 |url=https://books.google.com/books?id=of8KAAAAQBAJ&pg=PA454 |page=454}}</ref>

In short, there are two major facets of roundoff errors involved in numerical calculations:<ref name="Chapra_2012">{{cite book |author-last=Chapra |author-first=Steven |title=Applied Numerical Methods with MATLAB for Engineers and Scientists |publisher=[[McGraw Hill Education|McGraw-Hill]] |date=2012 |isbn=9780073401102 |edition=3rd}}</ref>
# The ability of computers to represent both magnitude and precision of numbers is inherently limited.
# Certain numerical manipulations are highly sensitive to roundoff errors. This can result from both mathematical considerations as well as from the way in which computers perform arithmetic operations.

== Representation error ==

The error introduced by attempting to represent a number using a finite string of digits is a form of roundoff error called '''representation error'''.<ref name="Laplante_2000">{{cite book |title=Dictionary of Computer Science, Engineering and Technology |first=Philip A. |last=Laplante |publisher=[[CRC Press]] |date=2000 |isbn=978-0-84932691-2 |page=420 |url=https://books.google.com/books?id=U1M3clUwCfEC&pg=PA420}}</ref> Here are some examples of representation error in decimal representations:

{| class="wikitable" style="margin:1em auto"
! Notation
! Representation
! Approximation
! Error
|-
|{{Fraction|1|7}}|| 0.{{overline|142 857}} || 0.142 857 || 0.000 000 {{overline|142 857}}
|-
|[[Natural logarithm|ln 2]] || 0.693 147 180 559 945 309 41... || 0.693 147 || 0.000 000 180 559 945 309 41...
|-
|[[Logarithm|log10 2]] || 0.301 029 995 663 981 195 21... || 0.3010 || 0.000 029 995 663 981 195 21...
|-
|[[cube root|{{radic|2|3}}]] || 1.259 921 049 894 873 164 76... || 1.25992 || 0.000 001 049 894 873 164 76...
|-
|[[square root|{{radic|2}}]] || 1.414 213 562 373 095 048 80... || 1.41421 || 0.000 003 562 373 095 048 80...
|-
|[[E (mathematical constant)|''e'']] || 2.718 281 828 459 045 235 36... || 2.718 281 828 459 045 || 0.000 000 000 000 000 235 36...
|-
|[[Pi|''π'']] || 3.141 592 653 589 793 238 46... || 3.141 592 653 589 793 || 0.000 000 000 000 000 238 46...
|}

Increasing the number of digits allowed in a representation reduces the magnitude of possible roundoff errors, but any representation limited to finitely many digits will still cause some degree of roundoff error for [[Countable|uncountably many]] real numbers. Additional digits used for intermediary steps of a calculation are known as [[guard digit]]s.<ref name="Higham_2002">{{cite book |title=Accuracy and Stability of Numerical Algorithms |edition=2 |author-first=Nicholas John |author-link=Nicholas Higham |author-last=Higham |publisher=[[Society for Industrial and Applied Mathematics]] (SIAM) |date=2002 |isbn=978-0-89871521-7 |pages=43–44 |url=https://books.google.com/books?id=epilvM5MMxwC&pg=PA43}}</ref>

Rounding multiple times can cause error to accumulate.<ref name="Volkov_1990">{{cite book |title=Numerical Methods |author-first=E. A. |author-last=Volkov |publisher=[[Taylor & Francis]] |date=1990 |isbn=978-1-56032011-1 |page=24 |url=https://books.google.com/books?id=ubfrNN8GGOIC&pg=PA24}}</ref> For example, if 9.945309 is rounded to two decimal places (9.95), then rounded again to one decimal place (10.0), the total error is 0.054691. Rounding 9.945309 to one decimal place (9.9) in a single step introduces less error (0.045309). This can occur, for example, when software performs arithmetic in [[Extended precision#x86 extended precision format|x86 80-bit floating-point]] and then rounds the result to [[Double-precision floating-point format|IEEE 754 binary64 floating-point]].

== Floating-point number system ==

Compared with the [[fixed-point arithmetic|fixed-point number system]], the [[floating-point arithmetic|floating-point number system]] is more efficient in representing real numbers so it is widely used in modern computers. While the real numbers <math>\mathbb{R}</math> are infinite and continuous, a floating-point number system <math>F</math> is finite and discrete. Thus, representation error, which leads to roundoff error, occurs under the floating-point number system.

=== Notation of floating-point number system ===
A floating-point number system <math>F</math> is characterized by <math>4</math> integers:
*<math> \beta </math>: base or radix
*<math>p</math>: precision
*<math> [L, U] </math>: exponent range, where <math>L</math> is the lower bound and <math>U</math> is the upper bound

Any <math>x \in F</math> has the following form:
<math display="block"> x = \pm (\underbrace{d_{0}.d_{1}d_{2}\ldots d_{p-1}}_\text{significand})_{\beta} \times \beta ^{\overbrace{E}^\text{exponent}} = \pm d_{0}\times \beta ^{E}+d_{1}\times \beta ^{E-1}+\ldots+ d_{p-1}\times \beta ^{E-(p-1)}</math>
where <math>d_{i}</math> is an integer such that <math>0 \leq d_{i} \leq \beta-1</math> for <math>i = 0, 1, \ldots, p-1</math>, and <math>E</math> is an integer such that <math>L \leq E \leq U</math>.

=== Normalized floating-number system ===

* A floating-point number system is normalized if the leading digit <math>d_{0}</math> is always nonzero unless the number is zero.<ref name="Forrester_2018"/> Since the [[significand]] is <math>d_{0}.d_{1}d_{2}\ldots d_{p-1}</math>, the significand of a nonzero number in a normalized system satisfies <math>1 \leq \text{significand} < \beta ^{p}</math>. Thus, the normalized form of a nonzero [[Institute of Electrical and Electronics Engineers|IEEE]] floating-point number is <math>\pm 1.bb \ldots b \times 2^{E}</math> where <math>b \in {0, 1}</math>. In binary, the leading digit is always <math>1</math> so it is not written out and is called the implicit bit. This gives an extra bit of precision so that the roundoff error caused by representation error is reduced.
* Since floating-point number system <math>F</math> is finite and discrete, it cannot represent all real numbers which means infinite real numbers can only be approximated by some finite numbers through [[rounding|rounding rule]]s. The floating-point approximation of a given real number <math>x</math> by <math>fl(x)</math> can be denoted.
** The total number of normalized floating-point numbers is <math display="block">2(\beta -1)\beta^{p-1} (U-L+1)+1,</math> where
*** <math>2</math> counts choice of sign, being positive or negative
*** <math>(\beta -1)</math> counts choice of the leading digit
*** <math>\beta^{p-1}</math> counts remaining significand digits
*** <math>U-L+1</math> counts choice of exponents
*** <math>1</math> counts the case when the number is <math>0</math>.

=== IEEE standard ===

In the [[Institute of Electrical and Electronics Engineers|IEEE]] standard the base is binary, i.e. <math>\beta = 2</math>, and normalization is used. The IEEE standard stores the sign, exponent, and significand in separate fields of a floating point word, each of which has a fixed width (number of bits). The two most commonly used levels of precision for floating-point numbers are single precision and double precision.
{| class="wikitable" style="margin:1em auto"
! Precision
! Sign (bits)
! Exponent (bits)
! Trailing Significand field (bits)
|-
|Single || 1 || 8 || 23
|-
|Double || 1 || 11 || 52
|}

== Machine epsilon ==

[[Machine epsilon]] can be used to measure the level of roundoff error in the floating-point number system. Here are two different definitions.<ref name="Forrester_2018"/>

* The machine epsilon, denoted <math>\epsilon_\text{mach}</math>, is the maximum possible [[approximation error|absolute relative error]] in representing a nonzero real number <math>x</math> in a floating-point number system. <math display="block">\epsilon_\text{mach} = \max_{x} \frac{|x-fl(x)|}{|x|}</math>
* The machine epsilon, denoted <math>\epsilon_\text{mach}</math>, is the smallest number <math>\epsilon</math> such that <math>fl(1+\epsilon) > 1</math>. Thus, <math>fl(1+\delta)=fl(1)=1</math> whenever <math>|\delta| < \epsilon_\text{mach}</math>.

== Roundoff error under different rounding rules ==

There are two common rounding rules, round-by-chop and round-to-nearest. The IEEE standard uses round-to-nearest.

* '''Round-by-chop''': The base-<math>\beta</math> expansion of <math>x</math> is truncated after the <math>(p-1)</math>-th digit.
** This rounding rule is biased because it always moves the result toward zero.
* '''Round-to-nearest''': <math>fl(x)</math> is set to the nearest floating-point number to <math>x</math>. When there is a tie, the floating-point number whose last stored digit is even (also, the last digit, in binary form, is equal to 0) is used.
** For IEEE standard where the base <math>\beta</math> is <math>2</math>, this means when there is a tie it is rounded so that the last digit is equal to <math>0</math>.
** This rounding rule is more accurate but more computationally expensive.
** Rounding so that the last stored digit is even when there is a tie ensures that it is not rounded up or down systematically. This is to try to avoid the possibility of an unwanted slow drift in long calculations due simply to a biased rounding.
* The following example illustrates the level of roundoff error under the two rounding rules.<ref name="Forrester_2018"/> The rounding rule, round-to-nearest, leads to less roundoff error in general.
{| class="wikitable" style="margin:1em auto"
! x
! Round-by-chop
! Roundoff Error
! Round-to-nearest
! Roundoff Error
|-
|1.649 || 1.6 || 0.049 || 1.6 || 0.049
|-
|1.650 || 1.6 || 0.050 || 1.6 || 0.050
|-
|1.651 || 1.6 || 0.051 || 1.7 || −0.049
|-
|1.699 || 1.6 || 0.099 || 1.7 || −0.001
|-
|1.749 || 1.7 || 0.049 || 1.7 || 0.049
|-
|1.750 || 1.7 || 0.050 || 1.8 || −0.050
|}

=== Calculating roundoff error in IEEE standard ===

Suppose the usage of round-to-nearest and IEEE double precision.

* Example: the decimal number <math>(9.4)_{10}=(1001.{\overline{0110}})_{2}</math> can be rearranged into <math display="block">+1.\underbrace{0010110011001100110011001100110011001100110011001100}_\text{52 bits}110 \ldots \times 2^{3}</math>
Since the 53rd bit to the right of the binary point is a 1 and is followed by other nonzero bits, the round-to-nearest rule requires rounding up, that is, add 1 bit to the 52nd bit. Thus, the normalized floating-point representation in IEEE standard of 9.4 is
<math display="block">fl(9.4)=1.0010110011001100110011001100110011001100110011001101 \times 2^{3}.</math>

* Now the roundoff error can be calculated when representing <math>9.4</math> with <math>fl(9.4)</math>.
This representation is derived by discarding the infinite tail <math display="block">0.{\overline{1100}} \times 2^{-52}\times 2^{3} = 0.{\overline{0110}} \times 2^{-51} \times 2^{3}=0.4 \times 2^{-48}</math>
from the right tail and then added <math>1 \times 2^{-52} \times 2^{3}=2^{-49}</math> in the rounding step.
:Then <math>fl(9.4) = 9.4-0.4 \times 2^{-48} + 2^{-49} = 9.4+(0.2)_{10} \times 2^{-49}</math>.
:Thus, the roundoff error is <math>(0.2 \times 2^{-49})_{10}</math>.

=== Measuring roundoff error by using machine epsilon ===

The machine epsilon <math>\epsilon_\text{mach}</math> can be used to measure the level of roundoff error when using the two rounding rules above. Below are the formulas and corresponding proof.<ref name="Forrester_2018"/> The first definition of machine epsilon is used here.

==== Theorem ====
# Round-by-chop: <math>\epsilon_\text{mach} = \beta^{1-p}</math>
# Round-to-nearest: <math>\epsilon_\text{mach} = \frac{1}{2}\beta^{1-p}</math>

==== Proof ====
Let <math>x=d_{0}.d_{1}d_{2} \ldots d_{p-1}d_{p} \ldots \times \beta^{n} \in \mathbb{R}</math> where <math>n \in [L, U]</math>, and let <math>fl(x)</math> be the floating-point representation of <math>x</math>.
Since round-by-chop is being used, it is
<math display="block"> \begin{align}
\frac{|x-fl(x)|}{|x|} &= \frac{|d_{0}.d_{1}d_{2}\ldots d_{p-1}d_{p}d_{p+1}\ldots \times \beta^{n} - d_{0}.d_{1}d_{2}\ldots d_{p-1} \times \beta^{n}|}{|d_{0}.d_{1}d_{2}\ldots \times \beta^{n}|}\\
&= \frac{|d_{p}.d_{p+1} \ldots \times \beta^{n-p}|}{|d_{0}.d_{1}d_{2}\ldots \times \beta^{n}|}\\
&= \frac{|d_{p}.d_{p+1}d_{p+2}\ldots|}{|d_{0}.d_{1}d_{2}\ldots|} \times \beta^{-p}
\end{align}</math>
In order to determine the maximum of this quantity, there is a need to find the maximum of the numerator and the minimum of the denominator. Since <math>d_{0}\neq 0</math> (normalized system), the minimum value of the denominator is <math>1</math>. The numerator is bounded above by <math>(\beta-1).(\beta-1){\overline{(\beta-1)}}=\beta </math>. Thus, <math>\frac{|x-fl(x)|}{|x|} \leq \frac{\beta}{1} \times \beta^{-p} = \beta^{1-p}</math>. Therefore, <math>\epsilon=\beta^{1-p}</math> for round-by-chop.
The proof for round-to-nearest is similar.
* Note that the first definition of machine epsilon is not quite equivalent to the second definition when using the round-to-nearest rule but it is equivalent for round-by-chop.

== Roundoff error caused by floating-point arithmetic ==

Even if some numbers can be represented exactly by floating-point numbers and such numbers are called '''machine numbers''', performing floating-point arithmetic may lead to roundoff error in the final result.

=== Addition ===

Machine addition consists of lining up the decimal points of the two numbers to be added, adding them, and then storing the result again as a floating-point number. The addition itself can be done in higher precision but the result must be rounded back to the specified precision, which may lead to roundoff error.<ref name="Forrester_2018"/>

* For example, adding <math>1</math> to <math>2^{-53}</math> in IEEE double precision as follows,{{Break}}<math>\begin{align}
1.00\ldots 0 \times 2^{0} + 1.00\ldots 0 \times 2^{-53} &= 1.\underbrace{00\ldots 0}_\text{52 bits} \times 2^{0} + 0.\underbrace{00\ldots 0}_\text{52 bits}1 \times 2^{0}\\
&= 1.\underbrace{00\ldots 0}_\text{52 bits}1\times 2^{0}.
\end{align}</math>{{Break}}This is saved as <math>1.\underbrace{00\ldots 0}_\text{52 bits}\times 2^{0}</math> since round-to-nearest is used in IEEE standard. Therefore, <math>1+2^{-53}</math> is equal to <math>1</math> in IEEE double precision and the roundoff error is <math>2^{-53}</math>.

This example shows that roundoff error can be introduced when adding a large number and a small number. The shifting of the decimal points in the significands to make the exponents match causes the loss of some of the less significant digits. The loss of precision may be described as '''absorption'''.<ref>{{cite book |last1=Biran |first1=Adrian B. |last2=Breiner |first2=Moshe |title=What Every Engineer Should Know About MATLAB and Simulink |date=2010 |publisher=[[CRC Press]] |publication-place=[[Boca Raton]], [[Florida]] |isbn=978-1-4398-1023-1 |pages=193–194 |chapter=5}}</ref>

Note that the addition of two floating-point numbers can produce roundoff error when their sum is an order of magnitude greater than that of the larger of the two.

* For example, consider a normalized floating-point number system with base <math>10</math> and precision <math>2</math>. Then <math>fl(62)=6.2 \times 10^{1}</math> and <math>fl(41) = 4.1 \times 10^{1}</math>. Note that <math>62+41=103</math> but <math>fl(103)=1.0 \times 10^{2}</math>. There is a roundoff error of <math>103-fl(103)=3</math>.

This kind of error can occur alongside an absorption error in a single operation.

=== Multiplication ===

In general, the product of two p-digit significands contains up to 2p digits, so the result might not fit in the significand.<ref name="Forrester_2018"/> Thus roundoff error will be involved in the result.

* For example, consider a normalized floating-point number system with the base <math>\beta=10</math> and the significand digits are at most <math>2</math>. Then <math>fl(77) = 7.7 \times 10^{1}</math> and <math>fl(88) = 8.8 \times 10^{1}</math>. Note that <math>77 \times 88=6776</math> but <math>fl(6776) = 6.7 \times 10^{3}</math> since there at most <math>2</math> significand digits. The roundoff error would be <math>6776 - fl(6776) = 6776 - 6.7 \times 10^{3}=76</math>.

=== Division ===

In general, the quotient of 2p-digit significands may contain more than p-digits.Thus roundoff error will be involved in the result.

* For example, if the normalized floating-point number system above is still being used, then <math>1/3=0.333 \ldots</math> but <math>fl(1/3)=fl(0.333 \ldots)=3.3 \times 10^{-1}</math>. So, the tail <math>0.333 \ldots - 3.3 \times 10^{-1}=0.00333 \ldots </math> is cut off.

=== Subtraction ===

Absorption also applies to subtraction.

* For example, subtracting <math>2^{-60}</math> from <math>1</math> in IEEE double precision as follows, <math display="block">\begin{align}
1.00\ldots 0 \times 2^{0} - 1.00\ldots 0 \times 2^{-60} &= \underbrace{1.00\ldots 0}_\text{60 bits} \times 2^{0} - \underbrace{0.00\ldots 01}_\text{60 bits} \times 2^{0}\\
&= \underbrace{0.11\ldots 1}_\text{60 bits}\times 2^{0}.
\end{align}</math> This is saved as <math>\underbrace{1.00\ldots 0}_\text{53 bits}\times 2^{0}</math> since round-to-nearest is used in IEEE standard. Therefore, <math>1-2^{-60}</math> is equal to <math>1</math> in IEEE double precision and the roundoff error is <math>-2^{-60}</math>.

The subtracting of two nearly equal numbers is called '''subtractive cancellation'''.<ref name="Forrester_2018"/>
When the leading digits are cancelled, the result may be too small to be represented exactly and it will just be represented as <math>0</math>.

* For example, let <math>|\epsilon| < \epsilon_\text{mach}</math> and the second definition of machine epsilon is used here. What is the solution to <math>(1+\epsilon) - (1-\epsilon)</math>?{{Break}} It is known that <math>1+\epsilon</math> and <math>1-\epsilon</math> are nearly equal numbers, and <math>(1+\epsilon) - (1-\epsilon)=1+\epsilon-1+\epsilon=2\epsilon</math>. However, in the floating-point number system, <math>fl((1+\epsilon) - (1-\epsilon))=fl(1+\epsilon)-fl(1-\epsilon)=1-1=0</math>. Although <math>2\epsilon</math> is easily big enough to be represented, both instances of <math>\epsilon</math> have been rounded away giving <math>0</math>.

Even with a somewhat larger <math>\epsilon</math>, the result is still significantly unreliable in typical cases. There is not much faith in the accuracy of the value because the most uncertainty in any floating-point number is the digits on the far right.

* For example, <math>1.99999 \times 10 ^{2}- 1.99998 \times 10^{2} = 0.00001\times10^{2} =1 \times 10^{-5}\times 10^{2}=1\times10^{-3}</math>. The result <math>1\times10^{-3}</math> is clearly representable, but there is not much faith in it.

This is closely related to the phenomenon of [[catastrophic cancellation]], in which the two numbers are ''known'' to be approximations.

== Accumulation of roundoff error ==

Errors can be magnified or accumulated when a sequence of calculations is applied on an initial input with roundoff error due to inexact representation.

=== Unstable algorithms ===

An algorithm or numerical process is called '''stable''' if small changes in the input only produce small changes in the output, and '''unstable''' if large changes in the output are produced.<ref name="Collins_2005">{{cite web |author-last=Collins |author-first=Charles |title=Condition and Stability |url=https://www.math.utk.edu/~ccollins/M577/Handouts/cond_stab.pdf |website=Department of Mathematics in University of Tennessee |date=2005 |access-date=2018-10-28}}</ref> For example, the computation of <math>f(x) = \sqrt{1 + x} - 1</math> using the "obvious" method is unstable near <math>x = 0</math> due to the large error introduced in subtracting two similar quantities, whereas the equivalent expression <math>\textstyle{f(x) = \frac{x}{\sqrt{1+x} + 1}}</math> is stable.<ref name="Collins_2005"/>

=== Ill-conditioned problems ===

Even if a stable algorithm is used, the solution to a problem may still be inaccurate due to the accumulation of roundoff error when the problem itself is '''ill-conditioned'''.

The [[condition number]] of a problem is the ratio of the relative change in the solution to the relative change in the input.<ref name="Forrester_2018"/> A problem is '''well-conditioned''' if small relative changes in input result in small relative changes in the solution. Otherwise, the problem is '''ill-conditioned'''.<ref name="Forrester_2018"/> In other words, a problem is '''ill-conditioned''' if its conditions number is "much larger" than 1.

The condition number is introduced as a measure of the roundoff errors that can result when solving ill-conditioned problems.<ref name="Chapra_2012"/>

== See also ==
* [[Precision (arithmetic)]]
* [[Truncation]]
* [[Rounding]]
* [[Loss of significance]]
* [[Floating point]]
* [[Kahan summation algorithm]]
* [[Machine epsilon]]
* [[Significant digits]]
* [[Wilkinson's polynomial]]

==References==
{{Reflist}}

== Further reading ==
* {{cite book |title=Humble Pi: When Math Goes Wrong in the Real World |year=2021 |author= Matt Parker |publisher=Riverhead Books |isbn=978-0593084694}}

== External links ==
* [http://mathworld.wolfram.com/RoundoffError.html Roundoff Error] at MathWorld.
* {{cite journal |author-first=David |author-last=Goldberg |title=What Every Computer Scientist Should Know About Floating-Point Arithmetic |journal=[[ACM Computing Surveys]] |date=March 1991 |volume=23 |issue=1 |pages=5–48 |doi=10.1145/103162.103163 |s2cid=222008826 |url=http://perso.ens-lyon.fr/jean-michel.muller/goldberg.pdf |access-date=2016-01-20}} ([http://www.validlab.com/goldberg/paper.pdf], [http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html])
* [http://www.devtopics.com/20-famous-software-disasters/ 20 Famous Software Disasters]

{{Authority control}}

{{DEFAULTSORT:RoundOff Error}}
[[Category:Numerical analysis]]

[[sv:Avrundningsfel]]

WavPack

2025-06-20T11:40:04Z

173.79.19.248: /* Technology */ is not and has never been such an article

{{Short description|Open-source lossless audio codec}}

{{distinguish|WAV}}{{Infobox file format
| name = WavPack file format
| icon = [[File:WavPack logo.svg|frameless|class=skin-invert]]
| logo =
| screenshot =
| caption =
| extension = .wv (for both lossless and hybrid files), .wvc (correction file for hybrid files only)
| mime = audio/x-wavpack (.wv), audio/x-wavpack-correction (.wvc)
| type code =
| uniform type =
| magic = wvpk
| container for =
| contained by = Matroska (not required)
| extended from =
| extended to =
| standard =
| open = Yes
| free = Yes
| type = [[Lossless compression|Lossless]] [[Audio file format|audio]]
}}

{{Infobox software
| name = WavPack software
| logo =
| screenshot = WVUNPACK screenshot.webp
| caption = Screenshot of <code>wvunpack</code> showing information of a .wv file
| developer = David Bryant
| latest release version = {{wikidata|property|preferred|references|edit|P348|P548=Q2804309}}
| latest release date = {{Start date and age|{{wikidata|qualifier|preferred|single|P348|P548=Q2804309|P577}}|df=yes}}
| operating_system = [[Cross-platform]]
| genre = [[Audio codec]] Container
| license = [[BSD licenses|BSD license]]
| website = [https://wavpack.com/ wavpack.com]
}}

'''WavPack''' is a [[free and open-source]] [[Audio compression (data)#Lossless|lossless audio compression]] [[file format|format]] and application implementing the format. It is unique in the way that it supports hybrid audio compression alongside normal compression which is similar to how [[FLAC]] works. It also supports compressing a wide variety of lossless formats, including various variants of [[Pulse-code modulation|PCM]] and also [[Direct Stream Digital|DSD]] as used in [[Super Audio CD|SACDs]], together with its support for surround audio.

==Features==
WavPack compression can compress (and losslessly restore) 8, 16, 24, and 32-bit [[fixed-point arithmetic|fixed-point]], and 32-bit [[floating-point]] [[PCM]] audio files in the .[[WAV]] file format. It can also handle DSD input in DSDIFF or DSF format.<ref name="wavpack_doc">{{cite web |title=WavPack User Documentation |url=https://www.wavpack.com/wavpack_doc.html |website=www.wavpack.com}}</ref> It also supports [[surround sound]] streams and high [[sampling rate]]s. Like other lossless compression schemes, the data reduction rate varies with the source, but it is generally between 30% and 70% for typical popular music and somewhat better than that for classical music and other sources with greater dynamic range.<ref>{{cite web
| last = Heijden
| first = Hans
| title = Compression and speed of lossless audio formats
| date = 11 July 2006
| url = http://web.inter.nl.net/users/hvdh/lossless/lossless.htm
| access-date = 17 July 2009
| archive-date = 3 July 2009
| archive-url = https://web.archive.org/web/20090703042443/http://web.inter.nl.net/users/hvdh/lossless/lossless.htm
| url-status = dead
}}</ref>

===Hybrid mode===
WavPack also incorporates a "hybrid" mode, which still provides the features of lossless compression, but creates two files: a relatively small, high-quality, lossy file (.wv) that can be used by itself; and a "correction" file (.wvc) that, when combined with the lossy file, provides full lossless restoration. This allows the use of lossy and lossless codecs together. The lossy algorithm is similar to [[ADPCM]].<ref>{{cite book |chapter=7.11 WavPack |last1=Bryant |first1=David |editor1-last=Solomon |editor1-first=David |title=Data Compression: The Complete Reference |page=773 |url=https://www.wavpack.com/WavPack.pdf}}</ref>

Hybrid mode can handle floating-point data, but only when "exception" values like infinities or NaNs are not present. It cannot handle DSD because there is no lossy algorithm for DSD.<ref name="wavpack_doc"/>

A similar "hybrid" feature is also offered by [[OptimFROG DualStream]], [[MPEG-4 SLS]] and [[DTS-HD Master Audio]].

===Summary===
* [[Open-source software|Open-source]], released under a [[BSD license|BSD]]-like license
* Multiplatform
* Error robustness
* Fast encoding
* Higher compression ratios than other widely used (FLAC/ALAC) open-source lossless audio codecs
* [[Streaming media|Streaming]] support
* Supports multichannel audio and high resolutions
* Native support in WavPack 5.x for compressing [[Direct Stream Digital]] without converting the source file to PCM.
* Hybrid/lossy mode
* Hardware support (provided by [[Rockbox]] firmware)
* Metadata support ([[ID3]], [[APEv2 tag|APE]] tags; APE tag is the preferred format)
* Supports [[RIFF (File format)|RIFF]] chunks
* Compatible with [[ReplayGain]]
* Ability to create self-extracting files for the Win32 platform
* Supports 32-bit floating-point streams
* Supports embedded [[Cue sheet (music software)|CUE sheets]]
* Includes [[MD5]] hashes for quick integrity checking
* Can encode in both symmetrical and asymmetrical (slower encoding to speed up decoding) modes

==History==
David Bryant started development on WavPack in mid-1998 with the release of version 1.0 (1998-08-15). This first version compressed and decompressed audio losslessly, and it already featured one of the best efficiency vs. speed ratios among lossless encoders.<ref>{{cite web
| last = Speek
| title = Performance comparison of lossless audio compressors
| date = 7 February 2005
| url = http://members.home.nl/w.speek/comparison.htm
| access-date = 17 July 2009
| archive-date = 25 November 2010
| archive-url = https://web.archive.org/web/20101125045330/http://members.home.nl/w.speek/comparison.htm
| url-status = dead
}}</ref>

Very soon after the release of version 1.0, v. 2.0 (2 September 1998) was released, featuring lossy encoding (using only quantization of prediction residue for data reduction{{snd}} no [[Psychoacoustics#Masking effects|psychoacoustic masking model]] was applied to the stream).

In 1999, version 3.0 (12 September 1999) was released, with a new "fast mode" (albeit with reduced compression ratio), compression of raw (headerless) [[Pulse-code modulation|PCM]] audio files, and error detection using a 32-bit [[cyclic redundancy check]].

A feature added in late 3.x versions is the "hybrid" mode where the encoder generates a lossy file and a correction file such that both can be decompressed back to a PCM stream that has same quality as the original.<ref>{{cite web |url=https://www.wavpack.com/#About |title = WavPack Audio Compression}}</ref>

In WavPack 4, a new file format structure is introduced. A "roadmap" is also published by the author with 4.40, containing possible hints on future development.<ref>{{cite web |url=http://www.hydrogenaud.io/forums/index.php?s=&showtopic=50911&view=findpost&p=456571 |title = WavPack 4.40 released}}</ref> WavPack 5 introduced support for compressing DSD.

==Support==
===Software===
Some software supports the format natively (like [[DeaDBeeF]], [[foobar2000]],<ref>{{Cite web|url=http://foobar2000.org/|title=foobar2000|website=foobar2000.org|access-date=2019-11-19}}</ref> and [[Jack! The Knife]]), while others require plugins. The official WavPack website offers plugins for [[Winamp]], [[Nero Burning ROM]], MediaChest 2.1, and several other applications, as well as a [[DirectShow]] filter.<ref name="wpdl">{{Cite web|url=https://www.wavpack.com/downloads.html|title=WavPack downloads|website=www.wavpack.com}}</ref> [https://www.dbpoweramp.com/cd-ripper.htm dBpoweramp CD-Ripper],<ref>{{Cite web|url=https://www.dbpoweramp.com/codec-central.htm|title=dBpoweramp Codec Central|website=www.dbpoweramp.com|access-date=2019-11-19}}</ref> by the author of foobar2000, as well as foobar2000<ref>{{Cite web|url=http://www.foobar2000.org/encoderpack|title=foobar2000: Free Encoder Pack|website=www.foobar2000.org|access-date=2019-11-19}}</ref> itself, and [[Asunder (software)|Asunder]] allow ripping Audio CDs directly into Wavpack files.

[[Linux]] support is available with a native port.

[[FFmpeg]] has a native WavPack encoder, which may be combined with software like [[GNU parallel]] to use multiple CPU cores to quickly transcode other lossless formats into WavPack, and from WavPack to any format that FFmpeg supports, without the need for additional software. However, FFMpeg's encoder is somewhat limited.

As of 2023, FFmpeg's WavPack encoder has some considerable limitations. It can only produce version 4 bitstreams, which do not support fast verification for file integrity checks, or more than 16 channels. It will also discard RIFF chunks and may not behave predictably with 24-bit input. It also defaults to less than optimal compression to achieve faster encoding. Due to lack of support for Hybrid mode, FFmpeg-based playback software will fail to consider the .wvc correction file if there is one present, and will play or transcode only the lossy section. (However, this is not the usual mode of operation for WavPack.) As native Wavpack support for Direct Stream Digital was added in version 5 of the reference encoder, FFmpeg WavPack also is limited to encoding PCM input.<ref>{{cite web | url=https://wiki.hydrogenaud.io/index.php?title=WavPack#FFmpeg.27s_WavPack_implementation:_differences_and_quirks | title=WavPack - Hydrogenaudio Knowledgebase }}</ref>

===Hardware===
Native support:
* [[Cowon A3]] PMP supports WavPack out of the box.{{Citation needed|date=May 2010}}

Non-native support:
* Apple [[iPod]] range of music players do not support WavPack out of the box, but can through open-source [[Rockbox]] firmware.
* [[iRiver H100 series]], through open-source Rockbox firmware.
* [[iRiver H300 series]], through open-source Rockbox firmware.<ref name="rockboxsoundcodecs">{{Cite web|url=https://www.rockbox.org/wiki/SoundCodecs|title=SoundCodecs < Main < Wiki|website=www.rockbox.org}}</ref>
* [[Android (operating system)|Android]] smartphones and tablets with the installation of third-party media-player software.
* [[ChromeOS]] devices using media-player software installed in the Linux subsystem or the Android Play Store.
The WavPack website also includes a plugin that allows support for the format on the [http://www.rokulabs.com/products/photobridge/features.php Roku PhotoBridge HD] {{Webarchive|url=https://web.archive.org/web/20050708003853/http://www.rokulabs.com/products/photobridge/features.php |date=2005-07-08 }}.<ref name="wpdl"/>

==Technology==
To ensure high-speed operation, WavPack uses a predictor that is implemented entirely in integer math.<ref>{{cite web
| last = Bryant
| first = David
| title = Forum comment by developer
| date = 21 March 2007
| url = http://www.hydrogenaud.io/forums/index.php?showtopic=52576&st=50&p=479913&hl=integer&#entry479913
| access-date = 17 July 2009}}</ref> In its "fast" mode the prediction is simply the arithmetic extrapolation of the previous two samples. For example, if the previous two samples were −10 and 20, then the prediction would be 50. For the default mode a simple adaptive factor is added to weigh the influence of the earlier sample on the prediction. In the example above the resulting prediction could then vary between 20 for no influence to 50 for full influence. This weight factor is constantly updated based on the audio data's changing spectral characteristics.

The prediction generated is then subtracted from the actual sample to be encoded to generate the error value. In mono mode this value is sent directly to the coder. However, [[stereophonic sound|stereo]] signals tend to have some correlation between the two channels that can be further exploited. Therefore, two error values are calculated that represent the difference and average of the left and right error values. In the "fast" mode of operation these two new values are simply sent to the coder instead of the left and right values. In the default mode, the difference value is always sent to the coder along with one of the other three values (average, left, or right). An adaptive algorithm continuously determines the most efficient of the three to send based on the changing balance of the channels.

Instead of [[Rice coding]], a special data encoder for WavPack is used. Rice coding is the optimal bit coding for this type of data, and WavPack's encoder is less efficient, but only by about 0.15 bits per sample (or less than 1% for 16-bit data). However, there are some advantages in exchange. The first one is that WavPack's encoder does not require the data to be buffered ahead of encoding; instead it converts each sample directly to bitcodes. This is more computationally efficient and is better in some applications where coding delay is critical. The second advantage is that it is easily adaptable to lossy encoding, since all significant bits (except the implied "one" [[Most significant bit|MSB]]) are transmitted directly. In this way it is possible to only transmit, for example, the 3 most significant bits (with sign) of each sample. In fact, it is possible to transmit only the sign and implied MSB for each sample with an average of only 3.65 bits per sample.

This coding scheme is used to implement the "lossy" mode of WavPack. In the "fast" mode the output of the non-adaptive decorrelator is simply rounded to the nearest codable value for the specified number of bits. In the default mode the adaptive decorrelator is used (which reduces the average noise about 1 [[decibel|dB]]), and both the current and the next sample are considered in choosing the better of the two available codes (which reduces noise another 1 dB).

No [[floating-point]] arithmetic is used in WavPack's data path because, according to the author, [[integer]] operations are less susceptible to subtle chip-to-chip variations that could corrupt the lossless nature of the compression (the [[Pentium FDIV bug|Pentium floating-point bug]] being an example). It is possible that a lossless compressor that used floating-point math could generate different output when running on that faulty Pentium. Even disregarding actual bugs, floating-point math is complicated enough that there could be subtle differences between "correct" implementations that could cause trouble for this type of application.<ref name="Goldberg_1991">{{cite journal |last=Goldberg |first=David |date=March 1991 |title=What Every Computer Scientist Should Know About Floating-Point Arithmetic |url=http://perso.ens-lyon.fr/jean-michel.muller/goldberg.pdf |journal=[[ACM Computing Surveys]] |volume=23 |issue=1 |pages=5–48 |doi=10.1145/103162.103163 |s2cid=222008826 |access-date=2016-01-20}} ([http://www.validlab.com/goldberg/paper.pdf], [http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html]).</ref> A 32-bit error detection code to the generated streams is included to maintain user confidence in the integrity of WavPack's compression.

WavPack source code is portable and has been compiled on several [[Unix]] and [[Unix-like]] operating systems ([[Linux]], [[Mac OS X]], [[Solaris (operating system)|Solaris]], [[FreeBSD]], [[OpenBSD]], [[NetBSD]], Compaq [[Tru64]], [[HP-UX]]...) as well as [[Microsoft Windows|Windows]], [[DOS]], [[Palm OS]], and [[OpenVMS]]. It works on many architectures, including [[x86]], [[ARM architecture|ARM]], [[PowerPC]], [[AMD64]], [[IA-64]], [[SPARC]], [[DEC Alpha|Alpha]], [[PA-RISC]], [[MIPS architecture|MIPS]] and [[68k|Motorola 68k]].

A cut-down version of WavPack was developed for the Texas Instruments TMS320 series Digital Signal Processor. This was aimed predominantly at encouraging manufacturers to incorporate WavPack compression (and de-compression) into portable memory audio recorders. This version supported features that were applicable only to embedded applications (stream compression in real-time, selectable compression rate) and dropped off features that only applied to full computer systems (self extraction, high compression modes, 32-bit floats). The TMS320 series DSPs are native integer devices, and support WavPack well. Some "special" features of the full WavPack software were included (ability to generate a correction "file" (stream), for example), and others were excluded. The port was based on version 4.

WavPack support was added to [[WinZip]] starting with version 11.0 beta, released in October 2006.<ref name="winzip">{{cite web
| url = http://www.winzip.com/comp_info.htm
| title = WinZip – Additional Compression Methods Specification
| access-date = 6 January 2008
| publisher = WinZip International LLC
| date = 15 November 2006
}}</ref> This extension to the [[ZIP (file format)|ZIP file format]] was included by [[PKZIP|PKWARE]], the maintainers of the format, in the official {{mono|APPNOTE.TXT}} description file starting with version 6.3.2, released on 28 September 2007.<ref name="appnote">{{cite web
| url = http://www.pkware.com/documents/casestudies/APPNOTE.TXT
| title = APPNOTE.TXT – .ZIP File Format Specification
| access-date = 6 January 2008
| publisher = PKWARE Inc
| date = 28 September 2007
}}</ref>

==See also==
{{Portal|Free and open-source software}}
*[[Comparison of audio formats]]
*[[MPEG-4 SLS]]
*[[Free Lossless Audio Codec|FLAC]]
*TTA
*[[Monkey's Audio]]
*[[Apple Lossless Audio Codec|ALAC]]
*[[Meridian Lossless Packing]]

==References==
{{Reflist}}

==External links==
*[https://www.wavpack.com/ Official website]
*[https://hydrogenaud.io/index.php/board,68.0.html WavPack forum at Hydrogenaudio Forums]
*[https://www.rarewares.org/rrw/wavpack.php Historical versions at ReallyRareWares]
*[https://wiki.hydrogenaud.io/index.php?title=Lossless_comparison A comparison of several Lossless Audio encoders] at Hydrogenaudio Wiki.
*[https://wiki.multimedia.cx/index.php?title=WavPack WavPack on MultimediaWiki]
*[https://members.home.nl/w.speek/wavpack.htm WavPack frontend]
*[https://www.wavpack.com/flash/wavpack.htm Flash WavPack player]
{{Compression formats}}
{{Compression software implementations}}

[[Category:Lossless audio codecs]]
[[Category:Computer file formats]]
[[Category:Free audio software]]
[[Category:Free audio codecs]]
[[Category:Open formats]]
[[Category:Software using the BSD license]]

Floating-point arithmetic

2025-06-20T01:30:23Z

173.79.19.248: has never been such an article

{{Redirect|Floating point}}
{{Short description|Computer approximation for real numbers}}
{{Use dmy dates|date=May 2019|cs1-dates=y}}
[[File:Z3 Deutsches Museum.JPG|thumb|200px|An early electromechanical programmable computer, the [[Z3 (computer)|Z3]], included floating-point arithmetic (replica on display at [[Deutsches Museum]] in [[Munich]]).]]
{{Floating-point}}

In [[computing]], '''floating-point arithmetic''' ('''FP''') is [[arithmetic]] on subsets of [[real number]]s formed by a ''[[significand]]'' (a [[Sign (mathematics)|signed]] sequence of a fixed number of digits in some [[Radix|base]]) multiplied by an [[integer power]] of that base.
Numbers of this form are called '''floating-point numbers'''.<ref name="Muller_2010"/>{{rp|3}}<ref name="sterbenz1974fpcomp">{{cite book
|last1=Sterbenz
|first1=Pat H.
|title=Floating-Point Computation
|date=1974
|publisher=Prentice-Hall
|location=Englewood Cliffs, NJ, United States
|isbn=0-13-322495-3
|url=https://archive.org/details/SterbenzFloatingPointComputation/mode/2up}}</ref>{{rp|10}}

For example, the number 2469/200 is a floating-point number in base ten with five digits:
<math display=block>2469/200 = 12.345 = \! \underbrace{12345}_\text{significand} \! \times \! \underbrace{10}_\text{base}\!\!\!\!\!\!\!\overbrace{{}^{-3}}^{\text{exponent}}</math>
However, 7716/625 = 12.3456 is not a floating-point number in base ten with five digits—it needs six digits.
The nearest floating-point number with only five digits is 12.346.
And 1/3 = 0.3333… is not a floating-point number in base ten with any finite number of digits.
In practice, most floating-point systems use [[Binary number|base two]], though base ten ([[decimal floating point]]) is also common.

Floating-point arithmetic operations, such as addition and division, approximate the corresponding real number arithmetic operations by [[rounding]] any result that is not a floating-point number itself to a nearby floating-point number.<ref name="Muller_2010"/>{{rp|22}}<ref name="sterbenz1974fpcomp"/>{{rp|10}}
For example, in a floating-point arithmetic with five base-ten digits, the sum 12.345 + 1.0001 = 13.3451 might be rounded to 13.345.

The term ''floating point'' refers to the fact that the number's [[radix point]] can "float" anywhere to the left, right, or between the [[significant digits]] of the number. This position is indicated by the exponent, so floating point can be considered a form of [[scientific notation]].

A floating-point system can be used to represent, with a fixed number of digits, numbers of very different [[Orders of magnitude (numbers)|orders of magnitude]] — such as the number of meters [[Orders of magnitude (length)#100_zettametres|between galaxies]] or [[Orders of magnitude (length)#10_femtometres|between protons in an atom]]. For this reason, floating-point arithmetic is often used to allow very small and very large real numbers that require fast processing times. The result of this [[dynamic range]] is that the numbers that can be represented are not uniformly spaced; the difference between two consecutive representable numbers varies with their exponent.<ref name="Smith_1997"/>

[[File:A number line representing single-precision floating point's numbers and numbers that it cannot display.png|500x500px|thumb|right|Single-precision floating-point numbers on a [[number line]]: the green lines mark representable values.]]
[[File:FloatingPointPrecisionAugmented.png|500x500px|thumb|right|Augmented version above showing both [[Signed number representations|signs]] of representable values]]

Over the years, a variety of floating-point representations have been used in computers. In 1985, the [[IEEE 754]] Standard for Floating-Point Arithmetic was established, and since the 1990s, the most commonly encountered representations are those defined by the IEEE.

The speed of floating-point operations, commonly measured in terms of [[FLOPS]], is an important characteristic of a [[computer system]], especially for applications that involve intensive mathematical calculations.

A [[floating-point unit]] (FPU, colloquially a math [[coprocessor]]) is a part of a computer system specially designed to carry out operations on floating-point numbers.

== Overview ==
=== Floating-point numbers ===
A [[number representation]] specifies some way of encoding a number, usually as a string of digits.

There are several mechanisms by which strings of digits can represent numbers. In standard mathematical notation, the digit string can be of any length, and the location of the [[radix point]] is indicated by placing an explicit [[Decimal separator|"point" character]] (dot or comma) there. If the radix point is not specified, then the string implicitly represents an [[integer]] and the unstated radix point would be off the right-hand end of the string, next to the least significant digit. In [[fixed-point arithmetic|fixed-point]] systems, a position in the string is specified for the radix point. So a fixed-point scheme might use a string of 8 decimal digits with the decimal point in the middle, whereby "00012345" would represent 0001.2345.

In [[scientific notation]], the given number is scaled by a [[power of 10]], so that it lies within a specific range—typically between 1 and 10, with the radix point appearing immediately after the first digit. As a power of ten, the scaling factor is then indicated separately at the end of the number. For example, the orbital period of [[Jupiter]]'s moon [[Io (moon)|Io]] is {{val|152853.5047|fmt=commas}} seconds, a value that would be represented in standard-form scientific notation as {{val|1.528535047|e=5|fmt=commas}} seconds.

Floating-point representation is similar in concept to scientific notation. Logically, a floating-point number consists of:
* A signed (meaning positive or negative) digit string of a given length in a given [[radix]] (or base). This digit string is referred to as the ''[[significand]]'', ''mantissa'', or ''coefficient''.<ref group="nb" name="NB_Significand"/> The length of the significand determines the ''precision'' to which numbers can be represented. The radix point position is assumed always to be somewhere within the significand—often just after or just before the most significant digit, or to the right of the rightmost (least significant) digit. This article generally follows the convention that the radix point is set just after the most significant (leftmost) digit.
* A signed integer [[exponent]] (also referred to as the ''characteristic'', or ''scale''),<ref group="nb" name="NB_Exponent"/> which modifies the magnitude of the number.

To derive the value of the floating-point number, the ''significand'' is multiplied by the ''base'' raised to the power of the ''exponent'', equivalent to shifting the radix point from its implied position by a number of places equal to the value of the exponent—to the right if the exponent is positive or to the left if the exponent is negative.

Using base-10 (the familiar [[Decimal representation|decimal]] notation) as an example, the number {{val|152853.5047|fmt=commas}}, which has ten decimal digits of precision, is represented as the significand {{val|1528535047|fmt=commas}} together with 5 as the exponent. To determine the actual value, a decimal point is placed after the first digit of the significand and the result is multiplied by {{10^|5}} to give {{val|1.528535047|e=5|fmt=commas}}, or {{val|152853.5047|fmt=commas}}. In storing such a number, the base (10) need not be stored, since it will be the same for the entire range of supported numbers, and can thus be inferred.

Symbolically, this final value is:
<math display=block>\frac{s}{b^{\,p-1}} \times b^e,</math>

where {{mvar|s}} is the significand (ignoring any implied decimal point), {{mvar|p}} is the precision (the number of digits in the significand), {{mvar|b}} is the base (in our example, this is the number ''ten''), and {{mvar|e}} is the exponent.

{{anchor|Base-4|Base-8|Base-256|Base-65536}}Historically, several number bases have been used for representing floating-point numbers, with base two ([[Binary numeral system|binary]]) being the most common, followed by base ten ([[decimal floating point]]), and other less common varieties, such as base sixteen ([[hexadecimal floating point]]<ref name="Zehendner_2008"/><ref name="Beebe_2017"/><ref group="nb" name="NB_9"/>), base eight (octal floating point<ref name="Muller_2010"/><ref name="Beebe_2017"/><ref name="Savard_2018"/><ref name="Zehendner_2008"/><ref group="nb" name="NB_8"/>), base four (quaternary floating point<ref name="Parkinson_2000"/><ref name="Beebe_2017"/><ref group="nb" name="NB_11"/>), base three ([[balanced ternary floating point]]<ref name="Muller_2010"/>) and even base 256<ref name="Beebe_2017"/><ref group="nb" name="NB_12"/> and base {{val|65536|fmt=commas}}.<ref name="Lazarus_1956"/><ref group="nb" name="NB_10"/>

A floating-point number is a [[rational number]], because it can be represented as one integer divided by another; for example {{val|1.45|e=3}} is (145/100)×1000 or {{val|145000|fmt=commas}}/100. The base determines the fractions that can be represented; for instance, 1/5 cannot be represented exactly as a floating-point number using a binary base, but 1/5 can be represented exactly using a decimal base ({{val|0.2}}, or {{val|2|e=-1}}). However, 1/3 cannot be represented exactly by either binary (0.010101...) or decimal (0.333...), but in [[ternary numeral system|base 3]], it is trivial (0.1 or 1×3−1) . The occasions on which infinite expansions occur [[positional notation#Infinite representations|depend on the base and its prime factors]].


The way in which the significand (including its sign) and exponent are stored in a computer is implementation-dependent. The common IEEE formats are described in detail later and elsewhere, but as an example, in the binary single-precision (32-bit) floating-point representation, <math>p = 24</math>, and so the significand is a string of 24 [[bit]]s. For instance, the number [[Pi|π]]'s first 33 bits are:
<math display=block>11001001\ 00001111\ 1101101\underline{0}\ 10100010\ 0.</math>

In this binary expansion, let us denote the positions from 0 (leftmost bit, or most significant bit) to 32 (rightmost bit). The 24-bit significand will stop at position 23, shown as the underlined bit {{val|0}} above. The next bit, at position 24, is called the ''round bit'' or ''rounding bit''. It is used to round the 33-bit approximation to the nearest 24-bit number (there are [[rounding#Tie-breaking|specific rules for halfway values]], which is not the case here). This bit, which is {{val|1=1}} in this example, is added to the integer formed by the leftmost 24 bits, yielding:
<math display=block>11001001\ 00001111\ 1101101\underline{1}.</math>

When this is stored in memory using the IEEE 754 encoding, this becomes the [[significand]] {{mvar|s}}. The significand is assumed to have a binary point to the right of the leftmost bit. So, the binary representation of π is calculated from left-to-right as follows:
<math display=block>\begin{align}
&\biggl(\sum_{n=0}^{p-1} \text{bit}_n \times 2^{-n}\biggr) \times 2^e \\
&\qquad{}= \left(1 \times 2^{-0} + 1 \times 2^{-1} + 0 \times 2^{-2} + 0 \times 2^{-3} + \cdots + 1 \times 2^{-23}\right) \times 2^1 \\[2mu]
&\qquad{} \approx 1.57079637 \times 2 \\[3mu]
&\qquad{} \approx 3.1415927
\end{align}</math>

where {{mvar|p}} is the precision ({{val|24}} in this example), {{mvar|n}} is the position of the bit of the significand from the left (starting at {{val|0}} and finishing at {{val|23}} here) and {{mvar|e}} is the exponent ({{val|1=1}} in this example).

{{anchor|Hidden bit}}It can be required that the most significant digit of the significand of a non-zero number be non-zero (except when the corresponding exponent would be smaller than the minimum one). This process is called ''normalization''. For binary formats (which uses only the digits {{val|0}} and {{val|1=1}}), this non-zero digit is necessarily {{val|1}}. Therefore, it does not need to be represented in memory, allowing the format to have one more bit of precision. This rule is variously called the ''leading bit convention'', the ''implicit bit convention'', the ''hidden bit convention'',<ref name="Muller_2010"/> or the ''assumed bit convention''.

=== Alternatives to floating-point numbers ===
The floating-point representation is by far the most common way of representing in computers an approximation to real numbers. However, there are alternatives:
* [[Fixed-point arithmetic|Fixed-point]] representation uses integer hardware operations controlled by a software implementation of a specific convention about the location of the binary or decimal point, for example, 6 bits or digits from the right. The hardware to manipulate these representations is less costly than floating point, and it can be used to perform normal integer operations, too. Binary fixed point is usually used in special-purpose applications on embedded processors that can only do integer arithmetic, but decimal fixed point is common in commercial applications.
* [[Logarithmic number system]]s (LNSs) represent a real number by the logarithm of its absolute value and a sign bit. The value distribution is similar to floating point, but the value-to-representation curve (''i.e.'', the graph of the logarithm function) is smooth (except at 0). Conversely to floating-point arithmetic, in a logarithmic number system multiplication, division and exponentiation are simple to implement, but addition and subtraction are complex. The ([[symmetric level-index arithmetic|symmetric]]) [[level-index arithmetic]] (LI and SLI) of Charles Clenshaw, [[Frank William John Olver|Frank Olver]] and Peter Turner is a scheme based on a [[generalized logarithm]] representation.
* [[Tapered floating-point representation]], used in [[Unum (number format)|Unum]].
* Some simple rational numbers (''e.g.'', 1/3 and 1/10) cannot be represented exactly in binary floating point, no matter what the precision is. Using a different radix allows one to represent some of them (''e.g.'', 1/10 in decimal floating point), but the possibilities remain limited. Software packages that perform [[fraction|rational arithmetic]] represent numbers as fractions with integral numerator and denominator, and can therefore represent any rational number exactly. Such packages generally need to use "[[bignum]]" arithmetic for the individual integers.
* [[Interval arithmetic]] allows one to represent numbers as intervals and obtain guaranteed bounds on results. It is generally based on other arithmetics, in particular floating point.
* [[Computer algebra system]]s such as [[Mathematica]], [[Maxima (software)|Maxima]], and [[Maple (software)|Maple]] can often handle irrational numbers like <math>\pi</math> or <math>\sqrt{3}</math> in a completely "formal" way ([[symbolic computation]]), without dealing with a specific encoding of the significand. Such a program can evaluate expressions like "<math>\sin (3\pi)</math>" exactly, because it is programmed to process the underlying mathematics directly, instead of using approximate values for each intermediate calculation.

== History ==
{{see also|IEEE 754#History}}

[[File:Quevedo 1917.jpg|thumb|upright=0.7|[[Leonardo Torres Quevedo]], in 1914, published an analysis of floating point based on the [[analytical engine]].]]
In 1914, the Spanish engineer [[Leonardo Torres Quevedo]] published ''Essays on Automatics'',<ref>Torres Quevedo, Leonardo. [https://quickclick.es/rop/pdf/publico/1914/1914_tomoI_2043_01.pdf Automática: Complemento de la Teoría de las Máquinas, (pdf)], pp. 575–583, Revista de Obras Públicas, 19 November 1914.</ref> where he designed a special-purpose electromechanical calculator based on [[Charles Babbage]]'s [[analytical engine]] and described a way to store floating-point numbers in a consistent manner. He stated that numbers will be stored in exponential format as ''n'' × 10<math>^m</math>, and offered three rules by which consistent manipulation of floating-point numbers by machines could be implemented. For Torres, "''n'' will always be the same number of [[Numerical digit|digits]] (e.g. six), the first digit of ''n'' will be of order of tenths, the second of hundredths, etc, and one will write each quantity in the form: ''n''; ''m''." The format he proposed shows the need for a fixed-sized significand as is presently used for floating-point data, fixing the location of the decimal point in the significand so that each representation was unique, and how to format such numbers by specifying a syntax to be used that could be entered through a [[typewriter]], as was the case of his [[Leonardo Torres y Quevedo#Analytical machines|Electromechanical Arithmometer]] in 1920.<ref>Ronald T. Kneusel. ''[https://books.google.com/books?id=eq4ZDgAAQBAJ&dq=leonardo+torres+quevedo++electromechanical+machine+essays&pg=PA84 Numbers and Computers],'' Springer, pp. 84–85, 2017. {{ISBN|978-3319505084}}</ref>{{Sfn|Randell|1982|pp=6, 11–13}}<ref>Randell, Brian. [https://dl.acm.org/doi/pdf/10.5555/1074100.1074334 Digital Computers, History of Origins, (pdf)], p. 545, Digital Computers: Origins, Encyclopedia of Computer Science, January 2003.</ref>

[[File:Konrad Zuse (1992).jpg|thumb|upright=0.7|right|[[Konrad Zuse]], architect of the [[Z3 (computer)|Z3]] computer, which uses a 22-bit binary floating-point representation]]

In 1938, [[Konrad Zuse]] of Berlin completed the [[Z1 (computer)|Z1]], the first binary, programmable [[mechanical computer]];<ref name="Rojas_1997"/> it uses a 24-bit binary floating-point number representation with a 7-bit signed exponent, a 17-bit significand (including one implicit bit), and a sign bit.<ref name="Rojas_2014"/> The more reliable [[relay]]-based [[Z3 (computer)|Z3]], completed in 1941, has representations for both positive and negative infinities; in particular, it implements defined operations with infinity, such as <math>^1/_\infty = 0</math>, and it stops on undefined operations, such as <math>0 \times \infty</math>.

Zuse also proposed, but did not complete, carefully rounded floating-point arithmetic that includes <math>\pm\infty</math> and NaN representations, anticipating features of the IEEE Standard by four decades.<ref name="Kahan_1997_JVNL"/> In contrast, [[John von Neumann|von Neumann]] recommended against floating-point numbers for the 1951 [[IAS machine]], arguing that fixed-point arithmetic is preferable.<ref name="Kahan_1997_JVNL"/>

The first ''commercial'' computer with floating-point hardware was Zuse's [[Z4 (computer)|Z4]] computer, designed in 1942–1945. In 1946, Bell Laboratories introduced the [[Model V|Model V]], which implemented [[decimal floating point|decimal floating-point numbers]].<ref name="Randell_1982_2"/>

The [[Pilot ACE]] has binary floating-point arithmetic, and it became operational in 1950 at [[National Physical Laboratory, UK]]. Thirty-three were later sold commercially as the [[English Electric DEUCE]]. The arithmetic is actually implemented in software, but with a one megahertz clock rate, the speed of floating-point and fixed-point operations in this machine were initially faster than those of many competing computers.

The mass-produced [[IBM 704]] followed in 1954; it introduced the use of a [[Exponent bias|biased exponent]]. For many decades after that, floating-point hardware was typically an optional feature, and computers that had it were said to be "scientific computers", or to have "[[scientific computation]]" (SC) capability (see also [[Extensions for Scientific Computation]] (XSC)). It was not until the launch of the Intel i486 in 1989 that ''general-purpose'' personal computers had floating-point capability in hardware as a standard feature.

The [[UNIVAC 1100/2200 series]], introduced in 1962, supported two floating-point representations:
* ''Single precision'': 36 bits, organized as a 1-bit sign, an 8-bit exponent, and a 27-bit significand.
* ''Double precision'': 72 bits, organized as a 1-bit sign, an 11-bit exponent, and a 60-bit significand.

The [[IBM 7094]], also introduced in 1962, supported single-precision and double-precision representations, but with no relation to the UNIVAC's representations. Indeed, in 1964, IBM introduced [[IBM hexadecimal floating-point|hexadecimal floating-point representations]] in its [[System/360]] mainframes; these same representations are still available for use in modern [[z/Architecture]] systems. In 1998, IBM implemented IEEE-compatible binary floating-point arithmetic in its mainframes; in 2005, IBM also added IEEE-compatible decimal floating-point arithmetic.

Initially, computers used many different representations for floating-point numbers. The lack of standardization at the mainframe level was an ongoing problem by the early 1970s for those writing and maintaining higher-level source code; these manufacturer floating-point standards differed in the word sizes, the representations, and the rounding behavior and general accuracy of operations. Floating-point compatibility across multiple computing systems was in desperate need of standardization by the early 1980s, leading to the creation of the [[IEEE 754]] standard once the 32-bit (or 64-bit) [[Word (computer architecture)|word]] had become commonplace. This standard was significantly based on a proposal from Intel, which was designing the [[Intel 8087|i8087]] numerical coprocessor; Motorola, which was designing the [[68000]] around the same time, gave significant input as well.

[[File:William Kahan 2008.jpg|thumb|upright=0.7|right|[[William Kahan]], principal architect of the [[IEEE 754]] floating-point standard]]

In 1989, mathematician and computer scientist [[William Kahan]] was honored with the [[Turing Award]] for being the primary architect behind this proposal; he was aided by his student Jerome Coonen and a visiting professor, [[Harold S. Stone|Harold Stone]].<ref name="Severance_1998"/>

Among the x86 (more specifically i8087) innovations are these:
* A precisely specified floating-point representation at the bit-string level, so that all compliant computers interpret bit patterns the same way. This makes it possible to accurately and efficiently transfer floating-point numbers from one computer to another (after accounting for [[endianness]]).
* A precisely specified behavior for the arithmetic operations: A result is required to be produced as if infinitely precise arithmetic were used to yield a value that is then rounded according to specific rules. This means that a compliant computer program would always produce the same result when given a particular input, thus mitigating the almost mystical reputation that floating-point computation had developed for its hitherto seemingly non-deterministic behavior.
* The ability of [[IEEE 754#Exception handling|exceptional conditions]] (overflow, [[Division by zero|divide by zero]], etc.) to propagate through a computation in a benign manner and then be handled by the software in a controlled fashion.

These features would be inherited into IEEE 754-1985 (with the exception of the encoding of special values and exceptions), though the extended internal precision of x87 means it requires explicit rounding of exact results directly to the destination precision in order to match standard IEEE 754 results.<ref name="Goldberg_1991"/> However, the behavior may not be the same as a rounding to the destination format due to a possible wider exponent range of the extended format.

== Range of floating-point numbers ==
A floating-point number consists of two [[Fixed-point arithmetic|fixed-point]] components, whose range depends exclusively on the number of bits or digits in their representation. Whereas components linearly depend on their range, the floating-point range linearly depends on the significand range and exponentially on the range of exponent component, which attaches outstandingly wider range to the number.

On a typical computer system, a ''[[Double-precision floating-point format|double-precision]]'' (64-bit) binary floating-point number has a coefficient of 53 bits (including 1 implied bit), an exponent of 11 bits, and 1 sign bit. Since 210 = 1024, the complete range of the positive normal floating-point numbers in this format is from 2−1022 ≈ 2 × 10−308 to approximately 21024 ≈ 2 × 10308.

The number of normal floating-point numbers in a system (''B'', ''P'', ''L'', ''U'') where
* ''B'' is the base of the system,
* ''P'' is the precision of the significand (in base ''B''),
* ''L'' is the smallest exponent of the system,
* ''U'' is the largest exponent of the system,
is <math>2 \left(B - 1\right) \left(B^{P-1}\right) \left(U - L + 1\right)</math>.

There is a smallest positive normal floating-point number,
: Underflow level = UFL = <math>B^L</math>,

which has a 1 as the leading digit and 0 for the remaining digits of the significand, and the smallest possible value for the exponent.

There is a largest floating-point number,
: Overflow level = OFL = <math>\left(1 - B^{-P}\right)\left(B^{U + 1}\right)</math>,

which has ''B'' − 1 as the value for each digit of the significand and the largest possible value for the exponent.

In addition, there are representable values strictly between −UFL and UFL. Namely, [[signed zero|positive and negative zeros]], as well as [[subnormal number]]s.

== IEEE 754: floating point in modern computers {{anchor|IEEE 754}} ==
{{Main|IEEE 754}}
{{Floating-point}}

The [[Institute of Electrical and Electronics Engineers|IEEE]] standardized the computer representation for binary floating-point numbers in [[IEEE 754]] (a.k.a. IEC 60559) in 1985. This first standard is followed by almost all modern machines. It was [[IEEE 754-2008 revision|revised in 2008]]. IBM mainframes support [[IBM hexadecimal floating point|IBM's own hexadecimal floating point format]] and IEEE 754-2008 [[decimal floating point]] in addition to the IEEE 754 binary format. The [[Cray T90]] series had an IEEE version, but the [[Cray SV1|SV1]] still uses Cray floating-point format.{{Citation needed|date=July 2020}}

The standard provides for many closely related formats, differing in only a few details. Five of these formats are called ''basic formats'', and others are termed ''extended precision formats'' and ''extendable precision format''. Three formats are especially widely used in computer hardware and languages:{{Citation needed|reason=Possibly wrong for double extended: OK for hardware, but for languages? Note that in C, long double may not correspond to double extended (see 32-bit ARM and PowerPC).|date=July 2020}}
* [[Single-precision floating-point format|Single precision]] (binary32), usually used to represent the "float" [[C data types#Basic types|type in the C language]] family. This is a binary format that occupies 32 bits (4 bytes) and its significand has a precision of 24 bits (about 7 decimal digits).
* [[Double-precision floating-point format|Double precision]] (binary64), usually used to represent the "double" [[C data types#Basic types|type in the C language]] family. This is a binary format that occupies 64 bits (8 bytes) and its significand has a precision of 53 bits (about 16 decimal digits).
* [[Extended precision|Double extended]], also ambiguously called "extended precision" format. This is a binary format that occupies at least 79 bits (80 if the hidden/implicit bit rule is not used) and its significand has a precision of at least 64 bits (about 19 decimal digits). The [[C99]] and [[C11 (C standard revision)|C11]] standards of the C language family, in their annex F ("IEC 60559 floating-point arithmetic"), recommend such an extended format to be provided as "[[long double]]".<ref name="C99"/> A format satisfying the minimal requirements (64-bit significand precision, 15-bit exponent, thus fitting on 80 bits) is provided by the [[x86]] architecture. Often on such processors, this format can be used with "long double", though extended precision is not available with MSVC.<ref name="MSVC"/> For [[Data structure alignment|alignment]] purposes, many tools store this 80-bit value in a 96-bit or 128-bit space.<ref name="GCC"/><ref name="float_128"/> On other processors, "long double" may stand for a larger format, such as quadruple precision,<ref name="ARM_2013_AArch64"/> or just double precision, if any form of extended precision is not available.<ref name="ARM_2013_Compiler"/>

Increasing the precision of the floating-point representation generally reduces the amount of accumulated [[round-off error]] caused by intermediate calculations.<ref name="Kahan_2004"/>
Other IEEE formats include:

* [[Decimal64 floating-point format|Decimal64]] and [[decimal128 floating-point format|decimal128]] floating-point formats. These formats (especially decimal128) are pervasive in financial transactions because, along with the [[Decimal32 floating-point format|decimal32]] format, they allow correct decimal rounding.
* [[Quadruple-precision floating-point format#IEEE 754 quadruple-precision binary floating-point format: binary128|Quadruple precision]] (binary128). This is a binary format that occupies 128 bits (16 bytes) and its significand has a precision of 113 bits (about 34 decimal digits).
* [[Half-precision floating-point format|Half precision]], also called binary16, a 16-bit floating-point value. It is being used in the NVIDIA [[Cg (programming language)|Cg]] graphics language, and in the openEXR standard (where it actually predates the introduction in the IEEE 754 standard).<ref name="OpenEXR"/><ref name="OpenEXR-half"/>


Any integer with absolute value less than 224 can be exactly represented in the single-precision format, and any integer with absolute value less than 253 can be exactly represented in the double-precision format. Furthermore, a wide range of powers of 2 times such a number can be represented. These properties are sometimes used for purely integer data, to get 53-bit integers on platforms that have double-precision floats but only 32-bit integers.

The standard specifies some special values, and their representation: positive [[infinity]] ({{math|+∞}}), negative infinity ({{math|−∞}}), a [[negative zero]] (−0) distinct from ordinary ("positive") zero, and "not a number" values ([[NaN]]s).

Comparison of floating-point numbers, as defined by the IEEE standard, is a bit different from usual integer comparison. Negative and positive zero compare equal, and every NaN compares unequal to every value, including itself. All finite floating-point numbers are strictly smaller than {{math|+∞}} and strictly greater than {{math|−∞}}, and they are ordered in the same way as their values (in the set of real numbers).

=== Internal representation ===
Floating-point numbers are typically packed into a computer datum as the sign bit, the exponent field, and a field for the significand, from left to right. For the [[IEEE 754]] binary formats (basic and extended) that have extant hardware implementations, they are apportioned as follows:

{| class="wikitable" style="text-align:right; border:0"
|-
!rowspan="2" |Format
!colspan="4" |Bits for the encoding
| rowspan="8" style="background:white; border:0"|
!rowspan="2" |Exponent bias
!rowspan="2" |Bits precision
!rowspan="2" |Number of decimal digits
|-
!Sign
!Exponent
!Significand
!Total
|-
|[[Half-precision floating-point format|Half]] (binary16)
|1
|5
|10
|16
|15
|11
|~3.3
|-
|[[Single-precision floating-point format|Single]] (binary32)
|1
|8
|23
|32
|127
|24
|~7.2
|-
|[[Double-precision floating-point format|Double]] (binary64)
|1
|11
|52
|64
|1023
|53
|~15.9
|-
|[[Extended precision#x86 extended-precision format|x86 extended]]
|1
|15
|64
|80
|16383
|64
|~19.2
|-
|[[Quadruple-precision floating-point format|Quadruple]] (binary128)
|1
|15
|112
|128
|16383
|113
|~34.0
|-
|[[Octuple-precision floating-point format|Octuple]] (binary256)
|1
|19
|236
|256
|262143
|237
|~71.3
|}

While the exponent can be positive or negative, in binary formats it is stored as an unsigned number that has a fixed "bias" added to it. Values of all 0s in this field are reserved for the zeros and [[subnormal number]]s; values of all 1s are reserved for the infinities and NaNs. The exponent range for normal numbers is [−126, 127] for single precision, [−1022, 1023] for double, or [−16382, 16383] for quad. Normal numbers exclude subnormal values, zeros, infinities, and NaNs.

In the IEEE binary interchange formats the leading bit of a normalized significand is not actually stored in the computer datum, since it is always 1. It is called the "hidden" or "implicit" bit. Because of this, the single-precision format actually has a significand with 24 bits of precision, the double-precision format has 53, quad has 113, and octuple has 237.

For example, it was shown above that π, rounded to 24 bits of precision, has:
* sign = 0 ; ''e'' = 1 ; ''s'' = 110010010000111111011011 (including the hidden bit)
The sum of the exponent bias (127) and the exponent (1) is 128, so this is represented in the single-precision format as
* 0 10000000 10010010000111111011011 (excluding the hidden bit) = 40490FDB<ref name="IEEE-754_Analysis"/> as a [[hexadecimal]] number.

An example of a layout for [[Single-precision floating-point format|32-bit floating point]] is
[[File:Float example.svg|none]]
and the [[Double-precision floating-point format|64-bit ("double")]] layout is similar.

== Other notable floating-point formats ==
In addition to the widely used [[IEEE 754]] standard formats, other floating-point formats are used, or have been used, in certain domain-specific areas.
* The [[Microsoft Binary Format|Microsoft Binary Format (MBF)]] was developed for the Microsoft BASIC language products, including Microsoft's first ever product the [[Altair BASIC]] (1975), [[TRS-80|TRS-80 LEVEL II]], [[CP/M]]'s [[MBASIC]], [[IBM PC 5150]]'s [[BASICA]], [[MS-DOS]]'s [[GW-BASIC]] and [[QuickBASIC]] prior to version 4.00. QuickBASIC version 4.00 and 4.50 switched to the IEEE 754-1985 format but can revert to the MBF format using the /MBF command option. MBF was designed and developed on a simulated [[Intel 8080]] by [[Monte Davidoff]], a dormmate of [[Bill Gates]], during spring of 1975 for the [[MITS Altair 8800]]. The initial release of July 1975 supported a single-precision (32 bits) format due to cost of the [[MITS Altair 8800]] 4-kilobytes memory. In December 1975, the 8-kilobytes version added a double-precision (64 bits) format. A single-precision (40 bits) variant format was adopted for other CPU's, notably the [[MOS 6502]] ([[Apple II]], [[Commodore PET]], [[Atari]]), [[Motorola 6800]] (MITS Altair 680) and [[Motorola 6809]] ([[TRS-80 Color Computer]]). All Microsoft language products from 1975 through 1987 used the [[Microsoft Binary Format]] until Microsoft adopted the IEEE 754 standard format in all its products starting in 1988 to their current releases. MBF consists of the MBF single-precision format (32 bits, "6-digit BASIC"),<ref name="Borland_1994_MBF"/><ref name="Steil_2008_6502"/> the MBF extended-precision format (40 bits, "9-digit BASIC"),<ref name="Steil_2008_6502"/> and the MBF double-precision format (64 bits);<ref name="Borland_1994_MBF"/><ref name="Microsoft_2006_KB35826"/> each of them is represented with an 8-bit exponent, followed by a sign bit, followed by a significand of respectively 23, 31, and 55 bits.
* The [[bfloat16 floating-point format|bfloat16 format]] requires the same amount of memory (16 bits) as the [[Half-precision floating-point format|IEEE 754 half-precision format]], but allocates 8 bits to the exponent instead of 5, thus providing the same range as a [[Single-precision floating-point format|IEEE 754 single-precision]] number. The tradeoff is a reduced precision, as the trailing significand field is reduced from 10 to 7 bits. This format is mainly used in the training of [[machine learning]] models, where range is more valuable than precision. Many machine learning accelerators provide hardware support for this format.
* The TensorFloat-32<ref name="Kharya_2020"/> format combines the 8 bits of exponent of the bfloat16 with the 10 bits of trailing significand field of half-precision formats, resulting in a size of 19 bits. This format was introduced by [[Nvidia]], which provides hardware support for it in the Tensor Cores of its [[Graphics processing unit|GPUs]] based on the Nvidia Ampere architecture. The drawback of this format is its size, which is not a power of 2. However, according to Nvidia, this format should only be used internally by hardware to speed up computations, while inputs and outputs should be stored in the 32-bit single-precision IEEE 754 format.<ref name="Kharya_2020"/>
* The [[Hopper (microarchitecture)|Hopper]] architecture GPUs provide two FP8 formats: one with the same numerical range as half-precision (E5M2) and one with higher precision, but less range (E4M3).<ref name="NVIDIA_Hopper"/><ref name="Micikevicius_2022"/>
* The [[Blackwell (microarchitecture)|Blackwell]] GPU architecture includes support for FP6 (E3M2 and E2M3) and FP4 (E2M1) formats. FP4 is the smallest floating-point format which allows for all IEEE 754 principles (see [[minifloat]]).

{| class="wikitable"
|+ Comparison of common floating-point formats
!Type
!Sign
!Exponent
!Significand
!Total bits
|-
|FP4
|1
|2
|1
|4
|-
|FP6 (E2M3)
|1
|2
|3
|6
|-
|FP6 (E3M2)
|1
|3
|2
|6
|-
|FP8 (E4M3)
|1
|4
|3
|8
|-
|FP8 (E5M2)
|1
|5
|2
|8
|-
|[[Half-precision floating-point format|Half-precision]]
|1
|5
|10
|16
|-
|[[bfloat16 floating-point format|bfloat16]]
|1
|8
|7
|16
|-
|[[TensorFloat-32]]
|1
|8
|10
|19
|-
|[[Single-precision floating-point format|Single-precision]]
|1
|8
|23
|32
|-
|[[Double-precision floating-point format|Double-precision]]
|1
|11
|52
|64
|-
|[[Quadruple-precision floating-point format|Quadruple-precision]]
|1
|15
|112
|128
|-
|[[Octuple-precision floating-point format|Octuple-precision]]
|1
|19
|236
|256
|}

==Representable numbers, conversion and rounding {{anchor|Representable numbers}}==

By their nature, all numbers expressed in floating-point format are [[rational number]]s with a terminating expansion in the relevant base (for example, a terminating decimal expansion in base-10, or a terminating binary expansion in base-2). Irrational numbers, such as [[Pi|π]] or <math display=inline>\sqrt{2}</math>, or non-terminating rational numbers, must be approximated. The number of digits (or bits) of precision also limits the set of rational numbers that can be represented exactly. For example, the decimal number 123456789 cannot be exactly represented if only eight decimal digits of precision are available (it would be rounded to one of the two straddling representable values, 12345678 × 101 or 12345679 × 101), the same applies to [[Repeating decimal|non-terminating digits]] (.{{overline|5}} to be rounded to either .55555555 or .55555556).

When a number is represented in some format (such as a character string) which is not a native floating-point representation supported in a computer implementation, then it will require a conversion before it can be used in that implementation. If the number can be represented exactly in the floating-point format then the conversion is exact. If there is not an exact representation then the conversion requires a choice of which floating-point number to use to represent the original value. The representation chosen will have a different value from the original, and the value thus adjusted is called the ''rounded value''.

Whether or not a rational number has a terminating expansion depends on the base. For example, in base-10 the number 1/2 has a terminating expansion (0.5) while the number 1/3 does not (0.333...). In base-2 only rationals with denominators that are powers of 2 (such as 1/2 or 3/16) are terminating. Any rational with a denominator that has a prime factor other than 2 will have an infinite binary expansion. This means that numbers that appear to be short and exact when written in decimal format may need to be approximated when converted to binary floating-point. For example, the decimal number 0.1 is not representable in binary floating-point of any finite precision; the exact binary representation would have a "1100" sequence continuing endlessly:
: ''e'' = −4; ''s'' = 1100110011001100110011001100110011...,
where, as previously, ''s'' is the significand and ''e'' is the exponent.

When rounded to 24 bits this becomes
: ''e'' = −4; ''s'' = 110011001100110011001101,
which is actually 0.100000001490116119384765625 in decimal.


As a further example, the real number [[Pi|π]], represented in binary as an infinite sequence of bits is
: 11.0010010000111111011010101000100010000101101000110000100011010011...
but is
: 11.0010010000111111011011
when approximated by [[rounding]] to a precision of 24 bits.

In binary single-precision floating-point, this is represented as ''s'' = 1.10010010000111111011011 with ''e'' = 1.
This has a decimal value of
: '''3.141592'''7410125732421875,
whereas a more accurate approximation of the true value of π is
: '''3.14159265358979323846264338327950'''...

The result of rounding differs from the true value by about 0.03 parts per million, and matches the decimal representation of π in the first 7 digits. The difference is the [[discretization error]] and is limited by the [[machine epsilon]].

The arithmetical difference between two consecutive representable floating-point numbers which have the same exponent is called a [[unit in the last place]] (ULP). For example, if there is no representable number lying between the representable numbers 1.45A70C2216 and 1.45A70C2416, the ULP is 2×16−8, or 2−31. For numbers with a base-2 exponent part of 0, i.e. numbers with an absolute value higher than or equal to 1 but lower than 2, an ULP is exactly 2−23 or about 10−7 in single precision, and exactly 2−53 or about 10−16 in double precision. The mandated behavior of IEEE-compliant hardware is that the result be within one-half of a ULP.

=== Rounding modes ===
Rounding is used when the exact result of a floating-point operation (or a conversion to floating-point format) would need more digits than there are digits in the significand. IEEE 754 requires ''correct rounding'': that is, the rounded result is as if infinitely precise arithmetic was used to compute the value and then rounded (although in implementation only three extra bits are needed to ensure this). There are several different [[rounding]] schemes (or ''rounding modes''). Historically, [[truncation]] was the typical approach. Since the introduction of IEEE 754, the default method (''[[rounding|round to nearest, ties to even]]'', sometimes called Banker's Rounding) is more commonly used. This method rounds the ideal (infinitely precise) result of an arithmetic operation to the nearest representable value, and gives that representation as the result.<ref group="nb" name="NB_1"/> In the case of a tie, the value that would make the significand end in an even digit is chosen. The IEEE 754 standard requires the same rounding to be applied to all fundamental algebraic operations, including square root and conversions, when there is a numeric (non-NaN) result. It means that the results of IEEE 754 operations are completely determined in all bits of the result, except for the representation of NaNs. ("Library" functions such as cosine and log are not mandated.)

Alternative rounding options are also available. IEEE 754 specifies the following rounding modes:
* round to nearest, where ties round to the nearest even digit in the required position (the default and by far the most common mode)
* round to nearest, where ties round away from zero (optional for binary floating-point and commonly used in decimal)
* round up (toward +∞; negative results thus round toward zero)
* round down (toward −∞; negative results thus round away from zero)
* round toward zero (truncation; it is similar to the common behavior of float-to-integer conversions, which convert −3.9 to −3 and 3.9 to 3)

Alternative modes are useful when the amount of error being introduced must be bounded. Applications that require a bounded error are multi-precision floating-point, and [[interval arithmetic]].
The alternative rounding modes are also useful in diagnosing numerical instability: if the results of a subroutine vary substantially between rounding to + and − infinity then it is likely numerically unstable and affected by round-off error.<ref name="Kahan_2006_Mindless"/>

=== Binary-to-decimal conversion with minimal number of digits ===
Converting a double-precision binary floating-point number to a decimal string is a common operation, but an algorithm producing results that are both accurate and minimal did not appear in print until 1990, with Steele and White's Dragon4. Some of the improvements since then include:
* David M. Gay's ''dtoa.c'', a practical open-source implementation of many ideas in Dragon4.<ref name="Gay_1990"/>
* Grisu3, with a 4× speedup as it removes the use of [[bignum]]s. Must be used with a fallback, as it fails for ~0.5% of cases.<ref name="Loitsch_2010"/>
* Errol3, an always-succeeding algorithm similar to, but slower than, Grisu3. Apparently not as good as an early-terminating Grisu with fallback.<ref name="mazong"/>
* Ryū, an always-succeeding algorithm that is faster and simpler than Grisu3.<ref name="Adams_2018"/>
* Schubfach, an always-succeeding algorithm that is based on a similar idea to Ryū, developed almost simultaneously and independently.<ref name="Giulietti"/> Performs better than Ryū and Grisu3 in certain benchmarks.<ref name="abolz"/>

Many modern language runtimes use Grisu3 with a Dragon4 fallback.<ref name="double_conversion_2020"/>

=== Decimal-to-binary conversion ===
The problem of parsing a decimal string into a binary FP representation is complex, with an accurate parser not appearing until Clinger's 1990 work (implemented in dtoa.c).<ref name="Gay_1990"/> Further work has likewise progressed in the direction of faster parsing.<ref name="Lemire_2021"/>

== Floating-point operations ==
For ease of presentation and understanding, decimal [[radix]] with 7 digit precision will be used in the examples, as in the IEEE 754 ''decimal32'' format. The fundamental principles are the same in any [[radix]] or precision, except that normalization is optional (it does not affect the numerical value of the result). Here, ''s'' denotes the significand and ''e'' denotes the exponent.

=== Addition and subtraction ===
A simple method to add floating-point numbers is to first represent them with the same exponent. In the example below, the second number (with the smaller exponent) is shifted right by three digits, and one then proceeds with the usual addition method:

123456.7 = 1.234567 × 10^5
101.7654 = 1.017654 × 10^2 = 0.001017654 × 10^5

Hence:
123456.7 + 101.7654 = (1.234567 × 10^5) + (1.017654 × 10^2)
= (1.234567 × 10^5) + (0.001017654 × 10^5)
= (1.234567 + 0.001017654) × 10^5
= 1.235584654 × 10^5

In detail:

e=5; s=1.234567 (123456.7)
+ e=2; s=1.017654 (101.7654)

e=5; s=1.234567
+ e=5; s=0.001017654 (after shifting)
--------------------
e=5; s=1.235584654 (true sum: 123558.4654)

This is the true result, the exact sum of the operands. It will be rounded to seven digits and then normalized if necessary. The final result is
e=5; s=1.235585 (final sum: 123558.5)

The lowest three digits of the second operand (654) are essentially lost. This is [[round-off error]]. In extreme cases, the sum of two non-zero numbers may be equal to one of them:

e=5; s=1.234567
+ e=−3; s=9.876543

e=5; s=1.234567
+ e=5; s=0.00000009876543 (after shifting)
----------------------
e=5; s=1.23456709876543 (true sum)
e=5; s=1.234567 (after rounding and normalization)

In the above conceptual examples it would appear that a large number of extra digits would need to be provided by the adder to ensure correct rounding; however, for binary addition or subtraction using careful implementation techniques only a ''guard'' bit, a ''rounding'' bit and one extra ''sticky'' bit need to be carried beyond the precision of the operands.<ref name="Goldberg_1991"/><ref name="Patterson-Hennessy_2014"/>{{rp|218–220}}

Another problem of loss of significance occurs when ''approximations'' to two nearly equal numbers are subtracted. In the following example ''e'' = 5; ''s'' = 1.234571 and ''e'' = 5; ''s'' = 1.234567 are approximations to the rationals 123457.1467 and 123456.659.

e=5; s=1.234571
− e=5; s=1.234567
----------------
e=5; s=0.000004
e=−1; s=4.000000 (after rounding and normalization)

The floating-point difference is computed exactly because the numbers are close—the [[Sterbenz lemma]] guarantees this, even in case of underflow when [[gradual underflow]] is supported. Despite this, the difference of the original numbers is ''e'' = −1; ''s'' = 4.877000, which differs more than 20% from the difference ''e'' = −1; ''s'' = 4.000000 of the approximations. In extreme cases, all significant digits of precision can be lost.<ref name="Goldberg_1991"/><ref name="Sierra_1962"/> This ''[[Catastrophic cancellation|cancellation]]'' illustrates the danger in assuming that all of the digits of a computed result are meaningful. Dealing with the consequences of these errors is a topic in [[numerical analysis]]; see also [[#Accuracy problems|Accuracy problems]].

=== Multiplication and division ===
To multiply, the significands are multiplied while the exponents are added, and the result is rounded and normalized.

e=3; s=4.734612
× e=5; s=5.417242
-----------------------
e=8; s=25.648538980104 (true product)
e=8; s=25.64854 (after rounding)
e=9; s=2.564854 (after normalization)

Similarly, division is accomplished by subtracting the divisor's exponent from the dividend's exponent, and dividing the dividend's significand by the divisor's significand.

There are no cancellation or absorption problems with multiplication or division, though small errors may accumulate as operations are performed in succession.<ref name="Goldberg_1991"/> In practice, the way these operations are carried out in digital logic can be quite complex (see [[Booth's multiplication algorithm]] and [[Division algorithm]]).<ref group="nb" name="NB_2"/>

=== Literal syntax ===
Literals for floating-point numbers depend on languages. They typically use <code>e</code> or <code>E</code> to denote [[scientific notation]]. The [[C (programming language)|C programming language]] and the [[IEEE 754]] standard also define a [[IEEE 754#Hexadecimal literals|hexadecimal literal syntax]] with a base-2 exponent instead of 10. In languages like [[C (programming language)|C]], when the decimal exponent is omitted, a decimal point is needed to differentiate them from integers. Other languages do not have an integer type (such as [[JavaScript]]), or allow overloading of numeric types (such as [[Haskell (programming language)|Haskell]]). In these cases, digit strings such as <code>123</code> may also be floating-point literals.

Examples of floating-point literals are:
* <code>99.9</code>
* <code>-5000.12</code>
* <code>6.02e23</code>
* <code>-3e-45</code>
* <code>0x1.fffffep+127</code> in C and IEEE 754

==Dealing with exceptional cases {{anchor|Floating point exception|Exception handling}}== 
{{further|IEEE 754#Exception handling}}

Floating-point computation in a computer can run into three kinds of problems:
* An operation can be mathematically undefined, such as ∞/∞, or [[division by zero]].
* An operation can be legal in principle, but not supported by the specific format, for example, calculating the [[square root]] of −1 or the inverse sine of 2 (both of which result in [[complex number]]s).
* An operation can be legal in principle, but the result can be impossible to represent in the specified format, because the exponent is too large or too small to encode in the exponent field. Such an event is called an overflow (exponent too large), [[arithmetic underflow|underflow]] (exponent too small) or [[Subnormal number|denormalization]] (precision loss).

Prior to the IEEE standard, such conditions usually caused the program to terminate, or triggered some kind of [[trap (computing)|trap]] that the programmer might be able to catch. How this worked was system-dependent, meaning that floating-point programs were not [[porting|portable]]. (The term "exception" as used in IEEE 754 is a general term meaning an exceptional condition, which is not necessarily an error, and is a different usage to that typically defined in programming languages such as a C++ or Java, in which an "[[Exception handling|exception]]" is an alternative flow of control, closer to what is termed a "trap" in IEEE 754 terminology.)

Here, the required default method of handling exceptions according to IEEE 754 is discussed (the IEEE 754 optional trapping and other "alternate exception handling" modes are not discussed). Arithmetic exceptions are (by default) required to be recorded in "sticky" status flag bits. That they are "sticky" means that they are not reset by the next (arithmetic) operation, but stay set until explicitly reset. The use of "sticky" flags thus allows for testing of exceptional conditions to be delayed until after a full floating-point expression or subroutine: without them exceptional conditions that could not be otherwise ignored would require explicit testing immediately after every floating-point operation. By default, an operation always returns a result according to specification without interrupting computation. For instance, 1/0 returns +∞, while also setting the divide-by-zero flag bit (this default of ∞ is designed to often return a finite result when used in subsequent operations and so be safely ignored).

The original IEEE 754 standard, however, failed to recommend operations to handle such sets of arithmetic exception flag bits. So while these were implemented in hardware, initially programming language implementations typically did not provide a means to access them (apart from assembler). Over time some programming language standards (e.g., [[C99]]/C11 and Fortran) have been updated to specify methods to access and change status flag bits. The 2008 version of the IEEE 754 standard now specifies a few operations for accessing and handling the arithmetic flag bits. The programming model is based on a single thread of execution and use of them by multiple threads has to be handled by a [[Concurrency (computer science)|means]] outside of the standard (e.g. [[C11 (C standard revision)|C11]] specifies that the flags have [[thread-local storage]]).

IEEE 754 specifies five arithmetic exceptions that are to be recorded in the status flags ("sticky bits"):
* '''inexact''', set if the rounded (and returned) value is different from the mathematically exact result of the operation.
* '''underflow''', set if the rounded value is tiny (as specified in IEEE 754) ''and'' inexact (or maybe limited to if it has denormalization loss, as per the 1985 version of IEEE 754), returning a subnormal value including the zeros.
* '''overflow''', set if the absolute value of the rounded value is too large to be represented. An infinity or maximal finite value is returned, depending on which rounding is used.
* '''divide-by-zero''', set if the result is infinite given finite operands, returning an infinity, either +∞ or −∞.
* '''invalid''', set if a finite or infinite result cannot be returned e.g. sqrt(−1) or 0/0, returning a quiet NaN.
[[File:Resistors in Parallel.svg|thumb|left|200px|Fig. 1: resistances in parallel, with total resistance <math>R_{tot}</math>]]The default return value for each of the exceptions is designed to give the correct result in the majority of cases such that the exceptions can be ignored in the majority of codes. ''inexact'' returns a correctly rounded result, and ''underflow'' returns a value less than or equal to the smallest positive normal number in magnitude and can almost always be ignored.<ref name="Kahan_1997_Status"/> ''divide-by-zero'' returns infinity exactly, which will typically then divide a finite number and so give zero, or else will give an ''invalid'' exception subsequently if not, and so can also typically be ignored. For example, the effective resistance of n resistors in parallel (see fig. 1) is given by <math>R_\text{tot}=1/(1/R_1+1/R_2+\cdots+1/R_n)</math>. If a short-circuit develops with <math>R_1</math> set to 0, <math>1/R_1</math> will return +infinity which will give a final <math>R_{tot}</math> of 0, as expected<ref name="Intel"/> (see the continued fraction example of [[Floating point#IEEE 754: floating point in modern computers|IEEE 754 design rationale]] for another example).
''Overflow'' and ''invalid'' exceptions can typically not be ignored, but do not necessarily represent errors: for example, a [[Zero of a function|root-finding]] routine, as part of its normal operation, may evaluate a passed-in function at values outside of its domain, returning NaN and an ''invalid'' exception flag to be ignored until finding a useful start point.<ref name="Kahan_1997_Status"/>

== Accuracy problems ==

The fact that floating-point numbers cannot accurately represent all real numbers, and that floating-point operations cannot accurately represent true arithmetic operations, leads to many surprising situations. This is related to the finite [[Precision (computer science)|precision]] with which computers generally represent numbers.

For example, the decimal numbers 0.1 and 0.01 cannot be represented exactly as binary floating-point numbers. In the IEEE 754 binary32 format with its 24-bit significand, the result of attempting to square the approximation to 0.1 is neither 0.01 nor the representable number closest to it. The decimal number 0.1 is represented in binary as {{math|1={{var|e}} = −4}}; {{math|1={{var|s}} = 110011001100110011001101}}, which is
{{block indent|1=0.100000001490116119384765625 exactly.}}
Squaring this number gives
{{block indent|1=0.010000000298023226097399174250313080847263336181640625 exactly.}}
Squaring it with rounding to the 24-bit precision gives
{{block indent|0.010000000707805156707763671875 exactly.}}
But the representable number closest to 0.01 is
{{block indent|0.009999999776482582092285156250 exactly.}}

Also, the non-representability of π (and π/2) means that an attempted computation of tan(π/2) will not yield a result of infinity, nor will it even overflow in the usual floating-point formats (assuming an accurate implementation of tan). It is simply not possible for standard floating-point hardware to attempt to compute tan(π/2), because π/2 cannot be represented exactly. This computation in C:
<syntaxhighlight lang="c">
/* Enough digits to be sure we get the correct approximation. */
double pi = 3.1415926535897932384626433832795;
double z = tan(pi/2.0);
</syntaxhighlight>
will give a result of 16331239353195370.0. In single precision (using the <code>tanf</code> function), the result will be −22877332.0.

By the same token, an attempted computation of sin(π) will not yield zero. The result will be (approximately) 0.1225{{e|−15}} in double precision, or −0.8742{{e|−7}} in single precision.<ref group="nb" name="NB_3"/>

While floating-point addition and multiplication are both [[commutative]] ({{math|{{var|a}} + {{var|b}} {{=}} {{var|b}} + {{var|a}}}} and {{math|{{var|a}} × {{var|b}} {{=}} {{var|b}} × {{var|a}}}}), they are not necessarily [[Associative property|associative]]. That is, {{math|({{var|a}} + {{var|b}}) + {{var|c}}}} is not necessarily equal to {{math|{{var|a}} + ({{var|b}} + {{var|c}})}}. Using 7-digit significand decimal arithmetic:
a = 1234.567, b = 45.67834, c = 0.0004

(a + b) + c:
1234.567 (a)
+ 45.67834 (b)
____________
1280.24534 rounds to 1280.245

1280.245 (a + b)
+ 0.0004 (c)
____________
1280.2454 rounds to '''1280.245''' ← (a + b) + c

a + (b + c):
45.67834 (b)
+ 0.0004 (c)
____________
45.67874

1234.567 (a)
+ 45.67874 (b + c)
____________
1280.24574 rounds to '''1280.246''' ← a + (b + c)

They are also not necessarily [[distributive property|distributive]]. That is, {{math|({{var|a}} + {{var|b}}) × {{var|c}}}} may not be the same as {{math|{{var|a}} × {{var|c}} + {{var|b}} × {{var|c}}}}:
1234.567 × 3.333333 = 4115.223
1.234567 × 3.333333 = 4.115223
4115.223 + 4.115223 = 4119.338
but
1234.567 + 1.234567 = 1235.802
1235.802 × 3.333333 = 4119.340

In addition to loss of significance, inability to represent numbers such as π and 0.1 exactly, and other slight inaccuracies, the following phenomena may occur:
{{ulist
|1= [[Catastrophic cancellation|Cancellation]]: subtraction of nearly equal operands may cause extreme loss of accuracy.<ref name="Harris"/><ref name="Sierra_1962"/> When we subtract two almost equal numbers we set the most significant digits to zero, leaving ourselves with just the insignificant, and most erroneous, digits.<ref name="Muller_2010"/>{{rp|124}} For example, when determining a [[derivative]] of a function the following formula is used:
<math display=block>Q(h) = \frac{f(a + h) - f(a)}{h}.</math>
Intuitively one would want an {{math|{{var|h}}}} very close to zero; however, when using floating-point operations, the smallest number will not give the best approximation of a derivative. As {{math|{{var|h}}}} grows smaller, the difference between {{math|{{var|f}}({{var|a}} + {{var|h}})}} and {{math|{{var|f}}({{var|a}})}} grows smaller, cancelling out the most significant and least erroneous digits and making the most erroneous digits more important. As a result the smallest number of {{math|{{var|h}}}} possible will give a more erroneous approximation of a derivative than a somewhat larger number. This is perhaps the most common and serious accuracy problem.
|2=Conversions to integer are not intuitive: converting (63.0/9.0) to integer yields 7, but converting (0.63/0.09) may yield 6. This is because conversions generally truncate rather than round. [[Floor and ceiling functions]] may produce answers which are off by one from the intuitively expected value.
|3=Limited exponent range: results might overflow yielding infinity, or underflow yielding a [[subnormal number]] or zero. In these cases precision will be lost.
|4=Testing for [[Division by zero#Computer arithmetic|safe division]] is problematic: Checking that the divisor is not zero does not guarantee that a division will not overflow.
|5=Testing for equality is problematic. Two computational sequences that are mathematically equal may well produce different floating-point values.<ref name="Barker"/>
}}

=== Incidents ===
* On 25 February 1991, a [[loss of significance]] in a [[MIM-104 Patriot]] missile battery [[MIM-104 Patriot#Failure at Dhahran|prevented it from intercepting]] an incoming [[Al Hussein (missile)|Scud]] missile in [[Dhahran]], [[Saudi Arabia]], contributing to the death of 28 soldiers from the U.S. Army's [[14th Quartermaster Detachment]].<ref name="GAO report IMTEC 92-26"/> The error was actually introduced by a [[Fixed-point arithmetic|fixed-point]] computation,<ref name="Skeel"/> but the underlying issue would have been the same with floating-point arithmetic.
* {{Clarify|date=November 2024|reason=It is not clear how this is an incident (the section title may have to be modified to cover more than incidents) and how this is due to floating-point arithmetic (rather than number approximations in general). The term 'invisible' may also be misleading without following explanations. |text=[[Salami slicing tactics#Financial schemes|Salami slicing]] is the practice of removing the 'invisible' part of a transaction into a separate account.}}

=== Machine precision and backward error analysis ===
''Machine precision'' is a quantity that characterizes the accuracy of a floating-point system, and is used in [[error analysis (mathematics)#Error analysis in numerical modeling|backward error analysis]] of floating-point algorithms. It is also known as unit roundoff or ''[[machine epsilon]]''. Usually denoted {{math|{{var|Ε}}mach}}, its value depends on the particular rounding being used.

With rounding to zero,
<math display=block>\Epsilon_\text{mach} = B^{1-P},\,</math>
whereas rounding to nearest,
<math display=block>\Epsilon_\text{mach} = \tfrac{1}{2} B^{1-P},</math>
where ''B'' is the base of the system and ''P'' is the precision of the significand (in base ''B'').

This is important since it bounds the ''[[relative error]]'' in representing any non-zero real number {{math|{{var|x}}}} within the normalized range of a floating-point system:
<math display=block>\left| \frac{\operatorname{fl}(x) - x}{x} \right| \le \Epsilon_\text{mach}.</math>

Backward error analysis, the theory of which was developed and popularized by [[James H. Wilkinson]], can be used to establish that an algorithm implementing a numerical function is numerically stable.<ref name="RalstonReilly2003"/> The basic approach is to show that although the calculated result, due to roundoff errors, will not be exactly correct, it is the exact solution to a nearby problem with slightly perturbed input data. If the perturbation required is small, on the order of the uncertainty in the input data, then the results are in some sense as accurate as the data "deserves". The algorithm is then defined as ''[[numerical stability#Forward, backward, and mixed stability|backward stable]]''. Stability is a measure of the sensitivity to rounding errors of a given numerical procedure; by contrast, the [[condition number]] of a function for a given problem indicates the inherent sensitivity of the function to small perturbations in its input and is independent of the implementation used to solve the problem.<ref name="Einarsson_2005"/>

As a trivial example, consider a simple expression giving the inner product of (length two) vectors <math>x</math> and <math>y</math>, then
<math display=block>\begin{align}
\operatorname{fl}(x \cdot y)
&= \operatorname{fl}\big(\operatorname{fl}(x_1 \cdot y_1) + \operatorname{fl}(x_2 \cdot y_2)\big), && \text{ where } \operatorname{fl}() \text{ indicates correctly rounded floating-point arithmetic} \\
&= \operatorname{fl}\big((x_1 \cdot y_1)(1 + \delta_1) + (x_2 \cdot y_2)(1 + \delta_2)\big), && \text{ where } \delta_n \leq \Epsilon_\text{mach}, \text{ from above} \\
&= \big((x_1 \cdot y_1)(1 + \delta_1) + (x_2 \cdot y_2)(1 + \delta_2)\big)(1 + \delta_3) \\
&= (x_1 \cdot y_1)(1 + \delta_1)(1 + \delta_3) + (x_2 \cdot y_2)(1 + \delta_2)(1 + \delta_3),
\end{align}</math>
and so
<math display=block>\operatorname{fl}(x \cdot y) = \hat{x} \cdot \hat{y},</math>

where

<math display=block>\begin{align}
\hat{x}_1 &= x_1(1 + \delta_1); & \hat{x}_2 &= x_2(1 + \delta_2);\\
\hat{y}_1 &= y_1(1 + \delta_3); & \hat{y}_2 &= y_2(1 + \delta_3),\\
\end{align}</math>

where

<math display=block>\delta_n \leq \Epsilon_\text{mach}</math>

by definition, which is the sum of two slightly perturbed (on the order of Εmach) input data, and so is backward stable. For more realistic examples in [[numerical linear algebra]], see Higham 2002<ref name="Higham_2002"/> and other references below.

=== Minimizing the effect of accuracy problems ===
Although individual arithmetic operations of IEEE 754 are guaranteed accurate to within half a [[unit in the last place|ULP]], more complicated formulae can suffer from larger errors for a variety of reasons. The loss of accuracy can be substantial if a problem or its data are [[condition number|ill-conditioned]], meaning that the correct result is hypersensitive to tiny perturbations in its data. However, even functions that are well-conditioned can suffer from large loss of accuracy if an algorithm [[numerical stability|numerically unstable]] for that data is used: apparently equivalent formulations of expressions in a programming language can differ markedly in their numerical stability. One approach to remove the risk of such loss of accuracy is the design and analysis of numerically stable algorithms, which is an aim of the branch of mathematics known as [[numerical analysis]]. Another approach that can protect against the risk of numerical instabilities is the computation of intermediate (scratch) values in an algorithm at a higher precision than the final result requires,<ref name="OliveiraStewart_2006"/> which can remove, or reduce by orders of magnitude,<ref name="Kahan_2005_ARITH17"/> such risk: [[Quadruple-precision floating-point format|IEEE 754 quadruple precision]] and [[extended precision]] are designed for this purpose when computing at double precision.<ref name="Kahan_2011_Debug"/><ref group="nb" name="NB_4"/>

For example, the following algorithm is a direct implementation to compute the function {{math|{{var|A}}({{var|x}}) {{=}} ({{var|x}}−1) / (exp({{var|x}}−1) − 1)}} which is well-conditioned at 1.0,<ref group="nb" name="NB_5"/> however it can be shown to be numerically unstable and lose up to half the significant digits carried by the arithmetic when computed near 1.0.<ref name="Kahan_2001_JavaHurt"/>
<syntaxhighlight lang="c" line highlight="3,7">
double A(double X)
{
double Y, Z; // [1]
Y = X - 1.0;
Z = exp(Y);
if (Z != 1.0)
Z = Y / (Z - 1.0); // [2]
return Z;
}
</syntaxhighlight>

If, however, intermediate computations are all performed in extended precision (e.g. by setting line [1] to [[C99]] {{code|long double}}), then up to full precision in the final double result can be maintained.<ref group="nb" name="NB_6"/> Alternatively, a numerical analysis of the algorithm reveals that if the following non-obvious change to line [2] is made:
<syntaxhighlight lang="c">
Z = log(Z) / (Z - 1.0);
</syntaxhighlight>
then the algorithm becomes numerically stable and can compute to full double precision.

To maintain the properties of such carefully constructed numerically stable programs, careful handling by the [[compiler]] is required. Certain "optimizations" that compilers might make (for example, reordering operations) can work against the goals of well-behaved software. There is some controversy about the failings of compilers and language designs in this area: C99 is an example of a language where such optimizations are carefully specified to maintain numerical precision. See the external references at the bottom of this article.

A detailed treatment of the techniques for writing high-quality floating-point software is beyond the scope of this article, and the reader is referred to,<ref name="Higham_2002"/><ref name="Kahan_2000_Marketing"/> and the other references at the bottom of this article. Kahan suggests several rules of thumb that can substantially decrease by orders of magnitude<ref name="Kahan_2000_Marketing"/> the risk of numerical anomalies, in addition to, or in lieu of, a more careful numerical analysis. These include: as noted above, computing all expressions and intermediate results in the highest precision supported in hardware (a common rule of thumb is to carry twice the precision of the desired result, i.e. compute in double precision for a final single-precision result, or in double extended or quad precision for up to double-precision results<ref name="Kahan_1981_WhyIEEE"/>); and rounding input data and results to only the precision required and supported by the input data (carrying excess precision in the final result beyond that required and supported by the input data can be misleading, increases storage cost and decreases speed, and the excess bits can affect convergence of numerical procedures:<ref name="Kahan_2001_LN"/> notably, the first form of the iterative example given below converges correctly when using this rule of thumb). Brief descriptions of several additional issues and techniques follow.

As decimal fractions can often not be exactly represented in binary floating-point, such arithmetic is at its best when it is simply being used to measure real-world quantities over a wide range of scales (such as the orbital period of a moon around Saturn or the mass of a [[proton]]), and at its worst when it is expected to model the interactions of quantities expressed as decimal strings that are expected to be exact.<ref name="Kahan_2005_ARITH17"/><ref name="Kahan_2000_Marketing"/> An example of the latter case is financial calculations. For this reason, financial software tends not to use a binary floating-point number representation.<ref name="Speleotrove_2012"/> The "decimal" data type of the [[C Sharp (programming language)|C#]] and [[Python (programming language)|Python]] programming languages, and the decimal formats of the [[IEEE 754-2008]] standard, are designed to avoid the problems of binary floating-point representations when applied to human-entered exact decimal values, and make the arithmetic always behave as expected when numbers are printed in decimal.

Expectations from mathematics may not be realized in the field of floating-point computation. For example, it is known that <math>(x+y)(x-y) = x^2-y^2\,</math>, and that <math>\sin^2{\theta}+\cos^2{\theta} = 1\,</math>, however these facts cannot be relied on when the quantities involved are the result of floating-point computation.

The use of the equality test (<code>if (x==y) ...</code>) requires care when dealing with floating-point numbers. Even simple expressions like <code>0.6/0.2-3==0</code> will, on most computers, fail to be true<ref name="Christiansen_Perl"/> (in IEEE 754 double precision, for example, <code>0.6/0.2 - 3</code> is approximately equal to {{val|-4.44089209850063e-16}}). Consequently, such tests are sometimes replaced with "fuzzy" comparisons (<code>if (abs(x-y) < epsilon) ...</code>, where epsilon is sufficiently small and tailored to the application, such as 1.0E−13). The wisdom of doing this varies greatly, and can require numerical analysis to bound epsilon.<ref name="Higham_2002"/> Values derived from the primary data representation and their comparisons should be performed in a wider, extended, precision to minimize the risk of such inconsistencies due to round-off errors.<ref name="Kahan_2000_Marketing"/> It is often better to organize the code in such a way that such tests are unnecessary. For example, in [[computational geometry]], exact tests of whether a point lies off or on a line or plane defined by other points can be performed using adaptive precision or exact arithmetic methods.<ref name="Shewchuk"/>

Small errors in floating-point arithmetic can grow when mathematical algorithms perform operations an enormous number of times. A few examples are [[matrix inversion]], [[eigenvector]] computation, and differential equation solving. These algorithms must be very carefully designed, using numerical approaches such as [[iterative refinement]], if they are to work well.<ref name="Kahan_1997_Cantilever"/>

Summation of a vector of floating-point values is a basic algorithm in [[Computational science|scientific computing]], and so an awareness of when loss of significance can occur is essential. For example, if one is adding a very large number of numbers, the individual addends are very small compared with the sum. This can lead to loss of significance. A typical addition would then be something like
3253.671
+ 3.141276
-----------
3256.812
The low 3 digits of the addends are effectively lost. Suppose, for example, that one needs to add many numbers, all approximately equal to 3. After 1000 of them have been added, the running sum is about 3000; the lost digits are not regained. The [[Kahan summation algorithm]] may be used to reduce the errors.<ref name="Higham_2002"/>

Round-off error can affect the convergence and accuracy of iterative numerical procedures. As an example, [[Archimedes]] approximated π by calculating the perimeters of polygons inscribing and circumscribing a circle, starting with hexagons, and successively doubling the number of sides. As noted above, computations may be rearranged in a way that is mathematically equivalent but less prone to error ([[numerical analysis]]). Two forms of the recurrence formula for the circumscribed polygon are:{{citation needed|reason=Not obvious formulas|date=June 2016}}
* <math display="inline">t_0 = \frac{1}{\sqrt{3}}</math>
* First form: <math display=inline>t_{i+1} = \frac{\sqrt{t_i^2+1}-1}{t_i}</math>
* Second form: <math display=inline>t_{i+1} = \frac{t_i}{\sqrt{t_i^2+1}+1}</math>
* <math>\pi \sim 6 \times 2^i \times t_i</math>, converging as <math>i \rightarrow \infty</math>

Here is a computation using IEEE "double" (a significand with 53 bits of precision) arithmetic:

i 6 × 2i × ti, first form 6 × 2i × ti, second form
---------------------------------------------------------
0 '''{{Fontcolor|purple|3}}'''.4641016151377543863 '''{{Fontcolor|purple|3}}'''.4641016151377543863
1 '''{{Fontcolor|purple|3}}'''.2153903091734710173 '''{{Fontcolor|purple|3}}'''.2153903091734723496
2 '''{{Fontcolor|purple|3.1}}'''596599420974940120 '''{{Fontcolor|purple|3.1}}'''596599420975006733
3 '''{{Fontcolor|purple|3.14}}'''60862151314012979 '''{{Fontcolor|purple|3.14}}'''60862151314352708
4 '''{{Fontcolor|purple|3.14}}'''27145996453136334 '''{{Fontcolor|purple|3.14}}'''27145996453689225
5 '''{{Fontcolor|purple|3.141}}'''8730499801259536 '''{{Fontcolor|purple|3.141}}'''8730499798241950
6 '''{{Fontcolor|purple|3.141}}'''6627470548084133 '''{{Fontcolor|purple|3.141}}'''6627470568494473
7 '''{{Fontcolor|purple|3.141}}'''6101765997805905 '''{{Fontcolor|purple|3.141}}'''6101766046906629
8 '''{{Fontcolor|purple|3.14159}}'''70343230776862 '''{{Fontcolor|purple|3.14159}}'''70343215275928
9 '''{{Fontcolor|purple|3.14159}}'''37488171150615 '''{{Fontcolor|purple|3.14159}}'''37487713536668
10 '''{{Fontcolor|purple|3.141592}}'''9278733740748 '''{{Fontcolor|purple|3.141592}}'''9273850979885
11 '''{{Fontcolor|purple|3.141592}}'''7256228504127 '''{{Fontcolor|purple|3.141592}}'''7220386148377
12 '''{{Fontcolor|purple|3.1415926}}'''717412858693 '''{{Fontcolor|purple|3.1415926}}'''707019992125
13 '''{{Fontcolor|purple|3.1415926}}'''189011456060 '''{{Fontcolor|purple|3.14159265}}'''78678454728
14 '''{{Fontcolor|purple|3.1415926}}'''717412858693 '''{{Fontcolor|purple|3.14159265}}'''46593073709
15 '''{{Fontcolor|purple|3.14159}}'''19358822321783 '''{{Fontcolor|purple|3.141592653}}'''8571730119
16 '''{{Fontcolor|purple|3.1415926}}'''717412858693 '''{{Fontcolor|purple|3.141592653}}'''6566394222
17 '''{{Fontcolor|purple|3.1415}}'''810075796233302 '''{{Fontcolor|purple|3.141592653}}'''6065061913
18 '''{{Fontcolor|purple|3.1415926}}'''717412858693 '''{{Fontcolor|purple|3.1415926535}}'''939728836
19 '''{{Fontcolor|purple|3.141}}'''4061547378810956 '''{{Fontcolor|purple|3.1415926535}}'''908393901
20 '''{{Fontcolor|purple|3.14}}'''05434924008406305 '''{{Fontcolor|purple|3.1415926535}}'''900560168
21 '''{{Fontcolor|purple|3.14}}'''00068646912273617 '''{{Fontcolor|purple|3.141592653589}}'''8608396
22 '''{{Fontcolor|purple|3.1}}'''349453756585929919 '''{{Fontcolor|purple|3.141592653589}}'''8122118
23 '''{{Fontcolor|purple|3.14}}'''00068646912273617 '''{{Fontcolor|purple|3.14159265358979}}'''95552
24 '''{{Fontcolor|purple|3}}'''.2245152435345525443 '''{{Fontcolor|purple|3.14159265358979}}'''68907
25 '''{{Fontcolor|purple|3.14159265358979}}'''62246
26 '''{{Fontcolor|purple|3.14159265358979}}'''62246
27 '''{{Fontcolor|purple|3.14159265358979}}'''62246
28 '''{{Fontcolor|purple|3.14159265358979}}'''62246
The true value is '''{{Fontcolor|purple|3.14159265358979323846264338327...}}'''

While the two forms of the recurrence formula are clearly mathematically equivalent,<ref group="nb" name="NB_7"/> the first subtracts 1 from a number extremely close to 1, leading to an increasingly problematic loss of [[significant digit]]s. As the recurrence is applied repeatedly, the accuracy improves at first, but then it deteriorates. It never gets better than about 8 digits, even though 53-bit arithmetic should be capable of about 16 digits of precision. When the second form of the recurrence is used, the value converges to 15 digits of precision.

=== "Fast math" optimization ===
The aforementioned lack of [[Associative property|associativity]] of floating-point operations in general means that [[compilers]] cannot as effectively reorder arithmetic expressions as they could with integer and fixed-point arithmetic, presenting a roadblock in optimizations such as [[common subexpression elimination]] and auto-[[Single instruction, multiple data|vectorization]].<ref name="Vectorizers"/> The "fast math" option on many compilers (ICC, GCC, Clang, MSVC...) turns on reassociation along with unsafe assumptions such as a lack of NaN and infinite numbers in IEEE 754. Some compilers also offer more granular options to only turn on reassociation. In either case, the programmer is exposed to many of the precision pitfalls mentioned above for the portion of the program using "fast" math.<ref name="FPM"/>

In some compilers (GCC and Clang), turning on "fast" math may cause the program to [[Subnormal_number#Disabling_subnormal_floats_at_the_code_level|disable subnormal floats]] at startup, affecting the floating-point behavior of not only the generated code, but also any program using such code as a [[Library (computing)|library]].<ref name="harmful"/>

In most [[Fortran]] compilers, as allowed by the ISO/IEC 1539-1:2004 Fortran standard, reassociation is the default, with breakage largely prevented by the "protect parens" setting (also on by default). This setting stops the compiler from reassociating beyond the boundaries of parentheses.<ref name="Gen"/> [[Intel Fortran Compiler]] is a notable outlier.<ref name="zheevd"/>

A common problem in "fast" math is that subexpressions may not be optimized identically from place to place, leading to unexpected differences. One interpretation of the issue is that "fast" math as implemented currently has a poorly defined semantics. One attempt at formalizing "fast" math optimizations is seen in ''Icing'', a verified compiler.<ref name="Becker-Darulova-Myreen-Tatlock_2019"/>

== See also ==
{{div col|colwidth=20em}}
* [[Arbitrary-precision arithmetic]]
* [[C99#IEEE 754 floating-point support|C99]] for code examples demonstrating access and use of IEEE 754 features.
* [[Computable number]]
* [[Coprocessor]]
* [[Decimal floating point]]
* [[Double-precision floating-point format]]
* [[Experimental mathematics]] – utilizes high precision floating-point computations
* [[Fixed-point arithmetic]]
* [[Floating-point error mitigation]]
* [[FLOPS]]
* [[Gal's accurate tables]]
* [[GNU MPFR]]
* [[Half-precision floating-point format]]
* [[IEEE 754]] – Standard for Binary Floating-Point Arithmetic
* [[IBM hexadecimal floating-point|IBM Floating Point Architecture]]
* [[Kahan summation algorithm]]
* [[Microsoft Binary Format]] (MBF)
* [[Minifloat]]
* [[Q (number format)]] for constant resolution
* [[Quadruple-precision floating-point format]] (including double-double)
* [[Significant figures]]
* [[Single-precision floating-point format]]
* [[Standard Apple Numerics Environment|Standard Apple Numerics Environment (SANE)]]
{{div col end}}

== Notes ==
{{reflist|group="nb"|refs=
<ref group="nb" name="NB_1">Computer hardware does not necessarily compute the exact value; it simply has to produce the equivalent rounded result as though it had computed the infinitely precise result.</ref>
<ref group="nb" name="NB_2">The enormous complexity of modern [[division algorithm]]s once led to a famous error. An early version of the [[Intel Pentium]] chip was shipped with a [[FDIV|division instruction]] that, on rare occasions, gave slightly incorrect results. Many computers had been shipped before the error was discovered. Until the defective computers were replaced, patched versions of compilers were developed that could avoid the failing cases. See ''[[Pentium FDIV bug]]''.</ref>
<ref group="nb" name="NB_3">But an attempted computation of cos(π) yields −1 exactly. Since the derivative is nearly zero near π, the effect of the inaccuracy in the argument is far smaller than the spacing of the floating-point numbers around −1, and the rounded result is exact.</ref>
<ref group="nb" name="NB_4">[[William Morton Kahan|William Kahan]] notes: "Except in extremely uncommon situations, extra-precise arithmetic generally attenuates risks due to roundoff at far less cost than the price of a competent error-analyst."</ref>
<ref group="nb" name="NB_5">The [[Taylor expansion]] of this function demonstrates that it is well-conditioned near 1: A(x) = 1 − (x−1)/2 + (x−1)^2/12 − (x−1)^4/720 + (x−1)^6/30240 − (x−1)^8/1209600 + ... for |x−1| < π.</ref>
<ref group="nb" name="NB_6">If [[long double]] is [[IEEE quad precision]] then full double precision is retained; if long double is [[IEEE double extended precision]] then additional, but not full precision is retained.</ref>
<ref group="nb" name="NB_7">The equivalence of the two forms can be verified algebraically by noting that the [[denominator]] of the fraction in the second form is the [[conjugate (algebra)|conjugate]] of the [[numerator]] of the first. By multiplying the top and bottom of the first expression by this conjugate, one obtains the second expression.</ref>
<ref group="nb" name="NB_8">Octal (base-8) floating-point arithmetic is used in the [[Ferranti Atlas]] (1962), [[Burroughs B5500]] (1964), [[Burroughs B5700]] (1971), [[Burroughs B6700]] (1971) and [[Burroughs B7700]] (1972) computers.</ref>
<ref group="nb" name="NB_9">[[Hexadecimal floating-point|Hexadecimal (base-16) floating-point]] arithmetic is used in the [[IBM System 360]] (1964) and [[IBM System 370|370]] (1970) as well as various newer IBM machines, in the [[RCA Spectra 70]] (1964), the Siemens 4004 (1965), 7.700 (1974), 7.800, 7.500 (1977) series mainframes and successors, the Unidata 7.000 series mainframes, the [[Manchester MU5]] (1972), the [[Heterogeneous Element Processor|HEP]] (1982) computers, and in 360/370-compatible mainframe families made by Fujitsu, Amdahl and Hitachi. It is also used in the [[Illinois ILLIAC III]] (1966), [[Data General Eclipse S/200]] (ca. 1974), [[Gould Powernode 9080]] (1980s), [[Interdata 8/32]] (1970s), the [[SEL System 85|SEL Systems 85]] and [[SEL System 86|86]] as well as the [[SDS Sigma 5]] (1967), [[SDS Sigma 7|7]] (1966) and [[Xerox Sigma 9]] (1970).</ref>
<ref group="nb" name="NB_10">Base-65536 floating-point arithmetic is used in the [[MANIAC II]] (1956) computer.</ref>
<ref group="nb" name="NB_11">Quaternary (base-4) floating-point arithmetic is used in the [[Illinois ILLIAC II]] (1962) computer. It is also used in the Digital Field System DFS IV and V high-resolution site survey systems.</ref>
<ref group="nb" name="NB_12">Base-256 floating-point arithmetic is used in the [[Rice Institute R1]] computer (since 1958).</ref>
<ref group="nb" name="NB_Significand">{{anchor|NB-Significand}}The ''[[significand]]'' of a floating-point number is also called ''mantissa'' by some authors—not to be confused with the [[mantissa (logarithm)|mantissa]] of a [[logarithm]]. Somewhat vague, terms such as ''coefficient'' or ''argument'' are also used by some. The usage of the term ''fraction'' by some authors is potentially misleading as well. The term ''characteristic'' (as used e.g. by [[Control Data Corporation|CDC]]) is ambiguous, as it was historically also used to specify some form of [[#NB-Exponent|exponent]] of floating-point numbers.</ref>
<ref group="nb" name="NB_Exponent">{{anchor|NB-Exponent}}The ''[[exponent]]'' of a floating-point number is sometimes also referred to as ''scale''. The term ''characteristic'' (for ''[[biased exponent]]'', ''exponent bias'', or ''excess n representation'') is ambiguous, as it was historically also used to specify the [[#NB-Significand|significand]] of floating-point numbers.</ref>
}}

== References ==
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<ref name="Rojas_2014">{{cite arXiv |eprint=1406.1886 |title=The Z1: Architecture and Algorithms of Konrad Zuse's First Computer |first=Raúl |last=Rojas |author-link=Raúl Rojas |date=2014-06-07|class=cs.AR }}</ref>
<ref name="Kahan_1997_JVNL">{{cite web |url=https://people.eecs.berkeley.edu/~wkahan/SIAMjvnl.pdf |archive-url=https://web.archive.org/web/20080905103125/http://www.cs.berkeley.edu/~wkahan/SIAMjvnl.pdf |archive-date=2008-09-05 |url-status=live |title=The Baleful Effect of Computer Languages and Benchmarks upon Applied Mathematics, Physics and Chemistry. John von Neumann Lecture |first=William Morton |last=Kahan |author-link=William Morton Kahan |date=1997-07-15 |page=3}}</ref>
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<ref name="Kahan_2001_JavaHurt">{{cite web |first1=William Morton |last1=Kahan |author-link1=William Morton Kahan |first2=Joseph |last2=Darcy |date=2001 |orig-date=1998-03-01 |url=https://people.eecs.berkeley.edu/~wkahan/JAVAhurt.pdf |archive-url=https://web.archive.org/web/20000816043653/http://www.cs.berkeley.edu/~wkahan/JAVAhurt.pdf |archive-date=2000-08-16 |url-status=live |title=How Java's floating-point hurts everyone everywhere |access-date=2003-09-05}}</ref>
<ref name="Kahan_2004">{{cite web |url=https://people.eecs.berkeley.edu/~wkahan/Qdrtcs.pdf |archive-url=https://web.archive.org/web/20060525111157/http://www.cs.berkeley.edu/~wkahan/Qdrtcs.pdf |archive-date=2006-05-25 |url-status=live |first=William Morton |last=Kahan |author-link=William Morton Kahan |title=On the Cost of Floating-Point Computation Without Extra-Precise Arithmetic |date=2004-11-20 |access-date=2012-02-19}}</ref>
<ref name="Kahan_2006_Mindless">{{cite web |url=https://people.eecs.berkeley.edu/~wkahan/Mindless.pdf |archive-url=https://web.archive.org/web/20041221020332/http://www.cs.berkeley.edu/~wkahan/Mindless.pdf |archive-date=2004-12-21 |url-status=live |title=How Futile are Mindless Assessments of Roundoff in Floating-Point Computation? |first=William Morton |last=Kahan |author-link=William Morton Kahan |date=2006-01-11}}</ref>
<ref name="Severance_1998">{{cite web |url=https://people.eecs.berkeley.edu/~wkahan/ieee754status/754story.html |title=An Interview with the Old Man of Floating-Point |first=Charles |last=Severance |author-link=Charles Severance (computer scientist) |date=1998-02-20}}</ref>
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<ref name="Higham_2002">{{cite book |title=Accuracy and Stability of Numerical Algorithms |edition=2nd |first=Nicholas John |author-link=Nicholas Higham |last=Higham |publisher=[[Society for Industrial and Applied Mathematics]] (SIAM) |date=2002 |isbn=978-0-89871-521-7 |id=0-89871-355-2 |pages=27–28, 110–123, 493 |url=https://books.google.com/books?id=epilvM5MMxwC}}</ref>
<ref name="OpenEXR">{{cite web |url=http://www.openexr.com/about.html |title=openEXR |publisher=openEXR |access-date=2012-04-25 |archive-date=2013-05-08 |archive-url=https://web.archive.org/web/20130508221152/http://www.openexr.com/about.html |url-status=dead |quote=Since the IEEE-754 floating-point specification does not define a 16-bit format, ILM created the "half" format. Half values have 1 sign bit, 5 exponent bits, and 10 mantissa bits.}}</ref>
<ref name="OpenEXR-half">{{cite web |url=https://openexr.com/en/latest/TechnicalIntroduction.html#the-half-data-type |title=Technical Introduction to OpenEXR – The half Data Type |publisher=openEXR |access-date=2024-04-16}}</ref>
<ref name="IEEE-754_Analysis">{{cite web|url=https://christophervickery.com/IEEE-754/|title=IEEE-754 Analysis|access-date=2024-08-29}}</ref>
<ref name="Goldberg_1991">{{cite journal |first=David |last=Goldberg |title=What Every Computer Scientist Should Know About Floating-Point Arithmetic |journal=[[ACM Computing Surveys]] |date=March 1991 |volume=23 |issue=1 |pages=5–48 |doi=10.1145/103162.103163 |doi-access=free |s2cid=222008826}} (With the addendum "Differences Among IEEE 754 Implementations": [https://web.archive.org/web/20171011072644/http://www.cse.msu.edu/~cse320/Documents/FloatingPoint.pdf], [https://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html])</ref>
<ref name="Harris">{{Cite journal |title=You're Going To Have To Think! |first=Richard |last=Harris |journal=[[Overload (magazine)|Overload]] |issue=99 |date=October 2010 |issn=1354-3172 |pages=5–10 |url=http://accu.org/index.php/journals/1702 |access-date=2011-09-24 |quote=Far more worrying is cancellation error which can yield catastrophic loss of precision.}} [http://accu.org/var/uploads/journals/overload99.pdf]</ref>
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<ref name="Skeel">{{citation |url=https://www-users.cse.umn.edu/~arnold/disasters/Patriot-dharan-skeel-siam.pdf |title=Roundoff Error and the Patriot Missile |last=Skeel |first=Robert |journal=SIAM News |volume=25 |issue=4 |page=11 |date=July 1992 |access-date=2024-11-15}}</ref>
<ref name="RalstonReilly2003">{{cite book |first=James Hardy |last=Wilkinson |author-link=James Hardy Wilkinson |editor-first1=Anthony |editor-last1=Ralston |editor-first2=Edwin D. |editor-last2=Reilly |editor-first3=David |editor-last3=Hemmendinger |chapter=Error Analysis |title=Encyclopedia of Computer Science |pages=669–674 |url=https://books.google.com/books?id=OLRwQgAACAAJ |access-date=2013-05-14 |date=2003-09-08 |publisher=[[Wiley (publisher)|Wiley]] |isbn=978-0-470-86412-8}}</ref>
<ref name="Einarsson_2005">{{cite book |first=Bo |last=Einarsson |title=Accuracy and reliability in scientific computing |url=https://books.google.com/books?id=sh4orx_qB_QC&pg=PA50 |access-date=2013-05-14 |date=2005 |publisher=[[Society for Industrial and Applied Mathematics]] (SIAM) |isbn=978-0-89871-815-7 |pages=50–}}</ref>
<ref name="Kahan_2005_ARITH17">{{cite conference |first=William Morton |last=Kahan |author-link=William Morton Kahan |date=2005-07-15 |title=Floating-Point Arithmetic Besieged by "Business Decisions" |type=Keynote Address |conference=IEEE-sponsored [[ARITH 17]], Symposium on Computer Arithmetic |pages=6, 18 |url=https://people.eecs.berkeley.edu/~wkahan/ARITH_17.pdf |access-date=2013-05-23 |url-status=live |archive-url=https://web.archive.org/web/20060317103619/http://www.cs.berkeley.edu/~wkahan/ARITH_17.pdf |archive-date=2006-03-17}} (NB. Kahan estimates that the incidence of excessively inaccurate results near singularities is reduced by a factor of approx. 1/2000 using the 11 extra bits of precision of [[extended precision|double extended]].)</ref>
<ref name="OliveiraStewart_2006">{{cite book |first1=Suely |last1=Oliveira |first2=David E. |last2=Stewart |title=Writing Scientific Software: A Guide to Good Style |url=https://books.google.com/books?id=E6a8oZOS8noC&pg=PA10 |date=2006-09-07 |publisher=[[Cambridge University Press]] |isbn=978-1-139-45862-7 |pages=10–}}</ref>
<ref name="Kahan_2011_Debug">{{cite conference |url=https://people.eecs.berkeley.edu/~wkahan/Boulder.pdf |archive-url=https://web.archive.org/web/20130620140729/http://www.eecs.berkeley.edu/~wkahan/Boulder.pdf |archive-date=2013-06-20 |url-status=live |title=Desperately Needed Remedies for the Undebuggability of Large Floating-Point Computations in Science and Engineering |conference=IFIP/SIAM/NIST Working Conference on Uncertainty Quantification in Scientific Computing, Boulder, CO |page=33 |first=William Morton |last=Kahan |author-link=William Morton Kahan |date=2011-08-03}}</ref>
<ref name="Kahan_1981_WhyIEEE">{{cite web |url=https://people.eecs.berkeley.edu/~wkahan/ieee754status/why-ieee.pdf |archive-url=https://web.archive.org/web/20041204070746/http://www.cs.berkeley.edu/~wkahan/ieee754status/why-ieee.pdf |archive-date=2004-12-04 |url-status=live |title=Why do we need a floating-point arithmetic standard? |page=26 |first=William Morton |last=Kahan |author-link=William Morton Kahan |date=1981-02-12}}</ref>
<ref name="Kahan_2001_LN">{{cite web|url=http://www.cims.nyu.edu/~dbindel/class/cs279/notes-06-04.pdf |archive-url=https://web.archive.org/web/20130517181356/http://www.cims.nyu.edu/~dbindel/class/cs279/notes-06-04.pdf |archive-date=2013-05-17 |url-status=live |first=William Morton |last=Kahan |author-link=William Morton Kahan |editor-first=David |editor-last=Bindel |title=Lecture notes of System Support for Scientific Computation |date=2001-06-04}}</ref>
<ref name="Speleotrove_2012">{{cite web |url=https://speleotrove.com/decimal/ |title=General Decimal Arithmetic |publisher=Speleotrove.com |access-date=2012-04-25}}</ref>
<ref name="Kahan_2000_Marketing">{{cite web |url=https://people.eecs.berkeley.edu/~wkahan/MktgMath.pdf |archive-url=https://web.archive.org/web/20030815150333/http://www.cs.berkeley.edu/~wkahan/MktgMath.pdf |archive-date=2003-08-15 |url-status=live |title=Marketing versus Mathematics |pages=15, 35, 47 |first=William Morton |last=Kahan |author-link=William Morton Kahan |date=2000-08-27}}</ref>
<ref name="Shewchuk">{{cite journal | title=Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates | journal=[[Discrete & Computational Geometry]] | volume=18 | pages=305–363 |first=Jonathan Richard |last=Shewchuk |date=1997 | issue=3 | doi=10.1007/PL00009321 | doi-access=free}}</ref>
<ref name="Christiansen_Perl">{{cite web |url=https://perldoc.perl.org/5.8.8/perlfaq4#Why-is-int()-broken? |title=perlfaq4 / Why is int() broken? |first1=Tom |last1=Christiansen |first2=Nathan |last2=Torkington |date=2006 |publisher=perldoc.perl.org |access-date=2011-01-11 |display-authors=etal}}</ref>
<ref name="Kahan_1997_Cantilever">{{cite web |url=https://people.eecs.berkeley.edu/~wkahan/Cantilever.pdf |archive-url=https://web.archive.org/web/20031205191038/http://www.cs.berkeley.edu/~wkahan/Cantilever.pdf |archive-date=2003-12-05 |url-status=live |title=Roundoff Degrades an Idealized Cantilever |first1=William Morton |last1=Kahan |author-link1=William Morton Kahan |first2=Melody Y. |last2=Ivory |date=1997-07-03}}</ref>
<ref name="Muller_2010">{{cite book |last1=Muller |first1=Jean-Michel |last2=Brisebarre |first2=Nicolas |last3=de Dinechin |first3=Florent |last4=Jeannerod |first4=Claude-Pierre |last5=Lefèvre |first5=Vincent |last6=Melquiond |first6=Guillaume |last7=Revol |first7=Nathalie|author7-link=Nathalie Revol |last8=Stehlé |first8=Damien |last9=Torres |first9=Serge |title=Handbook of Floating-Point Arithmetic |date=2010 |publisher=[[Birkhäuser]] |edition=1st |isbn=978-0-8176-4704-9 |doi=10.1007/978-0-8176-4705-6 |lccn=2009939668  |ref=muller_et_al_pg_16 |url=https://books.google.com/books?id=baFvrIOPvncC&pg=PA16}}</ref>
<ref name="Savard_2018">{{citation |title=The Decimal Floating-Point Standard |first=John J. G. |last=Savard |date=2018 |orig-date=2007 |work=quadibloc |url=http://www.quadibloc.com/comp/cp020302.htm |access-date=2018-07-16 |url-status=live |archive-url=https://web.archive.org/web/20180703002322/http://www.quadibloc.com/comp/cp020302.htm |archive-date=2018-07-03}}</ref>
<ref name="Zehendner_2008">{{cite web |type=Lecture script |title=Rechnerarithmetik: Fest- und Gleitkommasysteme |date=Summer 2008 |first=Eberhard |last=Zehendner |language=de |publisher=[[Friedrich-Schiller-Universität Jena]] |page=2 |url=https://users.fmi.uni-jena.de/~nez/rechnerarithmetik_5/folien/Rechnerarithmetik.2008.05.handout.pdf |access-date=2018-08-07 |url-status=live |archive-url=https://web.archive.org/web/20180807062449/https://users.fmi.uni-jena.de/~nez/rechnerarithmetik_5/folien/Rechnerarithmetik.2008.05.handout.pdf |archive-date=2018-08-07}} [https://web.archive.org/web/20180806175620/https://users.fmi.uni-jena.de/~nez/rechnerarithmetik_5/folien/Rechnerarithmetik.2008.komplett.pdf] (NB. This reference incorrectly gives the MANIAC II's floating point base as 256, whereas it actually is 65536.)</ref>
<ref name="Lazarus_1956">{{cite web |title=MANIAC II |first=Roger B. |last=Lazarus |date=1957-01-30 |orig-date=1956-10-01 |publisher=Los Alamos Scientific Laboratory of the University of California |location=Los Alamos, NM, USA |id=LA-2083 |page=14 |url=http://bitsavers.org/pdf/lanl/LA-2083_MANIAC_II_Oct56.pdf |access-date=2018-08-07 |archive-url=https://web.archive.org/web/20180807200914/http://bitsavers.org/pdf/lanl/LA-2083_MANIAC_II_Oct56.pdf |archive-date=2018-08-07 |url-status=live |quote=[…] the Maniac's floating base, which is 216 = 65,536. […] The Maniac's large base permits a considerable increase in the speed of floating point arithmetic. Although such a large base implies the possibility of as many as 15 lead zeros, the large word size of 48 bits guarantees adequate significance. […]}}</ref>
<ref name="Beebe_2017">{{cite book |first=Nelson H. F. |last=Beebe |title=The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library |chapter=Chapter H. Historical floating-point architectures |date=2017-08-22 |location=Salt Lake City, UT, USA |publisher=[[Springer International Publishing AG]] |edition=1st |lccn=2017947446 |isbn=978-3-319-64109-6 |doi=10.1007/978-3-319-64110-2 |page=948|s2cid=30244721 }}</ref>
<ref name="Parkinson_2000">{{cite book |title=High Resolution Site Surveys |first=Roger |last=Parkinson |publisher=[[CRC Press]] |date=2000-12-07 |chapter=Chapter 2 - High resolution digital site survey systems - Chapter 2.1 - Digital field recording systems |edition=1st |isbn=978-0-20318604-6 |page=24 |chapter-url=https://books.google.com/books?id=Ocip5vpLD4wC&pg=PA24 |access-date=2019-08-18 |quote=[…] Systems such as the [Digital Field System] DFS IV and DFS V were quaternary floating-point systems and used gain steps of 12 dB. […]}} (256 pages)</ref>
<ref name="MSVC">{{cite web |url=https://learn.microsoft.com/en-us/cpp/build/ieee-floating-point-representation |title=IEEE Floating-Point Representation |date=2021-08-03}}</ref>
<ref name="GCC">[https://gcc.gnu.org/onlinedocs/gcc/i386-and-x86-64-Options.html Using the GNU Compiler Collection, i386 and x86-64 Options] {{Webarchive |url=https://web.archive.org/web/20150116065447/http://gcc.gnu.org/onlinedocs/gcc/i386-and-x86-64-Options.html |date=2015-01-16}}.</ref>
<ref name="float_128">{{cite web |url=https://stackoverflow.com/questions/13516476 |title=long double (GCC specific) and __float128 |website=StackOverflow}}</ref>
<ref name="ARM_2013_AArch64">{{cite web |title=Procedure Call Standard for the ARM 64-bit Architecture (AArch64) |url=http://infocenter.arm.com/help/topic/com.arm.doc.ihi0055b/IHI0055B_aapcs64.pdf |archive-url=https://web.archive.org/web/20130731181404/http://infocenter.arm.com/help/topic/com.arm.doc.ihi0055b/IHI0055B_aapcs64.pdf |archive-date=2013-07-31 |url-status=live |date=2013-05-22 |access-date=2019-09-22}}</ref>
<ref name="ARM_2013_Compiler">{{cite web |title=ARM Compiler toolchain Compiler Reference, Version 5.03 |url=http://infocenter.arm.com/help/topic/com.arm.doc.dui0491i/DUI0491I_arm_compiler_reference.pdf |archive-url=https://web.archive.org/web/20150627210618/http://infocenter.arm.com/help/topic/com.arm.doc.dui0491i/DUI0491I_arm_compiler_reference.pdf |archive-date=2015-06-27 |url-status=live |at=Section 6.3 ''Basic data types'' |date=2013 |access-date=2019-11-08}}</ref>
<ref name="Kharya_2020">{{Cite web |title=TensorFloat-32 in the A100 GPU Accelerates AI Training, HPC up to 20x |url=https://blogs.nvidia.com/blog/2020/05/14/tensorfloat-32-precision-format/ |last=Kharya |first=Paresh |date=May 14, 2020 |access-date=May 16, 2020}}</ref>
<ref name="Sierra_1962">{{US patent reference |number=3037701A |issue-date=1962-06-05 |inventor=Huberto M Sierra |title=Floating decimal point arithmetic control means for calculator}}</ref>
<ref name="Barker">Christopher Barker: [https://www.python.org/dev/peps/pep-0485/ ''PEP 485 -- A Function for testing approximate equality'']</ref>
<ref name="C99">{{cite book |title=ISO/IEC 9899:1999 - Programming languages - C |publisher=Iso.org |at=§F.2, note 307 |quote="Extended" is IEC 60559's double-extended data format. Extended refers to both the common 80-bit and quadruple 128-bit IEC 60559 formats.}}</ref>
<ref name="Microsoft_2006_KB35826">{{cite web |title=IEEE vs. Microsoft Binary Format; Rounding Issues (Complete) |publisher=[[Microsoft]] |work=Microsoft Support |id=Article ID KB35826, Q35826 |date=2006-11-21 |url=https://www.betaarchive.com/wiki/index.php/Microsoft_KB_Archive/35826#IEEE_vs._Microsoft_Binary_Format.3B_Rounding_Issues_.28Complete.29 |access-date=2010-02-24 |url-status=live |archive-url=https://web.archive.org/web/20200828130651/https://www.betaarchive.com/wiki/index.php/Microsoft_KB_Archive/35826 |archive-date=2020-08-28}}</ref>
<ref name="Steil_2008_6502">{{cite web |title=Create your own Version of Microsoft BASIC for 6502 |first=Michael |last=Steil |publisher=pagetable.com |date=2008-10-20 |url=http://www.pagetable.com/?p=46 |access-date=2016-05-30 |url-status=live |archive-url=https://web.archive.org/web/20160530092603/http://www.pagetable.com/?p=46 |archive-date=2016-05-30}}</ref>
<ref name="Borland_1994_MBF">{{cite web |title=Converting between Microsoft Binary and IEEE formats |date=1998-07-02 |orig-date=1994-03-10 |id=ID 1400 |author=Borland staff |publisher=[[Embarcadero USA]] / [[Inprise]] (originally: [[Borland]]) |work=Technical Information Database |type=TI1431C.txt |url=https://community.embarcadero.com/index.php/article/technical-articles/162-programming/14799-converting-between-microsoft-binary-and-ieee-forma |access-date=2016-05-30 |url-status=live |archive-url=https://web.archive.org/web/20190220230417/https://community.embarcadero.com/index.php/article/technical-articles/162-programming/14799-converting-between-microsoft-binary-and-ieee-forma |archive-date=2019-02-20 |quote=[…] _fmsbintoieee(float *src4, float *dest4) […] MS Binary Format […] byte order => m3 {{!}} m2 {{!}} m1 {{!}} exponent […] m1 is [[most significant byte]] => sbbb{{!}}bbbb […] m3 is the [[least significant byte]] […] m = mantissa byte […] s = sign bit […] b = bit […] MBF is bias 128 and IEEE is bias 127. […] MBF places the [[decimal point]] before the [[assumed bit]], while IEEE places the decimal point after the assumed bit. […] ieee_exp = msbin[3] - 2; /* actually, msbin[3]-1-128+127 */ […] _dmsbintoieee(double *src8, double *dest8) […] MS Binary Format […] byte order => m7 {{!}} m6 {{!}} m5 {{!}} m4 {{!}} m3 {{!}} m2 {{!}} m1 {{!}} exponent […] m1 is most significant byte => smmm{{!}}mmmm […] m7 is the least significant byte […] MBF is bias 128 and IEEE is bias 1023. […] MBF places the decimal point before the assumed bit, while IEEE places the decimal point after the assumed bit. […] ieee_exp = msbin[7] - 128 - 1 + 1023; […]}}</ref>
<ref name="NVIDIA_Hopper">{{cite web |title=NVIDIA Hopper Architecture In-Depth |date=22 March 2022 |url=https://developer.nvidia.com/blog/nvidia-hopper-architecture-in-depth/}}</ref>
<ref name="Micikevicius_2022">{{cite arXiv |last1=Micikevicius |first1=Paulius |last2=Stosic |first2=Dusan |last3=Burgess |first3=Neil |last4=Cornea |first4=Marius |last5=Dubey |first5=Pradeep |last6=Grisenthwaite |first6=Richard |last7=Ha |first7=Sangwon |last8=Heinecke |first8=Alexander |last9=Judd |first9=Patrick |last10=Kamalu |first10=John |last11=Mellempudi |first11=Naveen |last12=Oberman |first12=Stuart |last13=Shoeybi |first13=Mohammad |last14=Siu |first14=Michael |last15=Wu |first15=Hao |eprint=2209.05433 |title=FP8 Formats for Deep Learning |class=cs.LG |date=2022-09-12}}</ref>
<ref name="Gay_1990">{{cite tech report |last=Gay |first=David M. |title=Correctly Rounded Binary-Decimal and Decimal-Binary Conversions |date=1990 |institution=NUMERICAL ANALYSIS MANUSCRIPT 90-10, AT&T BELL LABORATORIES |citeseerx=10.1.1.31.4049}} ([http://www.netlib.org/fp/dtoa.c dtoa.c in netlab])</ref>
<ref name="Loitsch_2010">{{cite conference
|last=Loitsch
|first=Florian
|title=Printing floating-point numbers quickly and accurately with integers
|conference=PLDI '10: ACM SIGPLAN Conference on Programming Language Design and Implementation
|book-title=Proceedings of the 31st ACM SIGPLAN Conference on Programming Language Design and Implementation
|date=2010
|pages=233–243
|doi=10.1145/1806596.1806623
|isbn=978-1-45030019-3
|s2cid=910409
|url=https://www.cs.tufts.edu/~nr/cs257/archive/florian-loitsch/printf.pdf
|archive-url=https://web.archive.org/web/20140729005717/http://www.cs.tufts.edu/~nr/cs257/archive/florian-loitsch/printf.pdf |archive-date=2014-07-29
|url-status=live
}}</ref>
<ref name="mazong">{{cite web |title=Added Grisu3 algorithm support for double.ToString(). by mazong1123 · Pull Request #14646 · dotnet/coreclr |url=https://github.com/dotnet/coreclr/pull/14646 |website=GitHub |language=en}}</ref>
<ref name="Adams_2018">{{cite journal |last=Adams |first=Ulf |title=Ryū: fast float-to-string conversion |journal=ACM SIGPLAN Notices |date=2 December 2018 |volume=53 |issue=4 |pages=270–282 |doi=10.1145/3296979.3192369 |s2cid=218472153 |doi-access=free}}</ref>
<ref name="Giulietti">{{cite web |last=Giulietti |first=Rafaello |title=The Schubfach way to render doubles |url=https://drive.google.com/file/d/1IEeATSVnEE6TkrHlCYNY2GjaraBjOT4f}}</ref>
<ref name="abolz">{{cite web |title=abolz/Drachennest |website=[[GitHub]] |url=https://github.com/abolz/Drachennest |date=10 November 2022}}</ref>
<ref name="double_conversion_2020">{{cite web |title=google/double-conversion |website=[[GitHub]] |url=https://github.com/google/double-conversion |date=21 September 2020}}</ref>
<ref name="Lemire_2021">{{cite journal |last=Lemire |first=Daniel |title=Number parsing at a gigabyte per second |journal=Software: Practice and Experience |date=22 March 2021 |volume=51 |issue=8 |pages=1700–1727 |doi=10.1002/spe.2984 |arxiv=2101.11408 |s2cid=231718830}}</ref>
<ref name="Patterson-Hennessy_2014">{{cite book |last1=Patterson |first1=David A. |last2=Hennessy |first2=John L. |title=Computer Organization and Design, The Hardware/Software Interface |publisher=Elsevier |series=The Morgan Kaufmann series in computer architecture and design |edition=5th |date=2014 |location=Waltham, Massachusetts, USA |pages=793 |language=en |isbn=978-9-86605267-5}}</ref>
<ref name="Vectorizers">{{cite web |title=Auto-Vectorization in LLVM |url=https://llvm.org/docs/Vectorizers.html |website=LLVM 13 documentation |quote=We support floating point reduction operations when -ffast-math is used.}}</ref>
<ref name="FPM">{{cite web |title=FloatingPointMath |url=https://gcc.gnu.org/wiki/FloatingPointMath |website=GCC Wiki}}</ref>
<ref name="harmful">{{cite web |title=55522 – -funsafe-math-optimizations is unexpectedly harmful, especially w/ -shared |url=https://gcc.gnu.org/bugzilla/show_bug.cgi?id=55522 |website=gcc.gnu.org}}</ref>
<ref name="Gen">{{cite web |title=Code Gen Options (The GNU Fortran Compiler) |url=https://gcc.gnu.org/onlinedocs/gfortran/Code-Gen-Options.html |website=gcc.gnu.org}}</ref>
<ref name="zheevd">{{cite web |title=Bug in zheevd · Issue #43 · Reference-LAPACK/lapack |url=https://github.com/Reference-LAPACK/lapack/issues/43 |website=GitHub |language=en}}</ref>
<ref name="Becker-Darulova-Myreen-Tatlock_2019">{{cite conference |last1=Becker |first1=Heiko |last2=Darulova |first2=Eva |last3=Myreen |first3=Magnus O. |last4=Tatlock |first4=Zachary |title=Icing: Supporting Fast-Math Style Optimizations in a Verified Compiler |conference=CAV 2019: Computer Aided Verification |date=2019 |volume=11562 |pages=155–173 |doi=10.1007/978-3-030-25543-5_10 |doi-access=free}}</ref>
}}

== Further reading ==
* {{cite book |first=James Hardy |last=Wilkinson |author-link=James Hardy Wilkinson |title=Rounding Errors in Algebraic Processes |date=1963 |publisher=[[Prentice-Hall, Inc.]] |location=Englewood Cliffs, New Jersey, USA |edition=1st |mr=161456 |url=https://books.google.com/books?id=yFogU9Ot-qsC |isbn=9780486679990}} (NB. Classic influential treatises on floating-point arithmetic.)
* {{cite book |first=James Hardy |last=Wilkinson |author-link=James Hardy Wilkinson |title=The Algebraic Eigenvalue Problem |series=Monographs on Numerical Analysis |date=1965 |edition=1st |publisher=[[Oxford University Press]] / [[Clarendon Press]] |url=https://books.google.com/books?id=N98IAQAAIAAJ&q=editions:ISBN0198534183 |access-date=2016-02-11|isbn=9780198534037 }}
* {{cite book |first=Pat H. |last=Sterbenz |title=Floating-Point Computation |date=1974 |edition=1st |series=Prentice-Hall Series in Automatic Computation |publisher=[[Prentice Hall]] |location=Englewood Cliffs, New Jersey, USA |isbn=978-0-13-322495-5}}
* {{cite book |first1=Gene F. |last1=Golub |first2=Charles F. |last2=van Loan |title=Matrix Computations |edition=3rd |publisher=[[Johns Hopkins University Press]] |date=1986 |isbn=978-0-8018-5413-2}}
* {{cite book |title=Numerical Recipes - The Art of Scientific Computing |title-link=Numerical Recipes |first1=William Henry |last1=Press |author-link1=William Henry Press |first2=Saul A. |last2=Teukolsky |author-link2=Saul A. Teukolsky |first3=William T. |last3=Vetterling |author-link3=William T. Vetterling |first4=Brian P. |last4=Flannery |author-link4=Brian P. Flannery |edition=3rd |date=2007 |orig-date=1986 |isbn=978-0-521-88407-5 |publisher=[[Cambridge University Press]]}} (NB. Edition with source code CD-ROM.)
* {{cite book |first=Donald Ervin |last=Knuth |author-link=Donald Ervin Knuth |title=[[The Art of Computer Programming]]'', Vol. 2: ''Seminumerical Algorithms |edition=3rd |publisher=[[Addison-Wesley]] |date=1997 |isbn=978-0-201-89684-8 |chapter=Section 4.2: Floating-Point Arithmetic |pages=214–264}}
* {{cite book |first1=Gerrit Anne |last1=Blaauw |author-link1=Gerrit Anne Blaauw |first2=Frederick Phillips |last2=Brooks, Jr. |author-link2=Frederick Phillips Brooks, Jr. |title=Computer Architecture: Concepts and Evolution |publisher=[[Addison-Wesley]] |date=1997 |edition=1st |isbn=0-201-10557-8}} (1213 pages) (NB. This is a single-volume edition. This work was also available in a two-volume version.)
* {{cite book | first1=Peter | last1=Kornerup | first2=David W. | last2=Matula | date=2010 | title=Finite Precision Number Systems and Arithmetic | isbn=978-0-521-76135-2 | publisher=[[Cambridge University Press]]}}
* {{citation |title=Floating-Point Formats |first=John J. G. |last=Savard |date=2018 |orig-date=2005 |work=quadibloc |url=http://www.quadibloc.com/comp/cp0201.htm |access-date=2018-07-16 |url-status=live |archive-url=https://web.archive.org/web/20180703001709/http://www.quadibloc.com/comp/cp0201.htm |archive-date=2018-07-03}}
* {{cite book |last1=Muller |first1=Jean-Michel |last2=Brunie |first2=Nicolas |last3=de Dinechin |first3=Florent |last4=Jeannerod |first4=Claude-Pierre |first5=Mioara |last5=Joldes |last6=Lefèvre |first6=Vincent |last7=Melquiond |first7=Guillaume |last8=Revol |first8=Nathalie|author8-link=Nathalie Revol |last9=Torres |first9=Serge |title=Handbook of Floating-Point Arithmetic |date=2018 |orig-date=2010 |publisher=[[Birkhäuser]] |edition=2nd |isbn=978-3-319-76525-9 |lccn=2018935254 |doi=10.1007/978-3-319-76526-6 |url=https://books.google.com/books?id=h3ZZDwAAQBAJ}}

== External links ==
* {{cite web |url=http://www.mrob.com/pub/math/floatformats.html |title=Survey of Floating-Point Formats}} (NB. This page gives a very brief summary of floating-point formats that have been used over the years.)
* {{cite journal |url=https://hal.science/hal-00128124/en/ |title=The pitfalls of verifying floating-point computations |first=David |last=Monniaux |journal=ACM Transactions on Programming Languages and Systems |publisher=[[Association for Computing Machinery]] (ACM) Transactions on programming languages and systems (TOPLAS) |date=May 2008 |volume=30 |issue=3 |pages=1–41 |doi=10.1145/1353445.1353446 |arxiv=cs/0701192 |s2cid=218578808}} (NB. A compendium of non-intuitive behaviors of floating point on popular architectures, with implications for program verification and testing.)
* [https://opencores.org/ OpenCores]. (NB. This website contains open source floating-point IP cores for the implementation of floating-point operators in FPGA or ASIC devices. The project ''double_fpu'' contains verilog source code of a double-precision floating-point unit. The project ''fpuvhdl'' contains vhdl source code of a single-precision floating-point unit.)
* {{cite web |url=http://msdn.microsoft.com/en-us/library/aa289157(v=vs.71).aspx |title=Microsoft Visual C++ Floating-Point Optimization |first=Eric |last=Fleegal |publisher=[[Microsoft Developer Network]] |date=2004 |archive-url=https://web.archive.org/web/20170706020455/http://msdn.microsoft.com/en-us/library/aa289157(v=vs.71).aspx |url-status=dead |archive-date=2017-07-06}}

{{Data types}}

[[Category:Floating point| ]]
[[Category:Computer arithmetic]]
[[Category:Articles with example C code]]

Associative property

2025-06-20T01:29:58Z

173.79.19.248: /* Nonassociativity of floating point calculation */ has never been such an article

{{Short description|Property of a mathematical operation}}
{{redirect|Associativity }}
{{redirect2|Associative|non-associative|associative and non-associative learning|Learning#Types}}
{{More citations needed|date=June 2009}}
{{Infobox mathematical statement
| name = Associative property
| image = [[File:Associativity of binary operations (without question marks).svg|300px|class=skin-invert-image]]
| caption = A visual graph representing associative operations; <math>(x\circ y)\circ z = x\circ(y\circ z)</math>
| type = [[Principle|Law]], [[rule of replacement]]
| field = {{Plainlist|
* [[Elementary algebra]]
* [[Boolean algebra]]
* [[Set theory]]
* [[Linear algebra]]
* [[Propositional calculus]]
}}
| statement =
| symbolic statement = {{Plainlist|
# Elementary algebra
#: <math>(x \,*\, y) \,*\, z = x \,*\, (y \,*\, z) \forall x,y,z \in S</math>
# Propositional calculus
#:<math>(P \lor (Q \lor R)) \Leftrightarrow ((P \lor Q) \lor R)</math>
#:<math>(P \land (Q \land R)) \Leftrightarrow ((P \land Q) \land R),</math>
}}
}}

In [[mathematics]], the '''associative property'''<ref>
{{cite book
|last=Hungerford
|first=Thomas W.
|year=1974 |edition=1st
|title=Algebra
|page=24
|publisher=[[Springer Science+Business Media|Springer]]
|isbn=978-0387905181
|quote=Definition 1.1 (i) a(bc) = (ab)c for all a, b, c in G.}}</ref> is a property of some [[binary operation]]s that rearranging the [[parentheses]] in an expression will not change the result. In [[propositional logic]], '''associativity''' is a [[Validity (logic)|valid]] [[rule of replacement]] for [[well-formed formula|expressions]] in [[Formal proof|logical proofs]].

Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the [[Operation (mathematics)|operations]] are performed does not matter as long as the sequence of the [[operand]]s is not changed. That is (after rewriting the expression with parentheses and in infix notation if necessary), rearranging the parentheses in such an expression will not change its value. Consider the following equations:

<math display="block">\begin{align}
(2 + 3) + 4 &= 2 + (3 + 4) = 9 \,\\
2 \times (3 \times 4) &= (2 \times 3) \times 4 = 24 .
\end{align}</math>

Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing addition and multiplication on any [[real number]]s, it can be said that "addition and multiplication of real numbers are associative operations".

Associativity is not the same as [[commutativity]], which addresses whether the order of two operands affects the result. For example, the order does not matter in the multiplication of real numbers, that is, {{math|1={{var|a}} × {{var|b}} = {{var|b}} × {{var|a}}}}, so we say that the multiplication of real numbers is a commutative operation. However, operations such as [[function composition]] and [[matrix multiplication]] are associative, but not (generally) commutative.

Associative operations are abundant in mathematics; in fact, many [[algebraic structure]]s (such as [[semigroup (mathematics)|semigroups]] and [[category (mathematics)|categories]]) explicitly require their binary operations to be associative. However, many important and interesting operations are non-associative; some examples include [[subtraction]], [[exponentiation]], and the [[vector cross product]]. In contrast to the theoretical properties of real numbers, the addition of [[floating point]] numbers in computer science is not associative, and the choice of how to associate an expression can have a significant effect on rounding error.

== Definition ==
[[File:Semigroup_associative.svg|thumbnail|A binary operation ∗ on the set ''S'' is associative when [[Commutative diagram|this diagram commutes]]. That is, when the two paths from {{math|{{var|S}}×{{var|S}}×{{var|S}}}} to {{mvar|S}} [[Function composition|compose]] to the same function from {{math|{{var|S}}×{{var|S}}×{{var|S}}}} to {{mvar|S}}.]]
Formally, a [[binary operation]] <math>\ast</math> on a [[Set (mathematics)|set]] {{mvar|S}} is called '''associative''' if it satisfies the '''associative law''':
:<math>(x \ast y) \ast z = x \ast (y \ast z)</math>, for all <math>x,y,z</math> in {{mvar|S}}.

Here, ∗ is used to replace the symbol of the operation, which may be any symbol, and even the absence of symbol ([[Juxtaposition#Mathematics|juxtaposition]]) as for [[multiplication]].
:<math>(xy)z = x(yz)</math>, for all <math>x,y,z</math> in {{mvar|S}}.

The associative law can also be expressed in functional notation thus: <math>(f \circ (g \circ h))(x) = ((f \circ g) \circ h)(x)</math>

==Generalized associative law==
[[Image:Tamari lattice.svg|thumb|In the absence of the associative property, five factors {{mvar|a}}, {{mvar|b}},{{mvar|c}}, {{mvar|d}}, {{mvar|e}} result in a [[Tamari lattice]] of order four, possibly different products.]]
If a binary operation is associative, repeated application of the operation produces the same result regardless of how valid pairs of parentheses are inserted in the expression.<ref>{{cite book |last=Durbin |first=John R. |title=Modern Algebra: an Introduction |year=1992 |publisher=Wiley |location=New York |isbn=978-0-471-51001-7 |page=78 |url=http://www.wiley.com/WileyCDA/WileyTitle/productCd-EHEP000258.html |edition=3rd |quote=If <math>a_1, a_2, \dots, a_n \,\, (n \ge 2)</math> are elements of a set with an associative operation, then the product <math>a_1 a_2 \cdots a_n</math> is unambiguous; this is, the same element will be obtained regardless of how parentheses are inserted in the product.}}</ref> This is called the '''generalized associative law'''.

The number of possible bracketings is just the [[Catalan number]], <math>C_n</math>
, for ''n'' operations on ''n+1'' values. For instance, a product of 3 operations on 4 elements may be written (ignoring permutations of the arguments), in <math>C_3 = 5</math> possible ways:

*<math>((ab)c)d</math>
*<math>(a(bc))d</math>
*<math>a((bc)d)</math>
*<math>(a(b(cd))</math>
*<math>(ab)(cd)</math>

If the product operation is associative, the generalized associative law says that all these expressions will yield the same result. So unless the expression with omitted parentheses already has a different meaning (see below), the parentheses can be considered unnecessary and "the" product can be written unambiguously as
:<math>abcd</math>
As the number of elements increases, the [[Catalan number#Applications in combinatorics|number of possible ways to insert parentheses]] grows quickly, but they remain unnecessary for disambiguation.

An example where this does not work is the [[logical biconditional]] {{math|↔}}. It is associative; thus, {{math|{{var|A}} ↔ ({{var|B}} ↔ {{var|C}})}} is equivalent to {{math|({{var|A}} ↔ {{var|B}}) ↔ {{var|C}}}}, but {{math|{{var|A}} ↔ {{var|B}} ↔ {{var|C}}}} most commonly means {{math|({{var|A}} ↔ {{var|B}}) and ({{var|B}} ↔ {{var|C}})}}, which is not equivalent.

==Examples==
[[File:Associativity of real number addition.svg|thumb|The addition of real numbers is associative.]]
Some examples of associative operations include the following.

{{unordered list
|1= The [[string concatenation|concatenation]] of the three strings <code>"hello"</code>, <code>" "</code>, <code>"world"</code> can be computed by concatenating the first two strings (giving <code>"hello "</code>) and appending the third string (<code>"world"</code>), or by joining the second and third string (giving <code>" world"</code>) and concatenating the first string (<code>"hello"</code>) with the result. The two methods produce the same result; string concatenation is associative (but not commutative).
|2= In [[arithmetic]], [[addition]] and [[multiplication]] of [[real number]]s are associative; i.e.,

<math display="block">
\left.
\begin{matrix}
(x+y)+z=x+(y+z)=x+y+z\quad
\\
(x\,y)z=x(y\,z)=x\,y\,z\qquad\qquad\qquad\quad\ \ \,
\end{matrix}
\right\}
\mbox{for all }x,y,z\in\mathbb{R}.
</math>

Because of associativity, the grouping parentheses can be omitted without ambiguity.
|3= The trivial operation {{math|1={{var|x}} ∗ {{var|y}} = {{var|x}}}} (that is, the result is the first argument, no matter what the second argument is) is associative but not commutative. Likewise, the trivial operation <math>x \circ y = y</math> (that is, the result is the second argument, no matter what the first argument is) is associative but not commutative.
|4= Addition and multiplication of [[complex number]]s and [[quaternion]]s are associative. Addition of [[octonion]]s is also associative, but multiplication of octonions is non-associative.
|5= The [[greatest common divisor]] and [[least common multiple]] functions act associatively.

<math display="block">
\left.
\begin{matrix}
\operatorname{gcd}(\operatorname{gcd}(x,y),z)=
\operatorname{gcd}(x,\operatorname{gcd}(y,z))=
\operatorname{gcd}(x,y,z)\ \quad
\\
\operatorname{lcm}(\operatorname{lcm}(x,y),z)=
\operatorname{lcm}(x,\operatorname{lcm}(y,z))=
\operatorname{lcm}(x,y,z)\quad
\end{matrix}
\right\}\mbox{ for all }x,y,z\in\mathbb{Z}.
</math>
|6= Taking the [[intersection (set theory)|intersection]] or the [[union (set theory)|union]] of [[Set (mathematics)|sets]]:
<math display="block">
\left.
\begin{matrix}
(A\cap B)\cap C=A\cap(B\cap C)=A\cap B\cap C\quad
\\
(A\cup B)\cup C=A\cup(B\cup C)=A\cup B\cup C\quad
\end{matrix}
\right\}\mbox{for all sets }A,B,C.
</math>
|7= If {{mvar|M}} is some set and {{mvar|S}} denotes the set of all functions from {{mvar|M}} to {{mvar|M}}, then the operation of [[function composition]] on {{mvar|S}} is associative:<math display="block">(f\circ g)\circ h=f\circ(g\circ h)=f\circ g\circ h\qquad\mbox{for all }f,g,h\in S.</math>
|8= Slightly more generally, given four sets {{mvar|M}}, {{mvar|N}}, {{mvar|P}} and {{mvar|Q}}, with {{math|{{var|h}} : {{var|M}} → {{var|N}}}}, {{math|{{var|g}} : {{var|N}} → {{var|P}}}}, and {{math|{{var|f}} : {{var|P}} → {{var|Q}}}}, then

<math display="block">(f\circ g)\circ h=f\circ(g\circ h)=f\circ g\circ h</math>

as before. In short, composition of maps is always associative.
|9= In [[category theory]], composition of morphisms is associative by definition. Associativity of functors and natural transformations follows from associativity of morphisms.
|10= Consider a set with three elements, {{mvar|A}}, {{mvar|B}}, and {{mvar|C}}. The following operation:

{{wikitable| class="wikitable" style="text-align:center"
|-
! × !! {{mvar|A}} !! {{mvar|B}} !! {{mvar|C}}
|-
! {{mvar|A}}
| {{mvar|A}} || {{mvar|A}} || {{mvar|A}}
|-
! {{mvar|B}}
| {{mvar|A}} || {{mvar|B}} || {{mvar|C}}
|-
! {{mvar|C}}
| {{mvar|A}} || {{mvar|A}} || {{mvar|A}}
}}

is associative. Thus, for example, {{math|1={{var|A}}({{var|B}}{{var|C}}) = ({{var|A}}{{var|B}}){{var|C}} = {{var|A}}}}. This operation is not commutative.
|11= Because [[Matrix (mathematics)|matrices]] represent [[linear map|linear function]]s, and [[matrix multiplication]] represents function composition, one can immediately conclude that matrix multiplication is associative.<ref>{{cite web|url=http://www.khanacademy.org/math/linear-algebra/matrix-transformations/composition-of-transformations/v/matrix-product-associativity|title=Matrix product associativity|publisher=Khan Academy|access-date=5 June 2016}}</ref>
|12= For [[real number]]s (and for any [[totally ordered set]]), the minimum and maximum operation is associative: <math display="block">\max(a, \max(b, c)) = \max(\max(a, b), c) \quad \text{ and } \quad \min(a, \min(b, c)) = \min(\min(a, b), c).</math>

}}

== Propositional logic ==
{{Transformation rules}}

=== Rule of replacement ===
In standard truth-functional propositional logic, ''association'',<ref>{{cite book |last1=Moore |first1=Brooke Noel |last2=Parker |first2=Richard |date=2017 |title=Critical Thinking |location=New York |publisher=McGraw-Hill Education |page=321 |isbn=9781259690877|edition=12th }}</ref><ref>{{cite book |last1=Copi |first1=Irving M. |last2=Cohen |first2=Carl |last3=McMahon |first3=Kenneth |date=2014 |title=Introduction to Logic |location=Essex |publisher=Pearson Education |page=387 |isbn=9781292024820|edition=14th }}</ref> or ''associativity''<ref>{{cite book |last1=Hurley |first1=Patrick J. |last2=Watson |first2=Lori |date=2016 |title=A Concise Introduction to Logic |location=Boston |publisher=Cengage Learning |page=427 |isbn=9781305958098|edition=13th }}</ref> are two [[Validity (logic)|valid]] [[rule of replacement|rules of replacement]]. The rules allow one to move parentheses in [[well-formed formula|logical expressions]] in [[formal proof|logical proofs]]. The rules (using [[Logical connective#In language|logical connectives]] notation) are:

<math display="block">(P \lor (Q \lor R)) \Leftrightarrow ((P \lor Q) \lor R)</math>

and

<math display="block">(P \land (Q \land R)) \Leftrightarrow ((P \land Q) \land R),</math>

where "<math>\Leftrightarrow</math>" is a [[metalogic]]al [[Symbol (formal)|symbol]] representing "can be replaced in a [[Formal proof|proof]] with".

=== Truth functional connectives ===
''Associativity'' is a property of some [[logical connective]]s of truth-functional [[propositional logic]]. The following [[logical equivalence]]s demonstrate that associativity is a property of particular connectives. The following (and their converses, since {{math|↔}} is commutative) are truth-functional [[tautology (logic)|tautologies]].{{citation needed|reason=Stack Exchange is not a reliable source?|date=June 2022}}

;Associativity of disjunction
:<math>((P \lor Q) \lor R) \leftrightarrow (P \lor (Q \lor R))</math>
;Associativity of conjunction
:<math>((P \land Q) \land R) \leftrightarrow (P \land (Q \land R))</math>
;Associativity of equivalence
:<math>((P \leftrightarrow Q) \leftrightarrow R) \leftrightarrow (P \leftrightarrow (Q \leftrightarrow R))</math>

[[Logical NOR|Joint denial]] is an example of a truth functional connective that is ''not'' associative.

== Non-associative operation==
A binary operation <math>*</math> on a set ''S'' that does not satisfy the associative law is called '''non-associative'''. Symbolically,

<math display="block">(x*y)*z\ne x*(y*z)\qquad\mbox{for some }x,y,z\in S.</math>

For such an operation the order of evaluation ''does'' matter. For example:
; [[Subtraction]]
:<math>
(5-3)-2 \, \ne \, 5-(3-2)
</math>
; [[Division (mathematics)|Division]]
:<math>
(4/2)/2 \, \ne \, 4/(2/2)
</math>
; [[Exponentiation]]
:<math>
2^{(1^2)} \, \ne \, (2^1)^2
</math>
; [[Vector cross product]]
:<math>\begin{align}
\mathbf{i} \times (\mathbf{i} \times \mathbf{j}) &= \mathbf{i} \times \mathbf{k} = -\mathbf{j} \\
(\mathbf{i} \times \mathbf{i}) \times \mathbf{j} &= \mathbf{0} \times \mathbf{j} = \mathbf{0}
\end{align}</math>

Also although addition is associative for finite sums, it is not associative inside infinite sums ([[series (mathematics)|series]]). For example,
<math display="block">
(1+-1)+(1+-1)+(1+-1)+(1+-1)+(1+-1)+(1+-1)+\dots = 0
</math>
whereas
<math display="block">
1+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+\dots = 1.
</math>

Some non-associative operations are fundamental in mathematics. They appear often as the multiplication in structures called [[non-associative algebra]]s, which have also an addition and a [[scalar multiplication]]. Examples are the [[octonion]]s and [[Lie algebra]]s. In Lie algebras, the multiplication satisfies [[Jacobi identity]] instead of the associative law; this allows abstracting the algebraic nature of [[infinitesimal transformation]]s.

Other examples are [[quasigroup]], [[quasifield]], [[non-associative ring]], and [[commutative non-associative magmas]].

===Nonassociativity of floating point calculation===

In mathematics, addition and multiplication of real numbers are associative. By contrast, in computer science, addition and multiplication of [[floating point]] numbers are ''not'' associative, as different rounding errors may be introduced when dissimilar-sized values are joined in a different order.<ref>Knuth, Donald, [[The Art of Computer Programming]], Volume 3, section 4.2.2</ref>

To illustrate this, consider a floating point representation with a 4-bit [[significand]]:

{{block indent|1=(1.0002×20 + 1.0002×20) +
1.0002×24 = 1.0002×2{{fontcolor|red|1}} + 1.0002×24 = 1.00{{fontcolor|red|1}}2×24}}
{{block indent|1=1.0002×20 + (1.0002×20 +
1.0002×24) = 1.0002×2{{fontcolor|red|0}} + 1.0002×24 = 1.00{{fontcolor|red|0}}2×24}}

Even though most computers compute with 24 or 53 bits of significand,<ref>{{Cite book |title=IEEE Standard for Floating-Point Arithmetic |author=IEEE Computer Society |date=29 August 2008 |id=IEEE Std 754-2008|doi=10.1109/IEEESTD.2008.4610935 |ref=CITEREFIEEE_7542008 |isbn=978-0-7381-5753-5}}</ref> this is still an important source of rounding error, and approaches such as the [[Kahan summation algorithm]] are ways to minimise the errors. It can be especially problematic in parallel computing.<ref>{{Citation
| last1 = Villa
| first1 = Oreste
| last2 = Chavarría-mir
| first2 = Daniel
| last3 = Gurumoorthi
| first3 = Vidhya
| last4 = Márquez
| first4 = Andrés
| last5 = Krishnamoorthy
| first5 = Sriram
| title = Effects of Floating-Point non-Associativity on Numerical Computations on Massively Multithreaded Systems
| url = http://cass-mt.pnnl.gov/docs/pubs/pnnleffects_of_floating-pointpaper.pdf
| access-date = 8 April 2014
| archive-url = https://web.archive.org/web/20130215171724/http://cass-mt.pnnl.gov/docs/pubs/pnnleffects_of_floating-pointpaper.pdf
| archive-date = 15 February 2013
| url-status = dead
}}</ref><ref name="Goldberg_1991">{{cite journal|last=Goldberg|first=David|date=March 1991|title=What Every Computer Scientist Should Know About Floating-Point Arithmetic|url=http://perso.ens-lyon.fr/jean-michel.muller/goldberg.pdf|journal=[[ACM Computing Surveys]]|volume=23|issue=1|pages=5–48|doi=10.1145/103162.103163|s2cid=222008826|access-date=20 January 2016|url-status=live|archive-url=https://web.archive.org/web/20220519083509/http://perso.ens-lyon.fr/jean-michel.muller/goldberg.pdf|archive-date=2022-05-19}}</ref>

=== Notation for non-associative operations ===
{{main|Operator associativity}}

In general, parentheses must be used to indicate the [[order of operations|order of evaluation]] if a non-associative operation appears more than once in an expression (unless the notation specifies the order in another way, like <math>\dfrac{2}{3/4}</math>). However, [[mathematician]]s agree on a particular order of evaluation for several common non-associative operations. This is simply a notational convention to avoid parentheses.

A '''left-associative''' operation is a non-associative operation that is conventionally evaluated from left to right, i.e.,

<math display="block">
\left.
\begin{array}{l}
a*b*c=(a*b)*c
\\
a*b*c*d=((a*b)*c)*d
\\
a*b*c*d*e=(((a*b)*c)*d)*e\quad
\\
\mbox{etc.}
\end{array}
\right\}
\mbox{for all }a,b,c,d,e\in S
</math>

while a '''right-associative''' operation is conventionally evaluated from right to left:

<math display="block">
\left.
\begin{array}{l}
x*y*z=x*(y*z)
\\
w*x*y*z=w*(x*(y*z))\quad
\\
v*w*x*y*z=v*(w*(x*(y*z)))\quad\\
\mbox{etc.}
\end{array}
\right\}
\mbox{for all }z,y,x,w,v\in S
</math>

Both left-associative and right-associative operations occur. Left-associative operations include the following:
; Subtraction and division of real numbers<ref>George Mark Bergman [https://math.berkeley.edu/~gbergman/misc/numbers/ord_ops.html "Order of arithmetic operations"]</ref><ref>[http://eduplace.com/math/mathsteps/4/a/index.html "The Order of Operations"]. Education Place.</ref><ref>[https://www.khanacademy.org/math/pre-algebra/pre-algebra-arith-prop/pre-algebra-order-of-operations/v/introduction-to-order-of-operations "The Order of Operations"], timestamp [https://www.youtube.com/watch?v=ClYdw4d4OmA&t=5m40s 5m40s]. [[Khan Academy]].</ref><ref>[http://www.doe.virginia.gov/instruction/mathematics/middle/algebra_readiness/curriculum_companion/order-operations.pdf#page=3 "Using Order of Operations and Exploring Properties"] {{Webarchive|url=https://web.archive.org/web/20220716062834/http://www.doe.virginia.gov/instruction/mathematics/middle/algebra_readiness/curriculum_companion/order-operations.pdf#page=3 |date=2022-07-16 }}, section 9. Virginia Department of Education.</ref><ref name="Bronstein_1987">Bronstein, ''[[:de:Taschenbuch der Mathematik]]'', pages 115-120, chapter: 2.4.1.1, {{ISBN|978-3-8085-5673-3}}</ref>
:<math>x-y-z=(x-y)-z</math>
:<math>x/y/z=(x/y)/z</math>
; Function application
:<math>(f \, x \, y) = ((f \, x) \, y)</math>
This notation can be motivated by the [[currying]] isomorphism, which enables partial application.

Right-associative operations include the following:
; [[Exponentiation]] of real numbers in superscript notation
:<math>x^{y^z}=x^{(y^z)}</math>Exponentiation is commonly used with brackets or right-associatively because a repeated left-associative exponentiation operation is of little use. Repeated powers would mostly be rewritten with multiplication:
:<math>(x^y)^z=x^{(yz)}</math>Formatted correctly, the superscript inherently behaves as a set of parentheses; e.g. in the expression <math>2^{x+3}</math> the addition is performed [[order of operations|before]] the exponentiation despite there being no explicit parentheses <math>2^{(x+3)}</math> wrapped around it. Thus given an expression such as <math>x^{y^z}</math>, the full exponent <math>y^z</math> of the base <math>x</math> is evaluated first. However, in some contexts, especially in handwriting, the difference between <math>{x^y}^z=(x^y)^z</math>, <math>x^{yz}=x^{(yz)}</math> and <math>x^{y^z}=x^{(y^z)}</math> can be hard to see. In such a case, right-associativity is usually implied.
; [[Function (mathematics)|Function definition]]
:<math>\mathbb{Z} \rarr \mathbb{Z} \rarr \mathbb{Z} = \mathbb{Z} \rarr (\mathbb{Z} \rarr \mathbb{Z})</math>
:<math>x \mapsto y \mapsto x - y = x \mapsto (y \mapsto x - y)</math>Using right-associative notation for these operations can be motivated by the [[Curry–Howard correspondence]] and by the [[currying]] isomorphism.

Non-associative operations for which no conventional evaluation order is defined include the following.
; Exponentiation of real numbers in infix notation<ref name="Codeplea_2016">[https://codeplea.com/exponentiation-associativity-options Exponentiation Associativity and Standard Math Notation] Codeplea. 23 August 2016. Retrieved 20 September 2016.</ref>
:<math>(x^\wedge y)^\wedge z\ne x^\wedge(y^\wedge z)</math>
; [[Knuth's up-arrow notation|Knuth's up-arrow operators]]
:<math> a \uparrow \uparrow (b \uparrow \uparrow c) \ne (a \uparrow \uparrow b) \uparrow \uparrow c</math>
:<math> a \uparrow \uparrow \uparrow (b \uparrow \uparrow \uparrow c) \ne (a \uparrow \uparrow \uparrow b) \uparrow \uparrow \uparrow c</math>
; Taking the [[cross product]] of three vectors
:<math>\vec a \times (\vec b \times \vec c) \neq (\vec a \times \vec b ) \times \vec c \qquad \mbox{ for some } \vec a,\vec b,\vec c \in \mathbb{R}^3</math>
; Taking the pairwise [[average]] of real numbers
:<math>{(x+y)/2+z\over2}\ne{x+(y+z)/2\over2} \qquad \mbox{for all }x,y,z\in\mathbb{R} \mbox{ with }x\ne z.</math>
; Taking the [[complement (set theory)|relative complement]] of sets
:<math>(A\backslash B)\backslash C \neq A\backslash (B\backslash C)</math>.(Compare [[material nonimplication]] in logic.)

== History ==

[[William Rowan Hamilton]] seems to have coined the term "associative property"<ref name="Hamilton">{{cite journal |author-link=William Rowan Hamilton |first=W.R. |last=Hamilton |year=1844–1850 |url=http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/OnQuat/ |title=On quaternions or a new system of imaginaries in algebra |journal=[[Philosophical Magazine]] |department=David R. Wilkins collection |publisher=[[Trinity College Dublin]]}}</ref> around 1844, a time when he was contemplating the non-associative algebra of the [[octonions]] he had learned about from [[John T. Graves]].<ref name="Baez">{{Cite journal | last1 = Baez | first1 = John C. | author-link = John Baez| title = The Octonions | journal = Bulletin of the American Mathematical Society | issn = 0273-0979 | volume = 39 | issue = 2 | pages = 145–205 | year = 2002 | url = https://www.ams.org/journals/bull/2002-39-02/S0273-0979-01-00934-X/S0273-0979-01-00934-X.pdf | doi = 10.1090/S0273-0979-01-00934-X | arxiv = math/0105155| mr = 1886087| s2cid = 586512}}</ref>

== Relationship with commutativity in certain special cases ==
In general, associative operations are not commutative. However, under certain special conditions, it may be the case that associativity implies commutativity. Associative operators defined on an interval of the real number line are commutative if they are continuous and injective in both arguments.<ref>{{Cite book |last=Aczél |first=J. |url=https://www.google.com/books/edition/Lectures_on_Functional_Equations_and_The/n7vckU_1tY4C |title=Lectures on Functional Equations and Their Applications |date=1966-01-01 |publisher=Academic Press |year=1966 |isbn=978-0-08-095525-4 |pages=267 |language=en |ol=46920179M |ol-access=free}}</ref> A consequence is that every continuous, associative operator on two real inputs that is strictly increasing in each of its inputs is commutative.<ref>{{Cite journal |last=Ling |first=Cho-Hsin |date=1 September 1964 |title=Representation of associative functions |url=https://scispace.com/pdf/representation-of-associative-functions-3chz5py4.pdf |journal=Publicationes Mathematicae |volume=12 |pages=189-212}}</ref>

==See also==
{{Wiktionary}}
* [[Light's associativity test]]
* [[Telescoping series]], the use of addition associativity for cancelling terms in an infinite [[series (mathematics)|series]]
* A [[semigroup]] is a set with an associative binary operation.
* [[Commutativity]] and [[distributivity]] are two other frequently discussed properties of binary operations.
* [[Power associativity]], [[alternativity]], [[flexible algebra|flexibility]] and [[N-ary associativity]] are weak forms of associativity.
* [[Moufang loop|Moufang identities]] also provide a weak form of associativity.

==References==
{{reflist}}

[[Category:Properties of binary operations]]
[[Category:Elementary algebra]]
[[Category:Functional analysis]]
[[Category:Rules of inference]]

David Goldberg

2025-06-20T01:29:35Z

173.79.19.248: 9 years later and there is still no article about this person (who is not mentioned at the one blue-link in this line), I think it's safe to remove

'''David Goldberg''' may refer to:

*[[David E. Goldberg]] (born 1953), American professor of engineering and computer science at University of Illinois at Urbana-Champaign
*[[David Theo Goldberg]] (born 1952), South African philosopher and director of the University of California Humanities Research Institute
*[[Dave Goldberg]] (1967–2015), American Internet entrepreneur
*[[David Goldberg (psychiatrist)]] (1934–2024), British academic and social psychiatrist

==See also==
*[[Adam David Goldberg]] (born 1980), American football offensive tackle
*[[Gary David Goldberg]] (1944–2013), American writer and producer for television and film
*[[Pizzagate conspiracy theory]], instigated by Tweet from @DavidGoldbergNY

{{hndis|name=Goldberg, David}}

Pythagorean means

2025-06-17T02:38:04Z

173.79.19.248: /* See also */ doesn’t seem to be linked?

{{short description|Classical averages studied in ancient Greece}}
{{More citations needed|date=July 2019}}
[[Image:MathematicalMeans.svg|thumb|right|A geometric construction of the quadratic mean and the Pythagorean means (of two numbers ''a'' and ''b''). Harmonic mean denoted by {{colorbox|purple}}{{nbsp}}''H'', geometric by {{colorbox|blue}}{{nbsp}}''G'', arithmetic by {{colorbox|red}}{{nbsp}}''A'' and quadratic mean (also known as [[root mean square]]) denoted by {{colorbox|green}}{{nbsp}}''Q''.]]
[[Image:Comparison_Pythagorean_means.svg|thumb|right|Comparison of the arithmetic, geometric and harmonic means of a pair of numbers. The vertical dashed lines are [[asymptote]]s for the harmonic means.]]
In mathematics, the three classical '''Pythagorean means''' are the [[arithmetic mean]] (AM), the [[geometric mean]] (GM), and the [[harmonic mean]] (HM). These [[mean]]s were studied with proportions by [[Pythagoreans]] and later generations of Greek mathematicians<ref>{{cite book|first=Thomas|last=Heath|title=History of Ancient Greek Mathematics}}</ref> because of their importance in geometry and music.

==Definition==
The three Pythagorean means are defined by the equations
<math display="block">\begin{align}
\operatorname{AM} \left( x_1,\; \ldots,\; x_n \right) &= \frac{ x_1 + \;\cdots\; + x_n }{n}, \\[9pt]
\operatorname{GM} \left( x_1,\; \ldots,\; x_n \right) &= \sqrt[n]{\left\vert x_1 \times \,\cdots\, \times x_n \right\vert}, \text{ and} \\[9pt]
\operatorname{HM} \left( x_1,\; \ldots,\; x_n \right) &= \frac{n}{\displaystyle \frac{1}{x_1} + \;\cdots\; + \frac{1}{x_n}} .
\end{align}</math>

==Properties==
Each mean, <math display="inline">\operatorname{M}</math>, has the following properties for positive real inputs:

; First-order [[homogeneous function|homogeneity]]: <math>\operatorname{M}(bx_1, \ldots, bx_n) = b \operatorname{M}(x_1, \ldots, x_n)</math>
; Invariance under exchange: <math>\operatorname{M}(\ldots, x_i, \ldots, x_j, \ldots) = \operatorname{M}(\ldots, x_j, \ldots, x_i, \ldots)</math> for any <math>i</math> and <math>j</math>.
; Monotonicity: if <math> a \leq b</math> then <math>\operatorname{M}(a,x_1,x_2,\ldots x_n) \leq \operatorname{M}(b,x_1,x_2,\ldots x_n)</math>
; [[Idempotent|Idempotence]]: <math> M(x,x,\ldots x) = x </math> for all <math>x</math>

Monotonicity and idempotence together imply that a mean of a set always lies between the extremes of the set:
<math display="block">\min(x_1, \ldots, x_n) \leq \operatorname{M}(x_1, \ldots, x_n) \leq \max(x_1, \ldots, x_n).</math>

The harmonic and arithmetic means are reciprocal duals of each other for positive arguments,
<math display="block">\operatorname{HM}\left(\frac{1}{x_1}, \ldots, \frac{1}{x_n}\right) = \frac{1}{\operatorname{AM}\left(x_1, \ldots, x_n\right)},</math>
while the geometric mean is its own reciprocal dual:
<math display="block">\operatorname{GM}\left(\frac{1}{x_1}, \ldots, \frac{1}{x_n}\right) = \frac{1}{\operatorname{GM}\left(x_1, \ldots, x_n\right)}.</math>

== Inequalities among means ==
{{QM_AM_GM_HM_inequality_visual_proof.svg}}
There is an ordering to these means (if all of the <math> x_i </math> are positive)
<math display="block">\min \leq \operatorname{HM} \leq \operatorname{GM} \leq \operatorname{AM} \leq \max</math>
with equality holding if and only if the <math>x_i</math> are all equal.

This is a generalization of the [[inequality of arithmetic and geometric means]] and a special case of an inequality for [[generalized mean]]s. The proof follows from the [[inequality of arithmetic and geometric means|arithmetic–geometric mean inequality]], <math>\operatorname{AM} \leq \max</math>, and reciprocal duality (<math>\min</math> and <math>\max</math> are also reciprocal dual to each other).

The study of the Pythagorean means is closely related to the study of [[majorization]] and [[Schur-convex function|Schur-convex functions]]. The harmonic and geometric means are concave symmetric functions of their arguments, and hence Schur-concave, while the arithmetic mean is a linear function of its arguments and hence is both concave and convex.

==History==

Almost everything that we know about the Pythagorean means came from arithmetic handbooks written in the first and second century. [[Nicomachus of Gerasa]] says that they were "acknowledged by all the ancients, Pythagoras, Plato and Aristotle."<ref>{{Cite book |author=Nicomachus of Gerasa |author-link=Nicomachus of Gerasa |url=https://books.google.com/books?id=wdhMAAAAMAAJ&dq=%22acknowledged+by+all+the+ancients%2C+Pythagoras%2C+Plato+and+Aristotle.%22&pg=PA266 |title=Introduction to Arithmetic |date=1926 |publisher=Macmillan |language=en}}</ref> Their earliest known use is a fragment of the Pythagorean philosopher [[Archytas of Tarentum]]:

{{Blockquote|text= There are three means in music: one is arithmetic, second is the geometric, third is sub-contrary, which they call harmonic. The mean is arithmetic when three terms are in proportion such that the excess by which the first exceeds the second is that by which the second exceeds the third. In this proportion it turns out that the interval of the greater terms is less, but that of the lesser terms greater. The mean is the geometric when they are such that as the first is to the second, so the second is to the third. Of these terms the greater and the lesser have the interval between them equal. Subcontrary, which we call harmonic, is the mean when they are such that, by whatever part of itself the first term exceeds the second, by that part of the third the middle term exceeds the third. It turns out that in this proportion the interval between the greater terms is greater and that between the lesser terms is less.|source=<ref>{{cite book
|last=Huffman
|first=Carl
|date=2005
|title=Archytas of Tarentum: Pythagorean, philosopher and mathematician king
|publisher=Cambridge University Press
|page= 163
|isbn=1139444077
}}</ref>|author=Archytas of Tarentum}}

The name "harmonic mean", according to [[Iamblichus]], was coined by Archytas and [[Hippasus]]. The Pythagorean means also appear in [[Plato]]'s [[Timaeus (dialogue)|Timaeus]]. Another evidence of their early use is a commentary by [[Pappus of Alexandria| Pappus]].

{{Blockquote|text= It was [...] [[Theaetetus (mathematician)| Theaetetus]] who distinguished the powers which are commensurable in length from those which are incommensurable, and who divided the more generally known irrational lines according to the different means, assigning the medial lines to geometry, the binomial to arithmetic, and the apotome to harmony, as is stated by [[Eudemus of Rhodes|Eudemus]], the Peripatetic.<ref>{{cite book
|last=Huffman
|first=Carl
|date=2014
|title=A History of Pythagoreanism
|publisher=Cambridge University Press
|page= 168
|isbn=978-1139915984
}}</ref>}}

The term "mean" (Ancient Greek μεσότης, ''mesótēs'') appears in the [[Neopythagoreanism|Neopythagorean]] arithmetic handbooks in connection with the term "proportion" (Ancient Greek ἀναλογία, ''analogía'').{{cn|date=April 2021}}

==Smallest distinct positive integer means==
{| class="wikitable infobox" style="width:1px;"
! ''a'' !! ''b'' !! HM !! GM !! AM
|-
| 5 || 45 || 9 || 15 || 25
|-
| 10 || 40 || 16 || 20 || 25

|}
[[File:Pythagorean_means_nomograms.svg|thumb|250px|[[Nomogram]]s to graphically calculate {{nowrap|arithmetic (1),}} {{nowrap|geometric (2)}} and {{nowrap|harmonic (3)}} means, ''z'' of {{nowrap|1=''x''=40}} and {{nowrap|1=''y''=10}} (red), and {{nowrap|1=''x''=45}} and {{nowrap|1=''y''=5}} (blue)]]
Of all pairs of different natural numbers of the form (''a'', ''b'') such that ''a'' < ''b'', the smallest (as defined by least value of ''a'' + ''b'') for which the arithmetic, geometric and harmonic means are all also natural numbers are (5, 45) and (10, 40).<ref>Virginia Tech Mathematics Department, [http://intranet.math.vt.edu/vtrmc/solutions/39th%20VTRMC,%202017,%20Solutions.pdf ''39th VTRMC, 2017, Solutions''], part 5</ref>

==See also==
* [[Arithmetic–geometric mean]]
* [[Average]]
* [[Golden ratio]]
* [[Kepler triangle]]
* [[QM-AM-GM-HM inequalities]]

==Notes==
{{reflist|group=note}}
==References==
{{reflist}}

==External links==
*{{MathWorld|urlname=PythagoreanMeans|title=Pythagorean Means|author=Cantrell, David W.}}

[[Category:Means]]
[[Category:Greek mathematics]]

Dhondo Keshav Karve

2025-06-14T01:33:45Z

173.79.19.248: Rv addition of very poor sources, see https://en.wikipedia.org/wiki/Wikipedia:Reliable_sources/Noticeboard/Archive_468#Observervoice.com

{{Short description|Indian social reformer (1858–1962)}}
{{Use dmy dates|date=March 2020}}
{{use Indian English|date=January 2018}}
{{Infobox person
| name = Dhondo Keshav Karve
| honorific_prefix = ''[[Maharshi]]''
| image = Dhondo Keshav Karve 1958 stamp of India.jpg
| caption = Karve on a 1958 stamp of India
| birth_date = {{Birth date|1858|4|18|df=y}}
| birth_place = [[Dapoli]], [[Bombay Presidency]], [[British Raj|British India]]
| death_date = {{death date and age|1962|11|9|1858|4|18|df=y}}
| death_place = [[Pune, Maharashtra]], India
| occupation = {{hlist|Professor|activist|writer|social worker}}
| spouse = Radhabai and Godubai
| children = 3, including [[Raghunath Karve]]
| awards = [[Bharat Ratna]] (1958) [[Padma Vibhushan]] (1955)
}}

'''Dhondo Keshav Karve''' (18 April 1858 – 9 November 1962) ({{Pronunciation|Dhondo_Keshav_Karve.wav|help=no}}), popularly known as '''Maharshi Karve''', was a social reformer in India in the field of [[women's rights|women's welfare]]. He advocated widow remarriage, and he himself remarried a widow as a widower. Karve was a pioneer in promoting widows' education. He founded the first women's university in India, the [[SNDT Women's University]] in 1916.<ref>{{Cite web |title=Remembering Maharshi Karve, the man who set up India's first university for women |url=https://www.indiatoday.in/education-today/gk-current-affairs/story/maharshi-karve-318508-2016-04-18 |access-date=2023-05-21 |website=India Today |language=en}}</ref> The [[Government of India]] awarded him with the highest civilian award, the [[Bharat Ratna]], in 1958, the year of his 100th birthday. He organized a conference against the practice of [[Devadasi|devdasi]]. He started 'Anath balikashram' an orphanage for girls. His intention was to give education to all women and make them stand on their own feet. Through his efforts, the first women university was set up in 20th century.

The appellation ''[[Maharshi]]'', which the Indian public often assigned to Karve, means "great sage".

== Biography ==

=== Early life and education ===
Dhondo Keshav Karve was born on 18 April 1858, at [[Shiravali|Sheravali]], in [[Ratnagiri district]] of [[Maharashtra]]. He belonged to a lower middle-class family and his father's name was Keshav Bapunna Karve.<ref>{{Cite web|date=2018-04-18|title=Dhondo Keshav Karve – The Great Indian Social Reformer who...|url=https://www.beaninspirer.com/dhondo-keshav-karve-the-great-indian-social-reformer-who-pioneered-womens-education-in-india-and-set-up-indias-first-school-for-widows-and-first-university-for-women/|access-date=2022-01-11|website=www.beaninspirer.com|language=en-us}}</ref>

In 1884, he graduated with a degree in mathematics from [[Elphinstone College]].<ref name=":0">{{Cite news |last=Dagar |first=Nisha |date=2019-11-09 |title=देश-विदेश घूमकर किया चंदा इकट्ठा और शुरू कर दी देश की पहली महिला यूनिवर्सिटी! |script-title= |language=hi |trans-title=Collected donations travelling across the country and the world, and established the country's first women's university |work=The Better India |url=https://hindi.thebetterindia.com/history-pages/history-dhondo-keshav-karve-maharashtra-widow-remarriage-sndt-first-women-university-india-news/ |access-date=2023-05-21}}</ref>

=== Career ===
During 1891–1914, Karve taught mathematics at [[Fergusson College]] in [[Pune, Maharashtra]].<ref name="FergussonCollegeMathematics">{{cite web |title=Fergusson College Department of Mathematics web page |url=http://www.fergusson.edu/courses/science/mathematics/ |access-date=11 August 2006 |archive-url=https://web.archive.org/web/20060619143154/http://www.fergusson.edu/courses/science/mathematics/
|archive-date=19 June 2006 |url-status=dead}}</ref><ref>{{cite web | url=https://www.thebetterindia.com/119398/dhondho-keshav-karve-maharashtra-sndt-first-women-university-india/ | title=Remembering the Selfless Man Who Set up India's First University for Women | date=27 October 2017 }}</ref>

In 1929, he visited Europe, America and Japan. During these travels, he met [[Albert Einstein]]. During this world tour, he also raised funds for the university.<ref name=":0" />

==Autobiographical works==
[[File:D_K_Karve.jpg|thumb|{{c.|1916}}]]
Karve wrote two autobiographical works: ''Ātmawrutta'' (1928) in Marathi, and ''Looking Back'' (1936) in English.

==Depictions in popular culture==
The Marathi play ''Himalayachi Saavli'' (हिमालयाची सावली) (literal meaning, "The Shadow of the Himalayas". Contextually it means, under the cover of Himalaya) by [[Vasant Kanetkar]], published in 1972, is loosely based on the life of Karve. The character of Nanasaheb Bhanu is a composite character based on Karve and other Marathi social reformers of the late 19th and early 20th century. The play itself depicts the tension between Bhanu/Karve's public life as a social reformer and his family life due to the social backlash and economic hardships his children and wife had to endure.

''The Story of Dr. Karve'' is a 1958 documentary film directed by Neil Gokhale and [[Ram Gabale]]. It was produced by the [[Government of India]]'s [[Films Division]].<ref>{{cite book|author=Jag Mohan|title=Documentary films and Indian Awakening|url=https://books.google.com/books?id=DfgADgAAQBAJ&pg=PT128|year=1990|publisher=[[Publications Division (India)|Publications Division]]|isbn=978-81-230-2363-2|page=128}}</ref>

The 2001 film ''[[Dhyaas Parva]]'' (ध्यास पर्व) by [[Amol Palekar]], based on the life of Karve's son [[Raghunath Dhondo Karve|Raghunath]], also depicts the Karve family, and their social reformation projects.<ref>{{Cite web|last1=Ahmed|first1=Afsana|date=August 20, 2001|first2=Smrity|last2=Sharma|title=Amol Palekar, Actor & Film-Maker on his latest film Dhyaas-Parva's|url=https://timesofindia.indiatimes.com/city/lucknow/Amol-Palekar-Actor-Film-Maker-on-his-latest-film-Dhyaas-Parvas/articleshow/1438446102.cms|access-date=2021-01-12|website=The Times of India|language=en}}</ref> Taluka Dapoli, a research based initiative, made a documentary on life of Maharshi Dhondo Keshav Karve in 2017.<ref name="talukadapoli.com">{{Cite web|url=http://www.talukadapoli.com/en/personalities/maharshi-dhondo-keshav-karve/|title=Maharshi Dhondo Keshav Karve|website=www.talukadapoli.com|access-date=1 July 2018}}</ref>

==Awards and honours==
*1942 – awarded [[Doctor of Letters]] (D.Litt.) by [[Banaras Hindu University]]
*1951 – awarded D.Litt. by [[Pune University]]
*1954 – awarded D.Litt. by [[SNDT Women's University|S.N.D.T. University]]
*1955 – awarded [[Padma Vibhushan]] by the Government of India<ref name="award">{{cite web|title=Padma Awards Directory (1954–2007) |url=http://www.mha.nic.in/pdfs/PadmaAwards1954-2007.pdf |publisher=Ministry of Home Affairs |access-date=26 November 2010 |url-status=dead |archive-url=https://web.archive.org/web/20090410024701/http://www.mha.nic.in/pdfs/PadmaAwards1954-2007.pdf |archive-date=10 April 2009 }}</ref>
*1957 – awarded LL.D. by [[University of Mumbai]]
*1958 – awarded [[Bharat Ratna]], the highest civilian award of India, by the Government of India<ref name="award" />

In his honour, Karvenagar in Pune was named after him & Queen's Road in Mumbai (Bombay) was renamed to Maharshi Karve Road.

==See also==

*[[Maharshi Karve Stree Shikshan Samstha]]
*The new Brahmans; five Maharashtrian families. Essays on D.K. Karve by his son and [[Irawati Karve]]<ref>{{Cite book |last=Karve |first=Dinakar Dhondo |url=http://archive.org/details/newbrahmansfivem00karv |title=The new Brahmans; five Maharashtrian families |date=1963 |publisher=Berkeley, University of California Press |others=Internet Archive}}</ref>

==References==
{{commons category|Dhondo Keshav Karve}}
{{Reflist}}

{{Bharat Ratna|state=expanded}}
{{Padma Vibhushan Awards}}
{{Authority control}}

{{Indian Independence Movement}}

{{DEFAULTSORT:Karve, Dhondu Keshav}}
[[Category:Recipients of the Bharat Ratna]]
[[Category:Recipients of the Padma Vibhushan in literature & education]]
[[Category:1858 births]]
[[Category:1962 deaths]]
[[Category:Hindu revivalists]]
[[Category:Indian men centenarians]]
[[Category:Women's education in India]]
[[Category:People from Ratnagiri district]]
[[Category:Indian social reformers]]
[[Category:19th-century Indian educators]]
[[Category:20th-century Indian educators]]
[[Category:Activists from Maharashtra]]
[[Category:Educators from Maharashtra]]
[[Category:People from Dapoli]]

Modular lattice

2025-06-08T01:26:48Z

173.79.19.248: /* History */

{{distinguish|unimodular lattice}}
[[File:2d modular lattice.svg|thumb|A modular lattice of [[order dimension]] 2. As with all finite 2-dimensional lattices, its [[Hasse diagram]] is an [[st-planar graph|''st''-planar graph]].]]

In the branch of mathematics called [[order theory]], a '''modular lattice''' is a [[lattice (order)|lattice]] that satisfies the following self-[[duality (order theory)|dual]] condition,
;'''Modular law''':{{Math|''a'' ≤ ''b''}} implies {{Math|1=''a'' ∨ (''x'' ∧ ''b'') = (''a'' ∨ ''x'') ∧ ''b''}}

where {{Math|''x'', ''a'', ''b''}} are arbitrary elements in the lattice,  ≤  is the [[partial order]], and  ∨  and  ∧ (called [[join and meet]] respectively) are the operations of the lattice. This phrasing emphasizes an interpretation in terms of projection onto the sublattice {{Math|[''a'', ''b'']}}, a fact known as the '''diamond isomorphism theorem'''.<ref>{{Cite web|url=https://math.stackexchange.com/q/443947 |title=Why are modular lattices important?|website=Mathematics Stack Exchange|access-date=2018-09-17}}</ref> An alternative but equivalent condition stated as an equation (see below) emphasizes that modular lattices form a [[variety (universal algebra)|variety]] in the sense of [[universal algebra]].

Modular lattices arise naturally in [[algebra]] and in many other areas of mathematics. In these scenarios, modularity is an abstraction of the [[Second isomorphism theorem|2nd Isomorphism Theorem]]. For example, the subspaces of a [[vector space]] (and more generally the submodules of a [[module over a ring]]) form a modular lattice.

In a not necessarily modular lattice, there may still be elements {{Mvar|b}} for which the modular law holds in connection with arbitrary elements {{Mvar|x}} and {{Mvar|a}} (for {{Math|''a'' ≤ ''b''}}). Such an element is called a '''right modular element'''. Even more generally, the modular law may hold for any {{Mvar|a}} and a fixed pair {{Math|(''x'', ''b'')}}. Such a pair is called a '''modular pair''', and there are various generalizations of modularity related to this notion and to [[semimodular lattice|semimodularity]].

Modular lattices are sometimes called '''Dedekind lattices''' after [[Richard Dedekind]], who discovered the modular identity in [[#History|several motivating examples]].

==Introduction==
The modular law can be seen as a restricted [[associativity|associative law]] that connects the two lattice operations similarly to the way in which the associative law λ(μ''x'') = (λμ)''x'' for vector spaces connects multiplication in the field and [[scalar multiplication]].

The restriction {{Math|''a'' ≤ ''b''}} is clearly necessary, since it follows from {{Math|1=''a'' ∨ (''x'' ∧ ''b'') = (''a'' ∨ ''x'') ∧ ''b''}}. In other words, no lattice with more than one element satisfies the unrestricted consequent of the modular law.

It is easy to see<ref>The following is true for any lattice: {{Math|''a'' ∨ (''x'' ∧ ''b'') ≤ (''a'' ∨ ''x'') ∧ (''a'' ∨ ''b'')}}. Also, whenever {{Math|''a'' ≤ ''b''}}, then {{Math|''a'' ∨ ''b'' {{=}} ''b''}}.</ref> that {{Math|''a'' ≤ ''b''}} implies {{Math|1=''a'' ∨ (''x'' ∧ ''b'') ≤ (''a'' ∨ ''x'') ∧ ''b''}} in every lattice. Therefore, the modular law can also be stated as

;'''Modular law (variant)''': {{Math|''a'' ≤ ''b''}} implies {{Math|(''a'' ∨ ''x'') ∧ ''b'' ≤ ''a'' ∨ (''x'' ∧ ''b'')}}.

The modular law can be expressed as an equation that is required to hold unconditionally. Since {{Math|''a'' ≤ ''b''}} implies {{Math|''a'' {{=}} ''a'' ∧ ''b''}} and since {{Math|''a'' ∧ ''b'' ≤ ''b''}}, replace {{Math|''a''}} with {{Math|''a'' ∧ ''b''}} in the defining equation of the modular law to obtain:
;'''Modular identity''': {{Math|1=(''a'' ∧ ''b'') ∨ (''x'' ∧ ''b'') = ((''a'' ∧ ''b'') ∨ ''x'') ∧ ''b''}}.
This shows that, using terminology from [[universal algebra]], the modular lattices form a subvariety of the [[variety (universal algebra)|variety]] of lattices. Therefore, all homomorphic images, [[sublattice]]s and direct products of modular lattices are again modular.

==Examples==
[[File:Smallest nonmodular lattice 2.svg|thumb|''N''5, the smallest non-modular lattice: {{nowrap|''x''∨(''a''∧''b'')}} = ''x''∨0 = ''x'' ≠ ''b'' = 1∧''b'' ={{nowrap|(''x''∨''a'')∧''b''}}.]]
The lattice of submodules of a [[module over a ring]] is modular. As a special case, the lattice of subgroups of an [[abelian group]] is modular.

The lattice of [[normal subgroup]]s of a [[group (mathematics)|group]] is modular. But in general the [[lattice of subgroups|lattice of all subgroups]] of a group is not modular. For an example, the lattice of subgroups of the [[dihedral group]] of order 8 is not modular.

The smallest non-modular lattice is the "pentagon" lattice ''N''5 consisting of five elements 0, 1, ''x'', ''a'', ''b'' such that 0 < ''x'' < ''b'' < 1, 0 < ''a'' < 1, and ''a'' is not comparable to ''x'' or to ''b''. For this lattice,
:''x'' ∨ (''a'' ∧ ''b'') = ''x'' ∨ 0 = ''x'' < ''b'' = 1 ∧ ''b'' = (''x'' ∨ ''a'') ∧ ''b''
holds, contradicting the modular law. Every non-modular lattice contains a copy of ''N''5 as a sublattice.<ref name=":0">{{Cite book |chapter-url=https://link.springer.com/book/10.1007%2Fb139095 |title=Lattices and Ordered Algebraic Structures |last=Blyth |first=T. S. |publisher=Springer |year=2005 |isbn=978-1-85233-905-0 |series=Universitext |location=London |at=Theorem 4.4 |chapter=Modular lattices |doi=10.1007/1-84628-127-X_4}}</ref>

== Properties ==
Every [[distributive lattice]] is modular.<ref>{{Cite book |chapter-url=https://link.springer.com/book/10.1007%2Fb139095 |title=Lattices and Ordered Algebraic Structures |last=Blyth |first=T. S. |publisher=Springer |year=2005 |isbn=978-1-85233-905-0 |series=Universitext |location=London |page=65 |chapter=Modular lattices |doi=10.1007/1-84628-127-X_4}}</ref><ref>In a distributive lattice, the following holds: <math>(a\wedge b)\vee (x\wedge b) = ((a\wedge b)\vee x)\wedge((a\wedge b)\vee b)</math>. Moreover, the absorption law, <math>(a\wedge b)\vee b = b</math>, is true for any lattice. Substituting this for the second conjunct of the right-hand side of the former equation yields the Modular Identity.</ref>

{{harvtxt|Dilworth|1954}} proved that, in every finite modular lattice, the number of join-irreducible elements equals the number of meet-irreducible elements. More generally, for every {{mvar|k}}, the number of elements of the lattice that cover exactly {{mvar|k}} other elements equals the number that are covered by exactly {{mvar|k}} other elements.<ref>{{citation
| last = Dilworth | first = R. P. | authorlink = Robert P. Dilworth
| doi = 10.2307/1969639
| journal = [[Annals of Mathematics]]
| mr = 0063348
| pages = 359–364
| series = Second Series
| title = Proof of a conjecture on finite modular lattices
| volume = 60
| issue = 2 | year = 1954| jstor = 1969639 }}. Reprinted in {{citation
| editor1-last = Bogart | editor1-first = Kenneth P.
| editor2-last = Freese | editor2-first = Ralph
| editor3-last = Kung | editor3-first = Joseph P. S.
| doi = 10.1007/978-1-4899-3558-8_21
| location = Boston
| pages = 219–224
| publisher = Birkhäuser
| series = Contemporary Mathematicians
| title = The Dilworth Theorems: Selected Papers of Robert P. Dilworth
| year = 1990| chapter = Proof of a Conjecture on Finite Modular Lattices
| isbn = 978-1-4899-3560-1
}}</ref>

A useful property to show that a lattice is not modular is as follows:

: A lattice {{Mvar|G}} is modular if and only if, for any {{Math|''a'', ''b'', ''c'' ∈ ''G''}},
::<math>\Big((c\leq a)\text{ and }(a\wedge b=c\wedge b)\text{ and }(a\vee b=c\vee b)\Big)\Rightarrow(a=c)</math>

Sketch of proof: Let G be modular, and let the premise of the implication hold. Then using absorption and modular identity:

: ''c'' = (''c''∧''b'') ∨ ''c'' = (''a''∧''b'') ∨ ''c'' = ''a'' ∧ (''b''∨''c'') = ''a'' ∧ (''b''∨''a'') = ''a''

For the other direction, let the implication of the theorem hold in G. Let ''a'',''b'',''c'' be any elements in G, such that ''c'' ≤ ''a''. Let ''x'' = (''a''∧''b'') ∨ ''c'', ''y'' = ''a'' ∧ (''b''∨''c''). From the modular inequality immediately follows that ''x'' ≤ ''y''. If we show that ''x''∧''b'' = ''y''∧''b'', ''x''∨''b'' = ''y''∨''b'', then using the assumption ''x'' = ''y'' must hold. The rest of the proof is routine manipulation with infima, suprema and inequalities.{{Citation needed|date=September 2018}}

== Diamond isomorphism theorem ==

For any two elements ''a'', ''b'' of a modular lattice, one can consider the intervals [''a'' ∧ ''b'', ''b''] and [''a'', ''a'' ∨ ''b'']. They are connected by order-preserving maps
::φ: [''a'' ∧ ''b'', ''b''] → [''a'', ''a'' ∨ ''b''] and
::ψ: [''a'', ''a'' ∨ ''b''] → [''a'' ∧ ''b'', ''b'']
that are defined by φ(''x'') = ''x'' ∨ ''a'' and ψ(''y'') = ''y'' ∧ ''b''.

<gallery heights="180px" widths="225px">
Image:Modular_pair.svg|In a modular lattice, the maps φ and ψ indicated by the arrows are mutually inverse isomorphisms.
Image:Not a modular pair.svg|Failure of the diamond isomorphism theorem in a non-modular lattice.
</gallery>

The composition ψφ is an order-preserving map from the interval [''a'' ∧ ''b'', ''b''] to itself which also satisfies the inequality ψ(φ(''x'')) = (''x'' ∨ ''a'') ∧ ''b'' ≥ ''x''. The example shows that this inequality can be strict in general. In a modular lattice, however, equality holds. Since the dual of a modular lattice is again modular, φψ is also the identity on [''a'', ''a'' ∨ ''b''], and therefore the two maps φ and ψ are isomorphisms between these two intervals. This result is sometimes called the '''diamond isomorphism theorem''' for modular lattices. A lattice is modular if and only if the diamond isomorphism theorem holds for every pair of elements.

The diamond isomorphism theorem for modular lattices is analogous to the second [[isomorphism theorem]] in algebra, and it is a generalization of the [[lattice theorem]].

==Modular pairs and related notions==
[[Image:Centred hexagon lattice D2.svg|thumb|right|The centred hexagon lattice ''S''7, also known as ''D''2, is M-symmetric but not modular.]]
In any lattice, a '''modular pair''' is a pair (''a, b'') of elements such that for all ''x'' satisfying ''a'' ∧ ''b'' ≤ ''x'' ≤ ''b'', we have (''x'' ∨ ''a'') ∧ ''b'' = ''x'', i.e. if one half of the diamond isomorphism theorem holds for the pair.<ref>The [[French language|French]] term for modular pair is ''couple modulaire''. A pair (''a, b'') is called a ''paire modulaire'' in French if both (''a, b'') and (''b, a'') are modular pairs.</ref> An element ''b'' of a lattice is called a '''right modular element''' if (''a, b'') is a modular pair for all elements ''a'', and an element ''a'' is called a '''left modular element''' if (''a, b'') is a modular pair for all elements ''b''.<ref>Modular element has been varying defined by different authors to mean right modular ({{harvtxt|Stern|1999|p=74}}), left modular ({{harvtxt|Orlik|Terao|1992|loc=Definition 2.25}}), both left and right modular (or dual right modular) ({{harvtxt|Sagan|1999}}, {{harvtxt|Schmidt|1994|p=43}}), or satisfying a modular rank condition ({{harvtxt|Stanley|2007|loc=Definition 4.12}}). These notions are equivalent in a semimodular lattice, but not in general.</ref>

A lattice with the property that if (''a, b'') is a modular pair, then (''b, a'') is also a modular pair is called an '''M-symmetric lattice'''.<ref>Some authors, e.g. Fofanova (2001), refer to such lattices as ''semimodular lattices''. Since every M-symmetric lattice is [[semimodular lattice|semimodular]] and the converse holds for lattices of finite length, this can only lead to confusion for infinite lattices.</ref> Thus, in an M-symmetric lattice, every right modular element is also left modular, and vice-versa. Since a lattice is modular if and only if all pairs of elements are modular, clearly every modular lattice is M-symmetric. In the lattice ''N''5 described above, the pair (''b, a'') is modular, but the pair (''a, b'') is not. Therefore, ''N''5 is not M-symmetric. The centred hexagon lattice ''S''7 is M-symmetric but not modular. Since ''N''5 is a sublattice of ''S''7, it follows that the M-symmetric lattices do not form a subvariety of the variety of lattices.

M-symmetry is not a self-dual notion. A '''dual modular pair''' is a pair which is modular in the [[duality (order theory)|dual]] lattice, and a lattice is called dually M-symmetric or '''M*-symmetric''' if its dual is M-symmetric. It can be shown that a finite lattice is modular if and only if it is M-symmetric and M*-symmetric. The same equivalence holds for infinite lattices which satisfy the [[ascending chain condition]] (or the descending chain condition).

Several less important notions are also closely related. A lattice is '''cross-symmetric''' if for every modular pair (''a, b'') the pair (''b, a'') is dually modular. Cross-symmetry implies M-symmetry but not M*-symmetry. Therefore, cross-symmetry is not equivalent to dual cross-symmetry. A lattice with a least element 0 is '''⊥-symmetric''' if for every modular pair (''a, b'') satisfying ''a'' ∧ ''b'' = 0 the pair (''b, a'') is also modular.

==History==

[[File:Free modular lattice with 3 generators (x,y,z).gif|thumb|Free modular lattice generated by three elements {''x'', ''y'', ''z''}]]
The definition of modularity is due to [[Richard Dedekind]], who published most of the relevant papers after his retirement.
In a paper published in 1894{{citation needed|date=August 2015}} he studied lattices, which he called ''dual groups'' ({{langx|de|Dualgruppen}}) as part of his "algebra of [[Module (mathematics)|modules]]" and observed that ideals satisfy what we now call the modular law. He also observed that for lattices in general, the modular law is equivalent to its dual.

In another paper in 1897, Dedekind studied the lattice of divisors with gcd and lcm as operations, so that the lattice order is given by divisibility.<ref name="Dedekind.1897">{{Citation | contribution-url=http://digisrv-1.biblio.etc.tu-bs.de:8080/docportal/servlets/MCRFileNodeServlet/DocPortal_derivate_00006737/V.C.1596.pdf | last1=Dedekind | first1=Richard | author1-link=Richard Dedekind | contribution=Über Zerlegungen von Zahlen durch ihre grössten gemeinsamen Theiler | title=Festschrift der Herzogl. Technischen Hochschule Carolo-Wilhelmina bei Gelegenheit der 69. Versammlung Deutscher Naturforscher und Ärzte in Braunschweig | publisher=Friedrich Vieweg und Sohn | year=1897 }}</ref>
In a digression he introduced and studied lattices formally in a general context.<ref name="Dedekind.1897"/>{{rp|10–18}} He observed that the lattice of submodules of a module satisfies the modular identity. He called such lattices ''dual groups of module type'' ({{lang|de|Dualgruppen vom Modultypus}}). He also proved that the modular identity and its dual are equivalent.<ref name="Dedekind.1897"/>{{rp|13}}

In the same paper, Dedekind also investigated the following stronger form<ref name="Dedekind.1897"/>{{rp|14}} of the modular identity, which is also self-dual:<ref name="Dedekind.1897"/>{{rp|9}}

: (''x'' ∧ ''b'') ∨ (''a'' ∧ ''b'') = [''x'' ∨ ''a''] ∧ ''b''.

He called lattices that satisfy this identity ''dual groups of ideal type'' ({{lang|de|Dualgruppen vom Idealtypus}}).<ref name="Dedekind.1897"/>{{rp|13}} In modern literature, they are more commonly referred to as [[distributive lattice]]s. He gave examples of a lattice that is not modular and of a modular lattice that is not of ideal type.<ref name="Dedekind.1897"/>{{rp|14}}

A paper published by Dedekind in 1900 had lattices as its central topic: He described the free modular lattice generated by three elements, a lattice with 28 elements (see picture).<ref>{{Citation | doi=10.1007/BF01448979 | last1=Dedekind | first1=Richard | title=Über die von drei Moduln erzeugte Dualgruppe | year=1900 | journal=Mathematische Annalen | volume=53 | issue=3 | pages=371–403 | s2cid=122529830 | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002257947 }}</ref>

==See also==
*[[Modular graph]], a class of graphs that includes the Hasse diagrams of modular lattices
*[[Young–Fibonacci lattice]], an infinite modular lattice defined on strings of the digits 1 and 2
*[[Orthomodular lattice]]
*[[Supersolvable lattice]]
*[[Iwasawa group]]

== Notes ==
<references />

==References==
* {{Citation | last1=Corry | first1=Leo | title=Modern algebra and the rise of mathematical structures | isbn=978-3-7643-7002-2 | edition=2nd | pages=121–129 | date=2003-11-27 }}
* {{springer| title=Semi-modular lattice| id=s/s084240| last=Fofanova| first=T. S.}}
* {{Citation | last1=Maeda | first1=Shûichirô | title=On the symmetry of the modular relation in atomic lattices | url=http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.hmj/1206139232&page=record | year=1965 | journal=Journal of Science of the Hiroshima University | volume=29 | pages=165–170}}
* {{Citation | last1= Orlik | first1=Peter | author1-link=Peter Orlik | last2=Terao | first2=Hiroaki | author2-link=Hiroaki Terao | title=Arrangements of Hyperplanes | year=1992 | publisher=Springer-Verlag | doi=10.1007/978-3-662-02772-1 | volume=300 | series= Grundlehren der mathematischen Wissenschaften| isbn=978-3-642-08137-8 }}
* {{Citation | last1=Rota | first1=Gian-Carlo | author1-link=Gian-Carlo Rota | title=The many lives of lattice theory | year=1997 | journal=[[Notices of the American Mathematical Society]] | issn=0002-9920 | volume=44 | issue=11 | pages=1440–1445 | url=https://www.ams.org/notices/199711/comm-rota.pdf}}
* {{springer| title=Modular lattice| id=m/m064460| last=Skornyakov| first=L. A.}}
*{{citation | last=Sagan|first=Bruce | author-link=Bruce Sagan | title=Why the characteristic polynomial factors | journal=[[Bulletin of the American Mathematical Society]] | year=1999 | volume=36 | issue=2 | pages=113–133 | doi=10.1090/S0273-0979-99-00775-2 | arxiv=math/9812136}}
*{{citation | last=Schmidt|first=Roland | title=Subgroup lattices of groups | publisher=Walter de Gruyter & Co. | year=1994 | volume=14 | series=de Gruyter Expositions in Mathematics | isbn=3-11-011213-2 | doi=10.1515/9783110868647}}
*{{Citation | last=Stanley | first=Richard P. | author-link=Richard P. Stanley | contribution=An Introduction to Hyperplane Arrangements | title=Geometric combinatorics | publisher=American Mathematical Society | year=2007 | volume=13 | series=IAS/Park City Mathematics Series | pages=389–496 | isbn= 978-0-8218-3736-8}}
* {{Citation | last1=Stern | first1=Manfred | title=Semimodular lattices | publisher=[[Cambridge University Press]] | isbn=978-0-521-46105-4 | year=1999 | url-access=registration | url=https://archive.org/details/semimodularlatti0000ster }}
* {{Citation |last=Weisstein |first=Eric W. |title=Modular Lattice |work=MathWorld—A Wolfram Web Resource |url=https://mathworld.wolfram.com/ModularLattice.html |access-date=November 26, 2024}}

==External links==
* {{planetmath reference|urlname=ModularLattice|title=Modular lattice}}
* {{OEIS el|A006981|Number of unlabeled modular lattices with ''n'' elements}}
* [https://www.source-code.biz/lattice Free Modular Lattice Generator] An open-source browser-based web application that can generate and visualize some free modular lattices.

[[Category:Lattice theory]]

Birnin Magaji/Kiyaw

2025-06-06T10:29:52Z

173.79.19.248: rv

{{for|the [[Romania]]n village of Pelişor, called Magaré in [[Hungarian language|Hungarian]]|Bârghiş}}
{{Use Nigerian English|date=January 2023}}
{{Infobox settlement
|official_name =Birnin Magaji/Kiyaw
|other_name =
|native_name =
|nickname =
|settlement_type = [[Local Government Areas of Nigeria|LGA]] and town
|motto =
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|pushpin_map =Nigeria
|pushpin_label_position =
|pushpin_mapsize=250
|pushpin_map_caption =Location in Nigeria
|subdivision_type = Country
|subdivision_name ={{flag|Nigeria}}
|subdivision_type1 = [[States of Nigeria|State]]
|subdivision_name1 =[[Zamfara State]]
|subdivision_type2 =
|subdivision_name2 =
|subdivision_type3 =
|subdivision_name3 =
|government_footnotes =
|government_type =
|leader_title =Local Government Chairman
|leader_name =Hon. Isiyaka Ibrahim
|established_title =
|established_date =


|area_magnitude =
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|area_footnotes =
|area_total_km2 = 1188
|area_land_km2 = 


|population_as_of = 2006 census
|population_footnotes =
|population_note =
|population_total = 178,619
|population_density_km2 =
|population_density_urban_sq_mi =
|population_blank1_title = Ethnicities
|population_blank1 =
|population_blank2_title = Religions
|population_blank2 =
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|population_density_blank1_sq_mi =

||timezone =[[West Africa Time|WAT]]
|utc_offset = +1
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|coordinates = {{coord|12|28|N|6|54|E|type:adm2nd_region:NG|display=inline,title}}
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|elevation_footnotes =
|elevation_m =
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|postal_code_type = 3-digit postal code prefix
|postal_code = 882
|area_code =
|iso_code = NG.ZA.BM
|website =
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}}
'''Birnin Magaji/Kiyaw''' is a [[Local Government Areas of Nigeria|Local Government Area]] in [[Zamfara State]], [[Nigeria]]. Its headquarters are in the town of Birnin Magaji (or Magare) in the north of the LGA at{{Coord|12|33|00|N|6|49|00|E}}. The LGA is also named from the town of Kiyaw (or Kiawa) to the south.

It has an area of 1,188 km{{sup|2}} and a population of 178,619 at the 2006 census.

The [[postal code]] of the area is 882.<ref>{{cite web|title=Post Offices- with map of LGA |publisher=NIPOST |url=http://www.nipost.gov.ng/PostCode.aspx |accessdate=2009-10-20 |url-status=dead |archiveurl=https://web.archive.org/web/20091007011423/http://www.nipost.gov.ng/PostCode.aspx |archivedate=2009-10-07 }}</ref>

== Climate ==
In Birnin Magaji, the dry season is windy and partly [[Cloud|cloudy]], and it is hot all year round. The wet season is oppressive and generally cloudy. The average annual [[temperature]] ranges from 59 to 101 [[Fahrenheit|degrees Fahrenheit]], and it is infrequently below 54 or beyond 106 [[Degree (temperature)|degrees]].<ref>{{Cite web |date=2023-09-29 |title=Weather Birnin Magaji |url=https://www.meteoblue.com/en/weather/week/birnin-magaji_nigeria_10194293 |access-date=2023-09-29 |website=meteoblue |language=en}}</ref><ref name=":0">{{Cite web |title=Birnin Magaji Climate, Weather By Month, Average Temperature (Nigeria) - Weather Spark |url=https://weatherspark.com/y/53016/Average-Weather-in-Birnin-Magaji-Nigeria-Year-Round#:~:text=The%20hottest%20month%20of%20the,temperature%20below%2088%C2%B0F. |access-date=2023-09-29 |website=weatherspark.com |language=en}}</ref>

The hot season lasts for 2.1 months, from March 15 to May 20, with an average daily high temperature of over 98°F. The hottest month of the year in Birnin Magaji is April, with an average high temperature of 101°F and low temperature of 77°F.<ref name=":0" />

A daily maximum temperature below 88°F is typical during the chilly season, which runs from December 10 to January 28. With an average low of 60°F and a high of 87°F, January is the coldest month of the year in Birnin Magaji.<ref>{{Cite web |title=Birnin Magaji June Weather, Average Temperature (Nigeria) - Weather Spark |url=https://weatherspark.com/m/53016/6/Average-Weather-in-June-in-Birnin-Magaji-Nigeria |access-date=2023-09-29 |website=weatherspark.com |language=en}}</ref><ref name=":0" />

Birnin Magaji Local Government Area experiences an average temperature of [[Celsius|34 degrees Celsius]] and spans an area of 1,188 square [[Kilometre|kilometers]]. The [[humidity]] is recorded at 19%, and the average wind speed is 12 km/h. This region has two primary seasons rainy and dry.<ref name=":1">{{Cite web |title=Birnin Magaji Local Government Area |url=https://www.manpower.com.ng/places/lga/800/birnin-magaji |access-date=2023-11-14 |website=www.manpower.com.ng}}</ref>

== Economy ==
The economic landscape of Birnin Magaji Local Government Area prominently involves [[agriculture]], encompassing the cultivation of various [[Cereal|cereals]] and [[Legume|legumes]]. Additionally, the residents actively participate in animal rearing, [[hunting]], and trade as significant economic pursuits.<ref name=":1" />

== Locality ==
Towns under Birnin Magaji Local Government Area:<ref>{{Cite web |title=Birnin Magaji-Kiyaw Local Government Area |url=https://www.finelib.com/listing/Birnin-MagajiKiyaw-Local-Government-Area/62586/ |access-date=2023-11-14 |website=www.finelib.com}}</ref>

# Birnin-Magaji Ward

# Darifami Ward

Sabon-Birnin

Gora

Gusami Gari

Gusami Hayi

Kiyawa

Modomawa East

Modomawa West

Nasarawa-Godol East

Nassarawa-Godol West

==References==
{{Reflist}}

{{LGAs and communities of Zamfara State}}

{{DEFAULTSORT:Birnin Magaji Kiyaw}}
[[Category:Local Government Areas in Zamfara State]]

{{Zamfara-geo-stub}}

List of Virgin Records artists

2025-06-04T14:21:49Z

173.79.19.248: Undid revision 1293830915 by Atisnakebite (talk) rv uncited

{{Short description|none}}
{{BLP sources|list|date=December 2012}}
<ref>{{Cite web|last=Age|first=Ez Mil Biography: Viral Pinoy Rapper’s|last2=Family|last3=Facts|last4=Details|first4=Other|date=2021-02-02|title=Ez Mil Biography: Viral Pinoy Rapper's Age, Family, Facts, Other Details|url=https://philnews.ph/2021/02/02/ez-mil-biography-viral-pinoy-rapper-age-family-facts-details/|access-date=2021-05-11|website=Philippine News|language=en-US}}</ref> The following artists ([[musicians]] or [[band (music)|bands]]) have had releases with [[Virgin Records]].
{{Inc-musong|date=October 2021}}
{{dynamic list|date=December 2012}}
{{compact ToC|side=yes|top=yes|num=yes}}

==#==
* [[Thirty Seconds to Mars]] ([[Immortal Records|Immortal]]/Virgin)
* [[The 69 Eyes]]
* [[311 (band)|311]]
* [[52nd Street (band)|52nd Street]]

==A==
* [[Aaliyah]] (Blackground/Virgin)
* [[Paula Abdul]]
* [[Albin Lee Meldau]]
* [[Adelitas Way]]
* Agung Gede
* [[A Perfect Circle]]
* [[After 7]]
* [[Aftershock (band)|Aftershock]]
* [[Air (French band)|Air]]
* [[Alice in Chains]]
* [[Alien (band)|Alien]]
* [[Brooke Allison]]
* [[Marc Almond]]
* [[The Almost]] (Tooth and Nail)
* [[Altan (band)|Altan]]
* [[Althea and Donna]]
* [[Amaral (band)|Amaral]]
* [[Amen (American band)|Amen]]
* [[American Music Club]]
* [[AM Taxi]]
* [[Amir Obè]] (After Platinum)
* [[Amorphis]]
* [[Carleen Anderson]]
* [[And One]]
* [[Animal Logic (band)|Animal Logic]]
* [[Antique (duo)|Antique]]
* [[Tasmin Archer]]
* [[The Ark (Swedish band)|The Ark]]
* [[Atomic Kitten]]
* [[The Avant Gardeners|The Avant Gardener]]
* [[Kevin Ayers]]
* [[Avicii]]
* [[:id:Ayushita|Ayushita Nugraha]] (Virgin Jogjakarta)

==B==
* [[Joan Baez]]
* [[Balaam and the Angel]]
* [[Bandolero (band)|Bandolero]]
* [[Tony Banks (musician)|Tony Banks]] (outside US/Canada)
* [[Toni Basil]] (Radialchoice/Virgin) (outside US/Canada)
* [[Bastille (band)|Bastille]]
* [[:id:Bukan Bintang Biasa|BBB]] (Virgin/Mercury Jogjakarta)
* [[Victoria Beckham]]
* [[David Bedford]]
* [[Beenie Man]]
* [[The Big Dish (band)|The Big Dish]]
* [[Benjamin Biolay]]
* [[Thom Bishop]]
* [[Bizzy Bone]] (After Platinum)
* [[Black Rebel Motorcycle Club]]
* [[Peter Blegvad]]
* [[Blind Guardian]]
* [[Blossoms (band)|Blossoms]]
* [[Blue (boy band)|Blue]]
* [[Blue Man Group]]
* [[The Blue Nile (band)|The Blue Nile]] (Linn/Virgin)
* [[Blur (band)|Blur]] (US only)
* [[Body Count (band)|Body Count]]
* [[Boxer (band)|Boxer]]
** [[David Bowie]] (US only)
* [[Boy George]] (More Protein/Virgin)
* [[Brake Brothers Band]]
* [[Breathe (British band)|Breathe]]
* [[Brother Cane]]
* [[Mel B|Melanie Brown]]
* [[Bruce Boniface]]
* [[Bubba Sparxxx]]
* [[Emma Bunton]]
* [[Burrell (1988)]]
* [[Brake Brothers Band]]

==C==
* [[Cabaret Voltaire (band)|Cabaret Voltaire]]
* [[Camper Van Beethoven]]
* [[Can (band)|Can]]
* [[Captain Beefheart]]
* [[Melanie C]]
* [[Chris Cagle]] (Virgin Nashville)
* [[Mariah Carey]] (Former)
* [[Belinda Carlisle]] (outside US & Canada until 1991, then worldwide in 1993)
* [[Cellophane (band)|Cellophane]]
* [[The Chemical Brothers]]
* [[Che'Nelle]]
* [[Neneh Cherry]]
* [[China Crisis]]
* [[Chvrches]]
* [[Anne Clark (poet)|Anne Clark]]
* [[Gilby Clarke]]
* [[College Boyz]]
* [[Phil Collins]] (UK and Ireland)
* [[Comateens]]
* [[The Constellations]]
* [[Cory Gunz]] (After Platinum)
* [[Nikka Costa]]
* [[Sagarika Mukherjee Da Costa]]
* [[Kevin Coyne]]
* [[Cracker (band)|Cracker]]
* [[The Cult]]
* [[Culture Club]]
* [[Curve (band)|Curve]]
* [[Ivor Cutler]]
* [[Cutting Crew]]
* [[Holger Czukay]]
* [[Mark Curry (rock musician)|Mark Curry]]

==D==
* [[Dave Gahan]] (US only)
* Deaf Pedestrians
* [[Aurra|Déjà]]
* [[Deutsch Amerikanische Freundschaft]]
* [[Daft Punk]]
* [[Étienne Daho]]
* [[Roger Daltrey]] (10/Virgin) (outside US/Canada)
* [[Danny Wilson (band)|Danny Wilson]]
* [[Clay Davidson]] (Virgin Nashville)
* [[D'Angelo]]
* [[Deadmau5]]
* [[Depeche Mode]] (Mute/Virgin) (US/France)
* [[Devo]] (UK/Europe only)
* [[Howard Devoto]]
* [[Divinyls]]
* [[Thomas Dolby]] (outside US/Canada)
* [[Does It Offend You, Yeah?]]
* [[Dreadzone]]
* [[dc Talk]] (ForeFront/Virgin)
* [[Hilary Duff]] (''[[4Ever Hilary]]'' only in Italy) (via [[Hollywood Records]])
* [[Stephen Tin Tin Duffy]]
* [[The Dukes of Stratosphear]]
* [[Dwele]]

==E==
* [[Eden xo]] (former)
* [[The Edge]]
* [[Efua Baker|Efua]]
* [[Electronic (band)|Electronic]] (outside the US and Canada)
* [[Emily the Strange|Emily and the Strangers]]
* [[Endgames (band)|Endgames]]
* [[Enigma (German band)|Enigma]]
* [[Eurythmics]]* (outside US/Australasia)
* [[Evanescence]]
* [[Everything but the Girl]] (except US & Canada)
* [[Every Little Thing (band)|Every Little Thing]] (Japan)
* [[The Exies]]
* [[Ella Eyre]]
* [[Emeli Sandé]]<ref>{{cite web|url=https://twitter.com/emelisande/status/10530369765|title=Emeli Sande on Twitter|publisher=|accessdate=1 July 2016}}</ref> (outside US)
* [[Ez Mil]]

==F==
* [[Ricky Fanté]]
* [[Perry Farrell]]
* [[Fat Joe]]
* [[Faust (band)|Faust]]
* [[Bryan Ferry]]
* [[A Fine Frenzy]]
* [[Fingerprintz]]
* [[Fivespeed]]
* [[Fishmans]]
* [[Florence + The Machine]]
* [[Tim Finn]]
* [[The Flying Lizards]]
* [[The Flying Pickets]]
* [[Julia Fordham]]
* [[Fountains of Wayne]]
* [[John Foxx]]
* [[Fra Lippo Lippi (band)|Fra Lippo Lippi]]
* [[Peter Frampton]]
* [[Frazier Chorus]]

==G==

* [[Peter Gabriel]] (outside US & Canada)
* [[Gang Starr]]
* [[General Public]]
* [[Genesis (band)|Genesis]] (outside US & Canada)
* [[Max A. George]]
* [[The Geraldine Fibbers]]
* [[Gillan (band)|Gillan]]
* [[Ian Gillan]]
* [[Girls Can't Help It]] (Outside USA) (USA - Sire/Virgin)
* [[The Golden Palominos]] (Germany only)
* [[Goldfrapp]]
* [[Goldrush (band)|Goldrush]]
* [[Gong (band)|Gong]]
* [[The Good, the Bad & the Queen]]
* [[Gorillaz]] (US only)
* [[Gorki (band)|Gorki]]
* [[Gravity Kills]] (Europe & Japan only)
* [[Loren Gray]]
* Great and Lady Soul
* [[Green River Ordinance (band)|Green River Ordinance]]
* [[Grizfolk]]
* [[David Guetta]]
* [[Guru (rapper)|Guru]]

==H==
* [[Steve Hackett]] (outside US)
* [[Geri Halliwell]]
* [[Françoise Hardy]]
* [[Ben Harper]]
* [[Lalah Hathaway]]
* [[Hawkwind]] ([[Charisma Records|Charisma]]; UK only)
* [[Murray Head]]
* [[Heaven 17]]
* [[Håkan Hellström]]
* [[Henry Cow]]
* [[Lauran Hibberd]]
* [[Steve Hillage]]
* [[Holly and the Italians]]
* [[Robert Holmes (musician)|Robert Holmes]]
* [[The Human League]]
* [[HRVY]]

==I==
* [[Iggy Pop]]
* [[James Iha]]
* [[Ima Robot]]
* [[IMx|Immature]]
* [[Inner City (band)|Inner City]]
* [[Interview (band)|Interview]]
* [[Mark Isham]]
* [[Ivan Iusco]]
* [[It Bites]]

==J==
* [[Jah Wobble]]
* [[Janet Jackson]]
* [[The Japanese Popstars]]
* [[Jay Chou]]
* [[Joe Jackson (musician)|Joe Jackson]]
* [[Japan (band)|Japan]]
* [[Jamie Scott]]
* [[Jamie T]]
* [[JBO (band)|JBO]]
* [[Jesus Loves You (band)|Jesus Loves You]]
* [[Jin (rapper)|Jin]]
* [[Johnny Hates Jazz]]
* [[Juliet Richardson|Juliet]]
* [[Jelena Karleuša]]

==K==
* [[Kavana (singer)|Kavana]]
* [[Kavinsky]]
* [[Kelis]]
* [[Jerry Kilgore (singer)|Jerry Kilgore]] (Virgin Nashville)
* [[Killing Joke]]
* [[King Crimson]]
* [[King Swamp]]
* [[Frankie Knuckles]]
* [[Konirata]]
* [[The Kooks]]
* [[Korn]]
* [[k-os]]
* [[Lenny Kravitz]]

==L==
* [[Shona Laing]]
* [[LANY]] (Sunset Garden)
* [[The Last Goodnight]]
* [[Latin Alliance]]
* [[Lauv]]
* [[Ava Leigh]]
* [[Julian Lennon]] (except US & Canada)
* [[LCD Soundsystem]]
* [[Les Rita Mitsouko]]
* [[Letoya Luckett|Le Toya]]
* [[Lewis Capaldi]]
* [[Lil' Eazy-E]]
* [[Linton Kwesi Johnson]]
* [[Courtney Love]]
* [[Loudhouse]]
* [[Cheryl Lynn]]
* [[Loren Gray]]
* [[Loose Ends (band)|Loose Ends]]
* [[Luniz]] (U.S. only)(Virgin/C-Note/[[Noo Trybe Records|Noo Trybe]])

==M==
* [[Kirsty MacColl]]
* [[Rita MacNeil]] (Virgin Canada)
* [[Madness (band)|Madness]] (outside the US & Canada)
* [[Madrugada (band)|Madrugada]]
* [[Magazine (band)|Magazine]]
* [[Cheb Mami]]
* [[Manfred Mann's Earth Band]]
* [[Mano Negra (band)|Mano Negra]]
* [[Manowar]]
* [[Lene Marlin]]
* [[Laura Marling]]
* [[Zeeteah Massiah]]
* [[Massive Attack]]
* [[Matia Bazar]]
* [[MC Skat Kat]]
* [[Malcolm McLaren]]
* [[Martha and the Muffins]] (Dinidisc/Virgin Canada)
* [[Holle Thee Maxwell]]
* [[Meat Loaf]] (outside US & Canada 1992–99; US & Canada 2007–10)
* [[Roy D. Mercer]] (Virgin Nashville)
* [[Miami Horror]]
* [[George Michael]] (outside US & Canada until 1999)
* [[Mick Karn]]
* [[Mickey 3D]]
* [[Mighty Diamonds]]
* [[Mike + The Mechanics]] (outside the US and Canada)
* [[Microdisney]]
* [[The Monochrome Set]]
* [[Gary Moore]]
* [[Moose (band)|Moose]] (US only)
* [[The Motors]]
* [[Bob Mould]]
* [[The Music]]
* [[My Favorite Highway]]

==N==
* [[Jimmy Nail]]
* [[Nazareth (band)|Nazareth]] ([[Mooncrest Records|Mooncrest]]/Charisma; UK only)
* [[NCT 127]] (Korean releases only)
* [[N*E*R*D]]
*[[Nettspend]]
* [[Tom Newman (musician)|Tom Newman]]
* [[Newsboys]] (StarSong/Forefront/Virgin)
* [[NF (rapper)|NF]]
* [[Now That's What I Call Music!]] (compilation series)
*Nikka Costa
* [[Michael Nyman]]

==O==
* [[Philip Oakey & Giorgio Moroder]]
* [[Liam O'Connor (Danish rapper)|Liam O'Connor]]
* [[Mary Margaret O'Hara]]
* [[Mike Oldfield]]
* [[Lisa Ono]] (1997–2006)
* [[Oomph!]]
* [[Roy Orbison]]
* [[Orchestral Manoeuvres in the Dark]]
* [[Stacie Orrico]]
* [[The Other Ones (Australian-German band)|The Other Ones]]
* [[Outside Edge]]

==P==
* [[The Pale Fountains]]
* [[Palladium (British band)|Palladium]]
* [[Paris Angels]]
* [[Hope Partlow]]
* [[Penetration (band)|Penetration]]
* [[Amanda Perez]]
* [[Anthony Phillips]]
* [[Sam Phillips]]
* [[Billie Piper]]
* [[Placebo (band)|Placebo]]
* [[Pekka Pohjola]]
* [[Porcelain Black]]
* [[Tristan Prettyman]]
* [[Pretty Poison (group)|Pretty Poison]]
* [[The Professionals (band)|The Professionals]]
* [[Professor Green]]
* [[Maxi Priest]]
* [[Public Image Ltd.]]
* [[Purple Ribbon All-Stars]]
* [[Pure Moods]] (compilation series)

==Q==
* [[Queen (band)|Queen]] (Virgin EMI) (outside US/Canada)

==R==
* [[The Railway Children (band)|The Railway Children]]
* [[Rain Tree Crow]]
* [[Rascalz]]
* [[RBD]]
* [[Reamonn]]
* [[The Records]]
* [[Axelle Red]]
* [[The Red Jumpsuit Apparatus]]
* [[Julie Reeves]] (Virgin Nashville)
* [[Priscilla Renea]] (Capitol/Virgin joining)
* [[Richard Ashcroft]]
* [[Keith Richards]]
* [[Juliet Richardson]]
* [[Rip Rig + Panic]]
* [[Rise Against]]
* [[River Road (band)|River Road]] (Virgin Nashville)
* [[The Rolling Stones]]
* [[Christy Carlson Romano]]
* [[Roxx Gang]]
* [[Roxy Music]] (E.G./Virgin)
* [[Röyksopp]]
* [[The Rutles]]
* [[The Ruts]]
* [[Ruth Lorenzo]] (2009–2010)
* [[Robbie Williams]] (2009–2010)

==S==
* [[Saosin]]
* [[Sacred Spirit]]
* [[Ryuichi Sakamoto]]
* [[Sandra (singer)|Sandra]]
* [[Savage Progress]]
* [[Saving Abel]]
* [[Boz Scaggs]]
* [[Scarlett and Black]]
* [[Jon Secada]]
* [[September Mourning]]
* [[The Sex Pistols]]
* [[Shaan (singer)|Shaan]]
* [[Shaggy (artist)|Shaggy]] (Former)
* [[Shane Harper]]
* [[Shapeshifters (band)|Shapeshifters]]
* [[Sharissa]]
* [[Feargal Sharkey]]
* [[SheSays]]
* [[Shinee|SHINee]] (Japan)
* [[Shonen Knife]] (North America/France)
* [[Shooting Star (band)|Shooting Star]]
* [[Sick Puppies]]
* [[Simple Minds]]
* [[Six (band)|Six]]
* [[Skunk Anansie]]
* [[Slapp Happy]]
* [[The Smashing Pumpkins]] (Hut [outside North America]/Virgin)
* [[Elliott Smith]]
* [[Sneaker Pimps]]
* [[Snow (musician)|Snow]]
* [[Something Happens]]
* [[Soul II Soul]]
* [[Source Direct]]
* [[Sparks (band)|Sparks]]
* [[Spice Girls]]
* [[Stacie Orrico]]
* [[The Starting Line]]
* [[Die Sterne]]
* [[Jermaine Stewart]] (10/Virgin)
* [[Joss Stone]]
* [[The Stone Roses]]
* [[Syd Straw]]
* [[Suicidal Tendencies]]
* [[Swedish House Mafia]] (former)
* [[Switched (band)|Switched]]
* [[David Sylvian]] and [[Robert Fripp]]

==T==
* [[The Table (punk band)|The Table]]
* [[Tangerine Dream]]
* [[Terror Squad (group)|Terror Squad]] (Terror Squad/Imperial/Virgin)
* [[Thalía]]
* [[That Petrol Emotion]]
* [[The Thrills]]
* [[Yann Tiersen]]
* [[Tin Tin Out]]
* [[Toto Coelo]] aka Total Coelo in USA (Radialchoice/Virgin) Chrysalis in USA
* [[Pete Townshend]]
* [[T'Pau (band)|T'Pau]] (Siren/Virgin)
* [[Traffic (band)|Traffic]]
* [[Trina]]
* [[Trypes]]
* [[KT Tunstall]] (Relentless/Virgin)
* [[Turbonegro]]
* [[Tina Turner]] (US only)
* [[Infamous Mobb|Twin Gambino]] (U.S. only)

==U==
* [[UB40]] (DEP International/Virgin)
* [[Unbelievable Truth]]
* [[Underoath]] (UK)
* [[Hikaru Utada]] (Japan; 1998–2005)

==V==
* [[Brooke Valentine]]
* [[The Vamps (British band)|The Vamps]]
* [[Velcra]]
* [[The Verve]]
* [[Angela Via]]
* [[Vicky Shu]] (Virgin Jogjakarta)

==W==
* [[Rick Wakeman]] (Charisma/Virgin)
* [[Scott Walker (singer)|Scott Walker]]
* [[The Wallflowers]]
* [[Warrior (band)|Warrior]]
* [[Delroy Washington]]
* [[We Are Scientists]]
* [[Brandi Wells]] (UK only)
* [[Wendy & Lisa]]
* [[Whale (band)|Whale]]
* [[When in Rome (band)|When in Rome]]
* [[The Who]]
* [[Wigwam (progressive rock)|Wigwam]]
* [[Steve Winwood]]
* [[Roger Wootton]]
* [[Working Week (band)|Working Week]]
* [[Link Wray]]
* [[Robert Wyatt]]

==X==
* [[XTC]]
* [[Xylina Spathia]]

==Y==
* [[Yanni]]
* [[Y-c'el]]
* [[You Am I]]
* [[You Me At Six]]
* [[Sydney Youngblood]]
* [[Yellowman]]

==Z==
* [[Warren Zevon]]

==References==
{{Reflist}}

{{portal|Lists|Music}}
{{clear}}

{{Virgin Records}}

{{DEFAULTSORT:Virgin Records Artists}}
[[Category:Lists of recording artists by label]]
[[Category:Virgin Records artists| ]]

Permutohedron

2025-06-04T13:56:07Z

173.79.19.248: /* External links */ this external link has no content not already included here

{{short description|Polyhedron whose vertices represent permutations}}
[[File:Symmetric group 4; permutohedron 3D; transpositions (1-based).png|thumb|300px|The permutohedron of order 4]]

In [[mathematics]], the '''permutohedron''' (also spelled '''permutahedron''') of order {{mvar|n}} is an {{math|(''n'' − 1)}}-dimensional [[polytope]] embedded in an {{mvar|n}}-dimensional space. Its [[vertex (geometry)|vertex]] coordinates (labels) are the [[permutation]]s of the first {{mvar|n}} [[natural number]]s. The edges identify the shortest possible paths (sets of [[Transposition (mathematics)|transpositions]]) that connect two vertices (permutations). Two permutations connected by an edge differ in only two places (one [[Transposition (mathematics)|transposition]]), and the numbers on these places are neighbors (differ in value by 1).

The image on the right shows the permutohedron of order 4, which is the [[truncated octahedron]]. Its vertices are the 24 permutations of {{math|(1, 2, 3, 4)}}. Parallel edges have the same edge color. The 6 edge colors correspond to the 6 possible [[Transposition (mathematics)|transpositions]] of 4 elements, i.e. they indicate in which two places the connected permutations differ. (E.g. red edges connect permutations that differ in the last two places.)

== History ==
According to {{harvs|txt|last=Ziegler|first=Günter M.|authorlink=Günter M. Ziegler|year=1995}}, permutohedra were first studied by {{harvs|txt|first=Pieter Hendrik|last=Schoute|authorlink=Pieter Hendrik Schoute|year=1911}}. The name ''permutoèdre'' was coined by {{harvs|txt|first1=Georges Th.|last1=Guilbaud|first2=Pierre|last2=Rosenstiehl|authorlink2=Pierre Rosenstiehl|year=1963}}. They describe the word as barbaric, but easy to remember, and submit it to the criticism of their readers.<ref>Original French: "le mot permutoèdre est barbare, mais il est facile à retenir; soumettons-le aux critiques des lecteurs."</ref>

The alternative spelling ''permut'''a'''hedron'' is sometimes also used.<ref>{{harvtxt|Thomas|2006}}.</ref> Permutohedra are sometimes called ''permutation polytopes'', but this terminology is also used for the related [[Birkhoff polytope]], defined as the [[convex hull]] of [[permutation matrix|permutation matrices]]. More generally, {{harvs|txt|last=Bowman|first=V. Joseph|year=1972}} uses that term for any polytope whose vertices have a [[bijection]] with the permutations of some set.

== Vertices, edges, and facets ==
{| align="right" style="border: 1px solid lightgray; margin: 3px 0 5px 10px;"
|style="text-align: center;"| {{color|red|vertices}}, {{color|green|edges}}, '''facets''', {{color|orange|faces}} Face dimension {{math|''d'' {{=}} ''n'' − ''k''}}.
|-
|
{{color|gray|{{mvar|k}} {{=}} 1 2 3 4 5}}
{{color|gray|{{mvar|n}}}}
{{color|gray|1}} {{color|red|1}} {{color|orange|1}}
{{color|gray|2}} {{color|green|1}} '''{{color|red|2}}''' {{color|orange|3}}
{{color|gray|3}} 1 '''{{color|green|6}}''' {{color|red|6}} {{color|orange|13}}
{{color|gray|4}} 1 '''14''' {{color|green|36}} {{color|red|24}} {{color|orange|75}}
{{color|gray|5}} 1 '''30''' 150 {{color|green|240}} {{color|red|120}} {{color|orange|541}}
|}

The permutohedron of order {{math|''n''}} has {{math|''n''!}} vertices, each of which is adjacent to {{math|''n'' − 1}} others.
The number of edges is {{math|{{sfrac|(''n'' − 1) ''n''!|2}}}}, and their length is {{math|[[square root of 2|{{radic|2}}]]}}.

Two connected vertices differ by swapping two coordinates, whose values differ by 1.<ref>{{harvtxt|Gaiha|Gupta|1977}}.</ref> The pair of swapped places corresponds to the direction of the edge.
(In [[:File:Symmetric group 4; permutohedron 3D; transpositions (1-based).png|the example image]] the vertices {{math|(3, 2, 1, 4)}} and {{math|(2, 3, 1, 4)}} are connected by a blue edge and differ by swapping 2 and 3 on the first two places. The values 2 and 3 differ by 1. All blue edges correspond to swaps of coordinates on the first two places.)

The number of [[Facet (mathematics)|facets]] is {{math|2''n'' − 2}}, because they correspond to non-empty proper [[subset]]s {{math|''S''}} of {{math|{{mset|1 ... ''n''}}}}.
The vertices of a facet corresponding to subset {{math|''S''}} have in common, that their coordinates on places in {{math|''S''}} are smaller than <abbr title="(smaller than the coordinates on positions not in S)">the rest</abbr>.<ref>{{harvtxt|Lancia|2018}}, p. 105 (see chapter [https://www.mimuw.edu.pl/~jarekw/pdf/Permutahedron.pdf ''The Permutahedron'']).</ref>

{| class="wikitable collapsible collapsed"
!colspan="5"| Facet examples
|-
|colspan="5"|<div style="width: 1250px;">
For order 3 the facets are the 6 edges, and for order 4 they are the 14 <abbr title="(here in the usual sense of 2-faces)">faces</abbr>. 
The coordinates on places {{math|''i''}} where {{math|''i'' ∈ ''S''}} are marked with a dark background. It can be seen, that they are smaller than the rest. 
The square on top corresponds to the subset {{math|{{mset|3, 4}}}}. In its four vertices the coordinates on the last two places have the smallest values (namely 1 and 2).</div>
|-
!colspan="2" style="border-right: 3px solid #aaa;"| order 3
!colspan="3"| order 4
|-
! 1-element subsets
!style="border-right: 3px solid #aaa;"| 2-element subsets
! 1-element subsets
! 2-element subsets
! 3-element subsets
|- style="text-align: center;"
| [[File:Permutohedron 3 subsets 1 (first).svg|250px]]
|style="border-right: 3px solid #aaa;"| [[File:Permutohedron 3 subsets 2 (first).svg|250px]]
| [[File:Permutohedron 4 subsets 1 (first).svg|250px]]
| [[File:Permutohedron 4 subsets 2 (first).svg|250px]]
| [[File:Permutohedron 4 subsets 3 (first).svg|250px]]
|}

More generally, the [[k-face|faces]] of dimensions 0 (vertices) to {{math|''n'' − 1}} (the permutohedron itself) correspond to the [[Weak ordering#Strict weak orderings|strict weak orderings]] of the set {{math|{{mset|1 ... ''n''}}}}. So the number of all faces is the {{math|''n''}}-th [[ordered Bell number]].<ref name="zface">See, e.g., {{harvtxt|Ziegler|1995}}, p. 18.</ref>
A face of dimension {{math|''d''}} corresponds to an ordering with {{math|''k'' {{=}} ''n'' − ''d''}} equivalence classes.

{| class="wikitable collapsible collapsed"
!colspan="2"| Face examples
|-
! order 3
! order 4
|- style="text-align: center;"
| [[File:Weak orderings 3 Hasse.svg|450px]]
| [[File:Weak orderings 4 Hasse.svg|550px]]
|-
|colspan="2"|
The images above show the [[face lattice]]s of the permutohedra of order 3 and 4 (without the empty faces). 
Each face center is labelled with a strict weak ordering.
E.g. the square on top as {{nowrap|3{{hsp}}4{{hsp}}{{!}}{{hsp}}1{{hsp}}2}}, which represents {{math|({{mset|3, 4}}, {{mset|1, 2}})}}. 
The orderings are [[partially ordered set|partially ordered]] by [[partition refinement|refinement]], with the finer ones being on the outside. 
Moving along an <abbr title="(an edge of the face lattice, not of the permutohedron)">edge</abbr> toward the inside means merging two neighbouring equivalence classes.

The vertex labels {{nowrap|a{{hsp}}{{!}}{{hsp}}b{{hsp}}{{!}}{{hsp}}c{{hsp}}{{!}}{{hsp}}d}} interpreted as permutations {{nowrap|(a,{{hsp}}b,{{hsp}}c,{{hsp}}d)}} are those forming the Cayley graph.

{| class="wikitable collapsible collapsed"
!colspan="2" style="color: gray;"| Facets among the faces
|-
|colspan="2"|
The before mentioned facets are among these faces, and correspond to orderings with two equivalence classes. 
The first one is used as the subset {{math|''S''}} assigned to the facet, so the ordering is {{math|(''S'', ''S''''c'')}}.

The images below show how the facets of the {{math|''n''}}-permutohedron correspond to the [[simplex|simplical]] projection of the {{math|''n''}}-[[hypercube|cube]]. 
The binary coordinate labels correspond to the subsets {{math|''S''}}, e.g. 0011 to {{math|{{mset|3, 4}}}}. (The vertex projected to the center does not correspond to a facet. Nor does its opposite vertex in the {{math|''n''}}-cube, which is not part of the projection.)
|- style="text-align: center;"
|style="width: 450px;"| [[File:Triangular cube shadow with bit patterns (first).svg|250px]]
|style="width: 550px;"| [[File:Tetrahedral tesseract shadow with bit patterns (first).svg|300px]]
|}

|}

The number of faces of dimension {{math|''d'' {{=}} ''n'' − ''k''}} in the permutohedron of order {{math|''n''}} is given by the triangle {{math|''T''}} {{OEIS|A019538}}:
<math display=block>T(n,k) = k! \cdot \left\{{n\atop k}\right\}</math>
with <math>\textstyle \left\{{n\atop k}\right\}</math> representing the [[Stirling numbers of the second kind]].

It is shown on the right together with its row sums, the [[ordered Bell number]]s.

== Other properties ==
[[File:Symmetric group 4; Cayley graph 1,2,6 (1-based).png|thumb|300px|The permutohedron-like Cayley graph of {{math|S4}} (see [[c:Category:Permutohedron of order 4 (raytraced)#Permutohedron vs. Cayley graph|here]] for a comparison with the permutohedron)]]
The permutohedron is [[vertex-transitive]]: the [[symmetric group]] {{math|S''n''}} [[Group action (mathematics)|acts]] on the permutohedron by permutation of coordinates.

The permutohedron is a [[zonotope]]; a translated copy of the permutohedron can be generated as the [[Minkowski sum]] of the {{math|[[triangular number|''n''(''n'' − 1)/2]]}} line segments that connect the pairs of the [[standard basis]] vectors.{{sfnp|Ziegler|1995|p=200}}

The vertices and edges of the permutohedron are [[graph isomorphism|isomorphic]] to one of the [[Cayley graph]]s of the [[symmetric group]], namely the one [[Generating set of a group|generated]] by the [[Transposition (mathematics)|transpositions]] that swap consecutive elements. The vertices of the Cayley graph are the [[Inverse function|inverse]] permutations of those in the permutohedron.<ref>This Cayley graph labeling is shown, e.g., by {{harvtxt|Ziegler|1995}}.</ref> The image on the right shows the Cayley graph of {{math|S4}}. Its edge colors represent the 3 generating transpositions: {{color|blue|(1, 2)}}, {{color|green|(2, 3)}}, {{color|red|(3, 4)}}.

This Cayley graph is [[Hamiltonian cycle|Hamiltonian]]; a Hamiltonian cycle may be found by the [[Steinhaus–Johnson–Trotter algorithm]].

== Tessellation of the space ==
{{multiple image
| align = right | total_width = 310
| image1 = Hexagonal tiling.svg
| image2 = Bitruncated cubic honeycomb.png
| footer = Tesselation of space by permutohedra of orders 3 and 4
}}
The permutohedron of order {{mvar|n}} lies entirely in the {{math|(''n'' − 1)}}-dimensional [[hyperplane]] consisting of all points whose coordinates sum to the number:

<math display=block>1 + 2 + \ldots + n = \frac{n(n+1)}{2}</math>

Moreover, this hyperplane can be [[tessellation|tiled]] by infinitely many [[Translation (geometry)|translated]] copies of the permutohedron. Each of them differs from the basic permutohedron by an element of a certain {{math|(''n'' − 1)}}-dimensional [[lattice (group)|lattice]], which consists of the {{mvar|n}}-tuples of integers that sum to zero and whose [[modular arithmetic|residues]] (modulo {{mvar|n}}) are all equal:
<math display=block> \begin{align}
& x_1 + x_2 + \ldots + x_n = 0 \\
& x_1 \equiv x_2 \equiv \ldots \equiv x_n \pmod n.
\end{align}</math>

This is the lattice <math>A_{n-1}^*</math>, the [[dual lattice]] of the [[Root system#An|root lattice <math>A_{n-1}</math>]]. In other words, the permutohedron is the [[Voronoi diagram|Voronoi cell]] for <math>A_{n-1}^*</math>. Accordingly, this lattice is sometimes called the permutohedral lattice.{{sfnp|Baek|Adams|Dolson|2013}}

Thus, the permutohedron of order 4 shown above tiles the 3-dimensional space by translation. Here the 3-dimensional space is the [[affine subspace]] of the 4-dimensional space {{tmath|\R^4}} with coordinates {{math|''x'', ''y'', ''z'', ''w''}} that consists of the 4-tuples of real numbers whose sum is 10,

<math display=block>x + y + z + w = 10.</math>

One easily checks that for each of the following four vectors,

<math display=block>(1,1,1,-3),\ (1,1,-3,1),\ (1,-3,1,1),\ (-3,1,1,1),</math>

the sum of the coordinates is zero and all coordinates are congruent to 1 (mod 4). Any three of these vectors [[Generating set of a group|generate]] the translation lattice.

The tessellations formed in this way from the order-2, order-3, and order-4 permutohedra, respectively, are the [[apeirogon]], the regular [[hexagonal tiling]], and the [[bitruncated cubic honeycomb]]. The dual tessellations contain all [[simplex]] facets, although they are not regular polytopes beyond order-3.

== Examples ==

{| class="wikitable" width=600
! Order 2
! Order 3
! Order 4
! Order 5
! Order 6
|- align=center
! 2 vertices
! 6 vertices
! 24 vertices
! 120 vertices
! 720 vertices
|- align=center
| [[File:Permutohedron order 2.svg|200px]]
| [[File:Permutohedron order 3.svg|200px]]
| [[File:Permutohedron.svg|200px]]
| [[File:Omnitruncated 5Cell as Permutohedron.svg|200px]]
| [[File:Omnitruncated Hexateron as Permutohedron.svg|200px]]
|-
| [[line segment]]
| [[hexagon]]
| [[truncated octahedron]]
| [[omnitruncated 5-cell]]
| [[omnitruncated 5-simplex]]
|}

== See also ==
{{Commons category|Permutohedra}}
* [[Associahedron]]
* [[Cyclohedron]]
* [[Permutoassociahedron]]

== Notes ==
{{reflist}}

== References ==
*{{citation|last1=Baek|first1=Jongmin|last2=Adams|first2=Andrew|last3=Dolson|first3=Jennifer|doi=10.1007/s10851-012-0379-2|issue=2|journal=Journal of Mathematical Imaging and Vision|mr=3061550|pages=211–237|title=Lattice-based high-dimensional Gaussian filtering and the permutohedral lattice|volume=46|year=2013|hdl=1721.1/105344|hdl-access=free}}
*{{citation|first=V. Joseph|last=Bowman|title=Permutation polyhedra|journal=[[SIAM Journal on Applied Mathematics]]|volume=22|issue=4|pages=580–589| year=1972| doi=10.1137/0122054|jstor=2099695|mr=0305800}}.
*{{citation|first1=Prabha|last1=Gaiha|first2=S. K.|last2=Gupta|title=Adjacent vertices on a permutohedron|journal=SIAM Journal on Applied Mathematics|volume=32|issue=2|pages=323–327|year=1977|doi=10.1137/0132025|jstor=2100417|mr=0427102}}.
*{{citation|first1=Georges Th.|last1=Guilbaud|first2=Pierre|last2=Rosenstiehl|authorlink2=Pierre Rosenstiehl| title=Analyse algébrique d'un scrutin|journal=Mathématiques et Sciences Humaines|volume=4|year=1963|pages=9–33|url=http://www.numdam.org/item?id=MSH_1963__4__9_0}}.
* {{citation| last = Lancia | first = Giuseppe | title = Compact extended linear programming models | publisher = Springer | location = Cham, Switzerland | year = 2018 | isbn = 978-3-319-63975-8}}.
*{{citation|first=Pieter Hendrik|last=Schoute|author-link=Pieter Hendrik Schoute|title=Analytic treatment of the polytopes regularly derived from the regular polytopes|journal=Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam|volume=11|issue=3|year=1911|pages=87 pp}} [https://books.google.com/books?id=qC5LAAAAYAAJ&dq=Analytic+treatment+of+the+polytopes+regularly+derived+from+the+regular+polytopes&pg=PA357 Googlebook, 370–381] Also online on the KNAW Digital Library at http://www.dwc.knaw.nl/toegangen/digital-library-knaw/?pagetype=publDetail&pId=PU00011495
*{{citation|title=Lectures in Geometric Combinatorics|title-link= Lectures in Geometric Combinatorics |volume=33|series=Student Mathematical Library: IAS/Park City Mathematical Subseries|first=Rekha R.|last=Thomas|author-link=Rekha R. Thomas|publisher=[[American Mathematical Society]]|year=2006|isbn=978-0-8218-4140-2|contribution=Chapter 9. The Permutahedron|pages=85–92}}.
*{{citation|first=Günter M.|last=Ziegler|author-link=Günter M. Ziegler|title=Lectures on Polytopes|publisher=Springer-Verlag, Graduate Texts in Mathematics 152|year=1995}}.

== Further reading ==
*{{citation|first=Cl.|last=Le Conte de Poly-Barbut|title=Le diagramme du treillis permutoèdre est intersection des diagrammes de deux produits directs d'ordres totaux|journal=Mathématiques, Informatique et Sciences Humaines|volume=112|pages=49–53|year=1990|ref=none}}.
*{{citation|last=Postnikov|first=Alexander|arxiv=math.CO/0507163|doi=10.1093/imrn/rnn153|issue=6|journal=International Mathematics Research Notices|mr=2487491|pages=1026–1106|title=Permutohedra, associahedra, and beyond|year=2009|volume=2009 }}
*{{citation|first=Joe|last=Santmyer|title=For all possible distances look to the permutohedron|journal=[[Mathematics Magazine]]|volume=80|issue=2|year=2007|pages=120–125|ref=none|doi=10.1080/0025570X.2007.11953465}}

[[Category:Permutations]]
[[Category:Polytopes]]

Bon Secours Wellness Arena

2025-05-31T12:07:52Z

173.79.19.248: /* History */ minor cleanup

{{Short description|Arena in Greenville, South Carolina, United States}}
{{Use mdy dates|date=April 2019}}
{{Infobox venue
| stadium_name = Bon Secours Wellness Arena
| nickname = The Well
| former_names = BI-LO Center (1998–2013)
| logo_image = Bon Secours Wellness Arena logo.svg
| image = BonSecoursArenaGRN.jpg
| caption = Bon Secours Wellness Arena in 2016
| pushpin_map = South Carolina#USA
| pushpin_map_caption = Location within South Carolina##Location within the United States
| pushpin_relief = 1
| pushpin_label = Bon Secours Wellness Arena
| address = 650 North Academy Street
| location = [[Greenville, South Carolina]], U.S.
| coordinates = {{coord|34.852789|-82.391458|type:landmark|display=inline,title}}
| broke_ground = March 7, 1996<ref>{{cite news |title=Bi-Lo Center Breaking Ground Today in Downtown Greenville|url=https://news.google.com/newspapers?id=z7YeAAAAIBAJ&sjid=bs8EAAAAIBAJ&pg=3873,1421851&dq=en|newspaper=[[Spartanburg Herald-Journal]]|date=March 7, 1996|access-date=September 23, 2011}}</ref>
| opened = September 3, 1998<ref>{{cite news |title=On the Town|first=Prentiss|last=Findlay|url=https://news.google.com/newspapers?id=PZpIAAAAIBAJ&sjid=PwoNAAAAIBAJ&pg=6091,717914&dq=en|newspaper=[[The Post and Courier]]|location=Charleston, SC|date=September 3, 1998|access-date=September 23, 2011}}</ref> (25 years ago)
| closed =
| demolished =
| owner = Greenville Arena District
| operator = Greenville Arena District
| construction_cost = $63 million (121 Million in 2024)
| architect = Odell Associates AMI Associates
| project_manager = International Facilities Group, LLC.<ref name="IFG">{{cite web |title=Bi-Lo Center|url=http://www.ifgroup.cc/content/?section=506&section2=508&section3=514&page=514|publisher=International Facilities Group|access-date=October 23, 2011}}</ref>
| structural engineer = [[Geiger Engineers|Geiger Engineers PC]]
| services engineer =
| general_contractor = [[Fluor Corporation|Fluor Daniel]]<ref name="IFG"/>
| tenants = [[Greenville Grrrowl]] ([[ECHL]]) (1998–2006) [[Carolina Rhinos]] ([[AF2]]) (2000–2002) [[Greenville Groove]] ([[National Basketball Development League|NBDL]]) (2001–2003) [[South Carolina Force]] ([[American Indoor Football Association|AIFA]]) (2009) [[Greenville Force]] ([[Southern Indoor Football League|SIFL]]) (2010) [[Greenville Swamp Rabbits]] (ECHL) (2010–present) [[Clemson Tigers men's basketball]] ([[National Collegiate Athletic Association|NCAA]]) (2015–2016)
| seating_capacity = Hockey: 13,951 Basketball: 15,000 Concert (Center Stage): 16,000 Concert (End Stage): 14,500
| website = {{url|bonsecoursarena.com}}
}}

'''Bon Secours Wellness Arena''' (formerly the '''BI-LO Center'''; {{A.k.a.}} The Well) is a [[multi-purpose arena]] in [[Greenville, South Carolina]], United States. The arena serves as the home of the [[Greenville Swamp Rabbits]] of the [[ECHL]].

==History==
The arena opened as the BI-LO Center in 1998, and cost [[United States dollar|US $]]63 million. It replaced Greenville's outdated and under-repaired [[Greenville Memorial Auditorium]], which was located across the street from the new arena and had been imploded on September 20, 1997.<ref>{{cite news |title=Greenville Memorial Auditorium Is History|agency=Associated Press|url=http://nl.newsbank.com/nl-search/we/Archives?p_product=CS&s_site=thestate&p_multi=CS&p_theme=realcities&p_action=search&p_maxdocs=200&p_topdoc=1&p_text_direct-0=0EB586B4639B5E49&p_field_direct-0=document_id&p_perpage=10&p_sort=YMD_date:D&s_trackval=GooglePM|newspaper=[[The State (newspaper)|The State]]|location=Greenville, SC|date=September 21, 1997|access-date=April 5, 2008}}</ref> The arena naming rights were purchased by Dutch grocer [[Ahold]], then-owner of [[BI-LO (United States)|BI-LO]], which had been founded in nearby [[Mauldin, South Carolina|Mauldin]] and was still based there at the time. When it was built, it passed [[Columbia, South Carolina|Columbia]]'s [[Carolina Coliseum]] as the largest arena in the state of South Carolina, a distinction it held until 2002, when [[Colonial Life Arena]] was built in [[Columbia, South Carolina|Columbia]].

On September 18, 2013, the BI-LO Center was officially renamed the Bon Secours Wellness Arena after the [[Bon Secours (Virginia & South Carolina)|Bon Secours Health System]] purchased the naming rights.

Bon Secours Wellness Arena was one of two regional hosts during the [[2023 NCAA Division I women's basketball tournament]], alongside [[Climate Pledge Arena]] in Seattle.<ref>{{Cite web |date=2023-03-28 |title=Women's NCAA Tournament Greenville Regionals TV schedule: Elite Eight scores, results, teams, seeds |url=https://www.cbssports.com/womens-college-basketball/news/womens-ncaa-tournament-greenville-regionals-tv-schedule-elite-eight-scores-results-teams-seeds/ |access-date=2023-11-14 |website=CBSSports.com |language=en}}</ref> This was the first season the NCAA had two regional brackets in each city, instead of the traditional set-up of four cities, one for each regional bracket.

===Seating capacity===
As a concert venue, the Bon Secours Wellness Arena can seat approximately 15,000 spectators, depending on the positioning of the stage. In addition, the arena features 30 luxury suites and 840 club seats.{{Citation needed|date=August 2021}}

==References==
{{Reflist}}

{{s-start}}
{{succession box
| title = Host of [[Against All Odds (2008)|Against All Odds]]
| years = 2008
| before = [[Impact Wrestling Zone|TNA Impact! Zone]]
| after = [[Impact Wrestling Zone|TNA Impact! Zone]]
}}
{{end}}
{{ECHL Arenas}}
{{Greenville, South Carolina}}
{{Authority control}}

[[Category:Sports venues completed in 1998]]
[[Category:Buildings and structures in Greenville, South Carolina]]
[[Category:Greenville Groove]]
[[Category:Ice hockey venues in South Carolina]]
[[Category:Ice hockey venues in the United States]]
[[Category:Sports venues in Greenville, South Carolina]]
[[Category:Indoor arenas in South Carolina]]

Georgy Gapon

2025-05-27T00:09:28Z

173.79.19.248: /* Bloody Sunday */ sentence structure

{{short description|Russian Orthodox priest (1870–1906)}}
{{Expand Russian|Гапон, Георгий Аполлонович|date=December 2024|topic=bio}}{{family name hatnote|Apollonovich|Gapon|lang=Eastern Slavic}}
{{Infobox clergy
| honorific_prefix = [[The Reverend]]
| name = Georgy Gapon
| image = Батька Гапон.jpg
| alt =
| caption = 
| native_name = Георгий Гапон
| native_name_lang = ru
| pronunciation =
| birth_name = Georgy Apollonovich Gapon
| birth_date = {{birth date|df=y|1870|2|17}}
| birth_place = Bilyky, [[Poltava Governorate]], Russian Empire
| death_date = {{death date and age|df=y|1906|4|10|1870|2|17}}
| death_place = [[Saint Petersburg]], Russian Empire
| death_cause = Assassination
| residence =
| alma_mater =
| occupation = {{hlist | Cleric | political activist}}
| years_active =
| spouse = Vera{{sfn|Lane|1995|p=336}} (died {{circa|1898}})
| children =
| parents =
| relatives = 
| website = 
| religion = {{nowrap|Christianity ([[Eastern Orthodox]])}}
| church = [[Russian Orthodox Church]]
| ordained =
| writings =
| congregations =
| offices_held =
| signature = Подпись Гапона.jpg
| signature_alt =
}}
'''Georgy Apollonovich Gapon'''{{efn|{{langx|ru|Гео́ргий Аполло́нович Гапо́н}}.}} ({{OldStyleDate|17 February|1870|5 February}} –{{OldStyleDate|10 April|1906|28 March}}) was a [[Russian Orthodox Church|Russian Orthodox]] priest of Ukrainian descent and a popular working-class leader before the [[1905 Russian Revolution]]. Father Gapon is mainly remembered as the leader of [[Bloody Sunday (1905)|peaceful crowds of protesters on Bloody Sunday]], when hundreds of them were killed by firing squads of the [[Imperial Russian Army]].

==Early life==
Georgy Apollonovich Gapon was born {{OldStyleDate|17 February|1870|5 February}}, in the village of Bilyky, [[Poltava Governorate]], [[Russian Empire]] (now Ukraine). He was the oldest son of a [[Cossacks|Cossack]] father and mother who hailed from the local peasantry.{{sfn|Sablinsky|1976|p=34}} Gapon's father, Apollon Fedorovich Gapon, had some formal education and served as an elected village elder and clerk in Bilyky.{{sfn|Sablinsky|1976|p=35}} His mother was illiterate and religiously devout and actively raised her son in the norms and traditions of the [[Russian Orthodox Church]].{{sfn|Sablinsky|1976|p=35}}

Gapon was an excellent primary school student and was offered a place at the Lower Ecclesiastical School in [[Poltava]], a seminary that offered Gapon the best prospect for advancing his formal education.{{sfn|Sablinsky|1976|p=36}} In his final year at this school, Gapon was first exposed to the radical philosophical teachings of [[Leo Tolstoy]] through one of his instructors, a devoted follower of the Russian writer.{{sfn|Sablinsky|1976|p=36}} This instructor, I. M. Tregubov, regarded Gapon as one of the top students at the school, serious and intelligent in demeanour and diligent and curious in his studies.{{sfn|Sablinsky|1976|p=36}}

Following his graduation from the Lower Ecclesiastical School, Gapon was admitted to {{interlanguage link|Poltava Seminary|ru|Полтавская духовная семинария}}, where he continued to be guided in his study of Tolstoy's ideas by other local followers.{{sfn|Sablinsky|1976|p=37}} Gapon was influenced by the [[Tolstoyan movement|Tolstoyan]] emphasis on working with the poor and with its philosophical criticism with the formalistic and hierarchical practices of the official church.{{sfn|Sablinsky|1976|p=37}} This brought him into conflict with certain seminary officials, who threatened to rescind his educational stipend.{{sfn|Sablinsky|1976|p=37}} Gapon met this threat by himself rejecting further aid and seeking to pay for his own education through work as a private tutor.{{sfn|Sablinsky|1976|pp=37–38}}

Gapon fell ill from typhus, which incapacitated him for a time, making it impossible to earn a living as a tutor and continue his studies effectively.{{sfn|Sablinsky|1976|p=38}} He decided to abandon plans for a career as a priest, seeking instead to attend [[Tomsk State University|Tomsk University]] to become a physician, so as to be of greater service to the ailing poor.{{sfn|Sablinsky|1976|p=38}} His plans were short-circuited, however, when the seminary issued him a less-than-perfect grade for behaviour, thereby effectively barring his path to further university education.{{sfn|Sablinsky|1976|p=38}}

At the age of 23 Gapon took a job in Poltava as a ''[[zemstvo]]'' statistician, supplementing his income with money earned working as a private tutor.{{sfn|Sablinsky|1976|p=39}} It was in this capacity that he met the daughter of a local merchant in a house in which he was giving private lessons.{{sfn|Sablinsky|1976|p=39}} The family objected to a proposed marriage due to Gapon's limited employment horizons, however, and as a means of overcoming this obstacle he again sought to become a priest.{{sfn|Sablinsky|1976|pp=39–40}} He made an appeal to Bishop {{interlanguage link|Ilarion (Yushenov)|ru|Иларион (Юшенов)|lt=Ilarion}} of Poltava, apologizing for past behaviour and promising to fulfill expectations of the church in the future.{{sfn|Sablinsky|1976|p=40}} The bishop was moved by the appeal and interceded with the family, winning the couple permission to marry.{{sfn|Sablinsky|1976|p=40}}

Gapon was placed on the fast track to priesthood, occupying a place as a church psalm reader for a year, followed by a ''[[pro forma]]'' promotion to deacon for just one day before being made priest of the Poltava cemetery church.{{sfn|Sablinsky|1976|p=40}} Gapon's services were innovative and informal, and his church rapidly grew in size, negatively impacting other more formalistic local churches, whose priests lodged complaints against him.{{sfn|Sablinsky|1976|p=40}} Nevertheless, Gapon continued to enjoy the support of the bishop in his position and was largely satisfied with his station in life.{{sfn|Sablinsky|1976|p=40}}

==Move to St. Petersburg==
[[File:St. Petersburg academy.jpg|thumb|St. Petersburg Theological Academy, attended by Georgy Gapon, was one of four religious academies of the Russian Orthodox Church. The school was a training facility for theologians.]]
Gapon and his wife had two children in rapid succession, but his wife fell ill following the 1898 birth of the second child, a boy.{{sfn|Sablinsky|1976|p=40}} She died not long afterward, and Gapon decided to leave Poltava to make a new life in the capital city of [[Saint Petersburg]].{{sfn|Sablinsky|1976|pp=40–41}} Bishop Ilarion made a strong recommendation to [[Konstantin Pobedonostsev]], Procurator of the Holy Synod, that Gapon be allowed to take the entrance examination to the [[Saint Petersburg Theological Academy]] despite his lack of the standard Seminary certificate.{{sfn|Sablinsky|1976|p=41}} Gapon placed 16th of 67 applicants and was subsequently awarded a scholarship reserved for the top prospects at the school.{{sfn|Sablinsky|1976|pp=41–42}} He would be one of 235 students to regularly attend classes at the school in 1898.{{sfn|Sablinsky|1976|p=42}}

During his first year at the St. Petersburg Academy, Gapon became involved in missionary work for the church through the Society for Religious and Moral Enlightenment in the Spirit of the Orthodox Church.{{sfn|Sablinsky|1976|p=42}} As part of this activity Gapon helped to conduct religious discussions in industrial shops, mess halls, and lodging houses, bringing him into close contact with the urban [[proletariat]] for the first time.{{sfn|Sablinsky|1976|p=42}} The tightly wound Gapon found the strain of missionary work plus the demands of academic life to be too great and fell into a state of acute depression and he began skipping classes.{{sfn|Sablinsky|1976|p=42}} He withdrew from school on a medical leave of absence{{sfn|Sablinsky|1976|p=45}} and spent almost a year in [[Crimea]] in an attempt to regain his psychological health.{{sfn|Sablinsky|1976|p=42}}

Gapon's status as a student at the St. Petersburg Theological Academy, one of the elite theological training institutions of the Orthodox Church, placed him in good graces with Bishop Nikolai of [[History of Crimea|Taurida]], who permitted Gapon to live in a monastery near [[Sevastopol|Sebastopol]] without having to take [[monastic vows]].{{sfn|Sablinsky|1976|p=43}} In Crimea he met several prominent members of the Tolstoyan movement, all of whom were intensely critical of the Orthodox Church and urged Gapon to leave the priesthood.{{sfn|Sablinsky|1976|p=43}} Gapon rejected this advice, choosing instead to return to course work in St. Petersburg in November 1899, renewed and reinvigorated.{{sfn|Sablinsky|1976|p=44}}

Gapon became a religious teacher at the St. Olga children's orphanage in 1900 and became involved in working with factory workers and families impoverished by unemployment.

==Bloody Sunday==
[[Image:Gapon u Narvskoy zastavy1.jpg|thumb|Gapon near [[Narva Gate]]]]
With the financial support of Colonel [[Akashi Motojiro]] of the [[Imperial Japanese Army]],{{sfn|Busch|1969|p=121}} Gapon organized the Assembly of Russian Factory and Mill Workers of St. Petersburg, which was also patronized by the Department of the Police and the St. Petersburg ''[[Okhrana]]'' and received support from [[Ivan Fullon]], then Governor-General of Saint Petersburg. The Assembly's objectives were to defend workers' rights and to elevate their moral and religious status. He was the person to lead the industrial workers to the capital of Russia during the year 1905. Only persons of Russian Orthodox faith were eligible to join the ranks. He received support from the police-sponsored trade unions established by Okhrana Chief [[Sergei Zubatov]] before their disbandment in 1903. The organization professed loyalty to the Russian Empire, beginning its meetings with the [[Lord's Prayer]] and concluding them with the imperial national anthem "[[God Save the Tsar!]]" At the same time, the Assembly had radical demands. Before the [[1905 Russian Revolution]], Gapon preached that the Tsar was a benevolent leader appointed by [[Divine right of kings|divine right]] who wanted to make fundamental reforms but was constantly thwarted by the [[Boyar|boyars]].<ref>Figes, Orlando (1996). ''A People's Tragedy: The Russian Revolution, 1891-1924''. London: Jonathan Cape. pp. 168–169. {{ISBN|0-224-04162-2}}. {{OCLC|35657827}}.</ref> Soon the organization had twelve branches and 8,000 members, and Gapon tried to expand activities to [[Kiev]] and [[Moscow]]. Gapon was not simply an obedient instrument of the police; cooperating with them, he tried to realize his plan for a positive change of the working class in Russia.{{Citation needed|date=December 2024}}

From the end of 1904, Gapon started to cooperate with radicals who championed the abolition of [[Tsarist autocracy]].{{Citation needed|date=December 2024}}

On {{OldStyleDate|22 January|1905|9 January}}, the day after a general strike burst out in St. Petersburg, Gapon organized a workers' procession to present an emotionally charged written [[The St. Petersburg workmen's petition to the Tsar (January 22, 1905)|petition to the Tsar.]] The demonstration ended tragically as police fired upon the workers; the incident came to be known as [[Bloody Sunday (1905)|Bloody Sunday]].<ref name="marx">{{cite encyclopedia |title=Gapon, Georgi Apollonovich (1870–1906) |url=http://www.marxists.org/glossary/people/g/a.htm |editor-last=Blunden |editor-first=Andy |editor-link=Andy Blunden |encyclopedia=Encyclopedia of Marxism |publisher=Marxists Internet Archive |access-date=12 January 2019 |archive-date=16 August 2021 |archive-url=https://web.archive.org/web/20210816155739/https://www.marxists.org/glossary/people/g/a.htm |url-status=live }}</ref> Gapon's life was saved by [[Pinhas Rutenberg]], who took him away from the gunfire and changed his priestly garments to a common man's. He then became the guest of [[Maxim Gorky]].{{sfn|Yedlin|1999|p=49}}

Following Bloody Sunday, Gapon [[Anathema#Orthodoxy|anathematized]] the Tsar and called upon the workers to take action against the regime, but soon after escaped abroad, where he had close ties with the [[Socialist Revolutionary Party]]. Gapon and Rutenberg were welcomed in Europe both by prominent Russian émigrés [[Georgi Plekhanov]], [[Vladimir Lenin]], [[Peter Kropotkin]] and [[French Left|French socialist]] leaders [[Jean Jaurès]] and [[Georges Clemenceau]]. He found sanctuary in [[Geneva]]<ref>{{cite web |title=George Gapon |url=http://www.spartacus.schoolnet.co.uk/RUSgapon.htm |publisher=Spartacus Educational |archive-url=https://web.archive.org/web/20070213024532/http://www.spartacus.schoolnet.co.uk/RUSgapon.htm |archive-date=13 February 2007 |access-date=21 February 2007}}</ref> and in London at 33 Dunstan House, [[Stepney]], with anarchists [[Peter Kropotkin]] and [[Rudolf Rocker]]. After the [[October Manifesto]], before the end of 1905, Gapon returned to Russia and resumed contact with the Okhrana.

==Death==
Gapon soon revealed to Rutenberg about his contacts with the police and tried to recruit him too, reasoning that dual loyalties were helpful to the workers' cause; however, Rutenberg reported this provocation to his party leaders, [[Yevno Azef]],<ref name=notes>{{cite web |last=Evans |first=Charles T. |title=Father Gapon |url=http://novaonline.nvcc.edu/eli/evans/HIS242/Notes/Gapon.html |website=novaonline.nvcc.edu |access-date=2 June 2014 |archive-date=26 October 2017 |archive-url=https://web.archive.org/web/20171026121139/http://novaonline.nvcc.edu/eli/evans/HIS242/Notes/Gapon.html |url-status=live }}</ref> who was himself a secret police spy, and [[Boris Savinkov]]. On 26 March 1906, Gapon arrived to meet Rutenberg in a rented cottage outside St. Petersburg, and after a month he was found there hanged. Rutenberg asserted later that Gapon was condemned by a comrades' court. In reality, three SR party combatants overheard their conversation from the next room. After Gapon had repeated his collaboration proposal, Rutenberg called the comrades into the room and left. When he returned, Gapon was dead.{{Citation needed|date=January 2018}} Gapon died {{OldStyleDate|10 April|1906|28 March}}.

==Works==
* [[s:The St. Petersburg workmen's petition to the Tsar, January 22, 1905|The St. Petersburg workmen's petition to the Tsar, 22 January 1905]]
* ''The Story of My Life'' (An autobiography by Gapon written just after the Bloody Sunday tragedy)

==Notes==
{{notelist}}

==References==
===Footnotes===
{{reflist|22em}}

===Bibliography===
{{refbegin|35em|indent=yes}}
* {{cite book
|last=Busch
|first=Noel F.
|year=1969
|title=The Emperor's Sword: Japan vs Russia in the Battle of Tsushima
|location=New York
|publisher=Funk & Wagnall
|oclc=12442
}}
* {{cite encyclopedia
|year=1995
|title=Gapon, Georgii Apollonovich
|editor-last=Lane
|editor-first=A. Thomas
|encyclopedia=Biographical Dictionary of European Labor Leaders
|volume=1
|location=Westport, Connecticut
|publisher=Greenwood Press
|pages=336ff
|isbn=978-0-313-29899-8
}}
* {{cite book
|last=Sablinsky
|first=Walter
|year=1976
|title=The Road to Bloody Sunday: Father Gapon and the St. Petersburg Massacre of 1905
|url=https://archive.org/details/roadtobloodysund0000sabl
|url-access=registration
|location=Princeton, New Jersey
|publisher=Princeton University Press
|isbn=978-0-691-10204-7
}}
* {{cite book
|last=Yedlin
|first=Tova
|year=1999
|title=Maxim Gorky: A Political Biography
|location=Westport, Connecticut
|publisher=Praeger Publishers
|isbn=0-275-96605-4
}}
{{refend}}

==Further reading==
{{refbegin|35em|indent=yes}}
* {{cite encyclopedia
|year=1914
|title=[[wikisource:The New Student's Reference Work/Gapon, George|Gapon, George]]
|editor1-last=Beach
|editor1-first=Chandler B.
|editor1-link=Chandler Beach
|editor2-last=McMurry
|editor2-first=Frank Morton
|editor2-link=Frank Morton McMurry
|encyclopedia=The New Student's Reference Work
|location=Chicago
|publisher=F. E. Compton and Company
|page=736
}}
* {{cite book
|last=Gapon
|first=George
|year=1906
|title=The Story of My Life
|url=https://archive.org/details/storymylifebyfa00gapogoog
|location=New York
|publisher=E. P. Dutton & Co.
|access-date=12 January 2019
}}
{{refend}}

==External links==
{{Commons category|George Gapon}}
* {{Internet Archive author |search=((Georgiĭ OR Gregory) AND Gapon)}}
* George Gapon, [https://archive.org/details/storymylifebyfa00gapogoog ''The Story of My Life''] (1906)
* {{Cite NSRW|short=x|wstitle=Gapon, George}}

{{Subject bar |portal=Biography |portal2=Christianity |commons=yes |commons-search=Category:George Gapon |s=yes |s-search=Author:George Gapon |d=yes |d-search=Q299681}}

{{Authority control}}
{{Use British English Oxford spelling|date=January 2019}}
{{Use dmy dates|date=January 2019}}

{{DEFAULTSORT:Gapon, Georgy}}
[[Category:1870 births]]
[[Category:1906 deaths]]
[[Category:19th-century Eastern Orthodox priests]]
[[Category:20th-century anarchists]]
[[Category:20th-century Eastern Orthodox priests]]
[[Category:Anarcho-pacifists]]
[[Category:Christian anarchists]]
[[Category:Clergy from Saint Petersburg]]
[[Category:Double agents]]
[[Category:Extrajudicial killings]]
[[Category:Okhrana informants]]
[[Category:People from Poltava Governorate]]
[[Category:People from Poltava Oblast]]
[[Category:People murdered in 1906]]
[[Category:People of the Russian Revolution of 1905]]
[[Category:Russian anarchists]]
[[Category:Russian anti-capitalists]]
[[Category:Russian Eastern Orthodox priests]]
[[Category:Russian pacifists]]
[[Category:Russian people of Ukrainian descent]]
[[Category:Saint Petersburg Theological Academy alumni]]
[[Category:Tolstoyans]]
[[Category:Unsolved murders in the Russian Empire]]

Okhrana

2025-05-27T00:02:26Z

173.79.19.248: Hyphen

{{short description|Secret police force of the Russian Empire}}
{{About|the secret police of the Russian Empire|the Slavophone militias in German-occupied Macedonia during World War II|Ohrana|the security company of the National Guard of Russia|Federal State Unitary Enterprise Guard (Russia)}}
{{use dmy dates|date=April 2024}}
{{Infobox government agency
| agency_name = Okhrana Guard Department Department for the Protection of Public Safety and Order
| nativename = {{lang|ru|Охрана}} {{lang|ru|Охранное отделение}} {{lang|ru|Отделение по охранению общественной безопасности и порядка}}
| nativename_a =
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| logo = Emblem of the Ministry of the Interior of the Russian Empire.svg
| logo_width =
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| picture = File:Okhranka group photo.jpg
| picture_width = 300px
| picture_caption = Okhrana group photograph in [[St. Petersburg]], [[Russian Empire]] in 1905
| formed = {{Start date and age|1881}}<ref name=AQA>{{cite book|last1= Corrin|first1= Chris|last2= Feihn|first2= Terry|title= AQA A-level History Tsarist and Communist Russia: 1855–1964|date= 31 July 2015|publisher= Hachette UK; Hodder Education; Dynamic Learning|isbn= 9781471837807|page= 44|url= https://books.google.com/books?id=YjTrCQAAQBAJ&pg=PT90|access-date= 8 November 2015|quote= In 1881 a new secret police – the ''Okhrana'' – was established.}}</ref>
| preceding1 =
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| dissolved = {{End date and age|1917}}
| superseding = [[Petrograd Military Revolutionary Committee|PVRK]]
| jurisdiction = {{flag| Russian Empire}}
| headquarters = [[Petrograd]]
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}}

The '''Department for the Protection of Public Safety and Order''' ({{langx|ru|Отделение по охранению общественной безопасности и порядка|Otdelenie po okhraneniyu obshchestvennoy bezopadnosti i poryadka}}), usually called the '''Guard Department''' ({{langx|ru|Охранное отделение|Okhrannoye otdelenie}}) and commonly abbreviated in modern English sources as the '''Okhrana''' ({{lang-rus |Охрана|p=ɐˈxranə|a=Ru-охрана.ogg|t= The Guard}}) was a [[secret police]] force of the [[Russian Empire]] and part of [[Police Department of Russia|the police department]] of the [[Ministry of Internal Affairs of the Russian Empire|Ministry of Internal Affairs]] (MVD) in the late 19th century and early 20th century, aided by the [[Special Corps of Gendarmes]].

==Overview==
Formed to combat political terrorism and [[left-wing politics]] and revolutionary activity,<ref>[http://www.britannica.com/eb/article-9056920/Okhranka#275729.hook Okhrana] ''Britannica Online''</ref> the Okhrana operated offices throughout the Russian Empire, as well as satellite agencies in a number of foreign countries. It concentrated on monitoring the activities of Russian revolutionaries abroad, including in Paris, where Okhrana agent [[Pyotr Rachkovsky]] (1853–1910) was based 1884–1902 before he returned to service in Saint Petersburg 1905–1906.

The Okhrana deployed multiple methods, including assassination, clandestine and covert operations, counterintelligence, espionage, HUMINT, manhunt for the capture or kill of high-value targets, and [[Wikt:perlustration|"perlustration"]]—the reading of private correspondence. The Okhrana's Foreign Agency also monitored revolutionary activity.<ref>
{{cite book
| last1 = Fischer
| first1 = Ben B.
| year = 1997
| title = Okhrana: The Paris Operations of the Russian Imperial Police
| url = https://books.google.com/books?id=szO2W7zF69kC
| publisher = DIANE Publishing
| publication-date = 1999
| page = 6
| isbn = 9780788183287
| access-date = 10 May 2019
| quote = The opening in 1883 of the Okhrana's Foreign Bureau, centred in Paris, was prompted by the shift of Russian revolutionary activity from the Russian Empire to Western and Central Europe.
}}
</ref> The Okhrana became notorious for its [[agent provocateur|agents provocateurs]], including [[Jacob Zhitomirsky]] (born 1880, a leading [[Bolshevik]] and close associate of [[Vladimir Lenin]]), [[Yevno Azef]] (1869–1918), [[Roman Malinovsky]] (1876–1918) and [[Dmitry Bogrov]] (1887–1911).

The Okhrana tried to compromise the [[labour movement]] by setting up police-run [[trade union]]s, a practice known as [[Sergei Zubatov|''zubatovshchina'']]. The Communists blamed the Okhrana in part for the [[Bloody Sunday 1905|Bloody Sunday]] event of January 1905,{{citation needed|date= May 2019}} when Tsarist troops<ref>{{cite book | title= Black Night White Snow | author= Salisbury, Harrison E. | publisher= Da Capo Press | year= 1981 | isbn= 0-306-80154-X | pages= 122–123}}</ref> killed hundreds of unarmed protesters who were marching during a demonstration organized by [[Father Gapon]].<ref>Evans, Charles T. Father Gapon</ref>{{request quotation|date=October 2020}}) and with the participation of [[Pinhas Rutenberg|Pyotr Rutenberg]].

Many historians, such as the German [[Konrad Heiden]]<ref name=Freund2000>[https://reason.com/archives/2000/02/01/forging-protocols "Forging Protocols"] by Charles Paul Freund. ''Reason Magazine'', February 2000</ref> and the [[Russian people|Russian]] historian Mikhail Lepekhine<ref>{{cite news |last1 =Bishop |first1= Patrick |title= 'Protocols of Zion' forger named |url= https://www.telegraph.co.uk/htmlContent.jhtml?html=/archive/1999/11/19/wzion19.html |access-date= 8 November 2015 |work= [[The Daily Telegraph]] |issue= 1638 |date= 19 November 1999 |archive-url= https://web.archive.org/web/20080528005157/http://www.telegraph.co.uk/htmlContent.jhtml?html=%2Farchive%2F1999%2F11%2F19%2Fwzion19.html |archive-date= 28 May 2008 |url-status= dead |location= [[Paris]] |quote= Research by a leading Russian historian, Mikhail Lepekhine, in recently opened archives has found the forgery to be the work of [[Mathieu Golovinski]], opportunistic [[wikt:scion|scion]] of an aristocratic but rebellious family who drifted into a life of [[espionage]] and [[propaganda]] work.}}</ref> maintain that [[Matvei Golovinski]], a writer and Okhrana agent, fabricated the first edition of ''[[The Protocols of the Elders of Zion]]'' (1903). The organization also fabricated documentation connected with the [[History of antisemitism|antisemitic]] [[Menahem Mendel Beilis|Beilis trial]] of 1913.

Suspects captured by the Okhrana were passed to the [[judicial system of the Russian Empire]].

The Okhrana was perpetually underfunded and understaffed; before 1914 it had just 49 employees split between seven offices and never had more than 2,000 informants at any one time. It never received more than 10% of the total police budget.<ref>
{{cite book
| last1 = Lauchlan
| first1 = Iain
| chapter = The Okhrana: security policing in late imperial Russia
| editor1-last = McKean
| editor1-first = Robert B.
| editor2-last = Thatcher
| editor2-first = Ian D.
| title = Late Imperial Russia: Problems and Prospects
| url = https://books.google.com/books?id=g7s8kE1aObAC
| series = Manchester University Press Series
| date = 3 September 2005
| location = Manchester
| publisher = Manchester University Press
| publication-date = 2005
| page = 50
| isbn = 9780719067877
| access-date = 10 May 2019
| quote = The high level of secrecy meant that revolutionaries could only guess at the size and nature of the Okhrana. Consequently the opposition seem to have over-estimated the omniscience of the secret police. Most thought that there was a Black Cabinet in every city and even many towns of the empire. When one Soviet historian dredged the archives he only found evidence of seven such offices with a grand total of 49 employees before 1914; reports of others, he noted, 'were sheer hallucinations'. Activists in the political underground imagined the cities to be infested with watchers and informers, and feared that their ranks were riddled with traitors. Early detractors of the Okhrana estimated that it employed up to 40,000 spies and referred to it as the most important prop to the tsarist regime. Yet when the police archives fell into the hands of the Provisional Government in 1917 they only managed to uncover 600 informers. Recent surveys of the archives have revealed that the Department of Police never employed more than 2,000 informers at any one time and most of these were not high-level spies. The entire Okhrana budget usually accounted for less than 10 per cent of the total expenditure on police, reaching a peak of around five million rubles in 1914 [...].
}}
</ref>

===Use of torture===
Despite the reforms<ref>
Russia disallowed the use of torture in 1774: [https://ru.wikisource.org/wiki/%D0%AD%D0%A1%D0%91%D0%95/%D0%9F%D1%8B%D1%82%D0%BA%D0%B0 8 ноября 1774 г. последовало секретное Высочайшее повеление о том, чтобы присутственные места ни под каким видом не допускали при допросах телесных истязаний «для познания о действиях истины».]
</ref>
in the early 19th century, the practice of [[torture]] was never truly abolished.<ref name="PT13">{{cite book
|author= Malcolm D. Evans, Rod Morgan
|title= Preventing Torture
|url= https://archive.org/details/preventingtortur0000evan
|url-access= registration
|publisher= Oxford University Press
|year= 1999
|page= [https://archive.org/details/preventingtortur0000evan/page/13 13]
|isbn= 0-19-826257-4
}}</ref> Possibly, the formation of the Okhrana led to increasing use of torture,<ref>{{Cite web |url=http://www.hrsolidarity.net/mainfile.php/2002vol12no04/2228/ |title=Patterns of Torture |access-date=2008-04-24 |archive-url=https://web.archive.org/web/20090114024536/http://www.hrsolidarity.net/mainfile.php/2002vol12no04/2228/ |archive-date=2009-01-14 |url-status=usurped | quote = After 1881, the Russian czar Alexander III created a secret police, the Okhrana, to fight terrorism, and the use of torture increased even more. }}</ref> due to the Okhrana using methods such as [[arbitrary arrest and detention|arbitrary arrest]], detention and torture to gain information.<ref>[http://www.hsc.csu.edu.au/modern_history/national_studies/russia/russia_concept/page101.htm Russia and the Soviet Union 1917–1941: Glossary] {{Webarchive|url= https://web.archive.org/web/20080803114257/http://hsc.csu.edu.au/modern_history/national_studies/russia/russia_concept/page101.htm |date= 2008-08-03 }} ''[[Charles Sturt University]]''</ref> Claims persisted the Okhrana had operated [[torture chamber]]s in places like [[Warsaw]], [[Riga]], [[Odessa]] and most of the urban centres.<ref name=Marxist>
[http://www.marxists.org/archive/serge/1926/repression/ch1b.htm The Russian Okhrana] ''[[Marxists.org]]'' – Section XVIII: "The cost of an execution" – "After 1905, the Okhrana had torture chambers in Warsaw, Riga, Odessa and apparently in most of the great urban centres."
</ref>

==History==
Forerunners of the Okhrana as a Russian security service included the Secret Prikaz ({{ill|Taynyy Prikaz|ru|Тайный приказ}}) (1654–1676), the {{ill|Preobrazhensky Prikaz|ru|Преображенский приказ}} (1686–1726), the {{ill|Secret Chancellery|ru|Тайная канцелярия}} (1731–1762), the {{ill|Secret Expedition|ru|Тайная экспедиция }} (1762–1801), and the [[Third Section of His Imperial Majesty's Own Chancellery]] (1826–1880).

The first special security department was the Department on Protecting the Order and Public Peace under the Head of [[Saint Petersburg|St. Petersburg]], set up in 1866 after a failed assassination attempt on Emperor [[Alexander II of Russia|Alexander II]], with a staff of 12 investigators. Its street address, [[Fontanka Embankment|Fontanka]], 16, was publicly known in the Russian Empire. After another failed assassination attempt, on August 6, 1880, the Emperor, acting on proposals made by [[Count Loris-Melikov]], established the Department of State Police under [[Ministry of Internal Affairs (Russia)|Ministry of the Interior (MVD)]] and transferred part of the [[Special Corps of Gendarmes]] and the [[Third Section of His Imperial Majesty's Own Chancellery|Third Section of the Imperial Chancellery]] to the new body. The position of Chief of Gendarmes was merged with that of the Minister, and Commander of the Corps was assigned as a Deputy of the Minister. Still, these measures did not prevent the assassination of Alexander II in March 1881.

In an attempt to implement preventive security measures, Emperor [[Alexander III of Russia|Alexander III]] ({{reign | 1881 | 1894}}) immediately set up two more Security and Investigation (охранно-розыскные) secret-police stations, supervised by Gendarme officers, in [[Moscow]] and [[Warsaw]]; they became the basis of the later Okhrana. The Imperial Gendarmerie still operated as security police in the rest of the country through their [[Guberniya|Gubernial]] and [[Uyezd]] Directorates. The Emperor also established the Special Conference under the MVD (1881), which had the right to declare a State of Emergency Security in various parts of the Empire (which was actively used in the time of [[Russian Revolution of 1905|1905 Revolution]]) and subordinated all of the imperial police forces to the Commander of the Gendarmes (1882).

The rise of the socialist movements led to the integration of security forces. From 1898 the Special Section (Особый отдел) of the Department of Police succeeded the Gendarmes in the role of gaining information from domestic and foreign agents and "{{linktext|perlustration}}". Following the Socialist-Revolutionary Party's assassination of MVD Minister [[Dmitry Sipyagin]] on April 2, 1902, the new Minister [[Vyacheslav von Plehve]] gradually relieved the Directorates of Gendarmes of their investigation power in favor of Security and Investigation Stations (Охранно-розыскное отделение) under respective Mayors and Governors (who as a matter of fact were subordinate to the MVD Minister).

===Pre-1905===
The Okhrana used many seemingly unorthodox methods in the pursuit of its mission to defend the Tsarist monarchy; indeed, some of the Okhrana's activities even contributed to the wave of domestic unrest and revolutionary terror that they were intended to quell. Perhaps most paradoxical of all was the Okhrana's collaboration with revolutionary organizations.<ref>Fredric S. Zuckerman, "Political Police and Revolution: The Impact of the 1905 Revolution on the Tsarist Secret Police", ''Journal of Contemporary History'' 27 (1992): 281.</ref> Early Okhrana agents to work alongside revolutionaries included Lieutenant-Colonel [[Georgy Sudeykin]] of the [[St. Petersburg]] Special Section, who, in 1882, set up an illegal printing operation to publish the revolutionary [[Narodnaya Volya|People's Will]] literature with Okhrana funds. Sudeykin and his colleague, a revolutionary-turned-police-informant named [[Sergey Degayev]], passed drafts of the publication through Okhrana censors before printing. This episode marked the beginning of the Okhrana's efforts to surreptitiously observe, but also influence and undermine, revolutionary movements.<ref>[[Ronald Hingley]], ''The Russian Secret Police: Muscovite, Imperial Russian, and Soviet Political Security Operations'' (New York: Dorset, 1970), 75–6.</ref> This focus on infiltrating and influencing revolutionary groups, rather than merely identifying and arresting their members, intensified with the innovations of one Okhrana bureau chief, [[Sergei Vasilyevich Zubatov|Sergey Zubatov]]. While [[Pyotr Rachkovsky|P.I. Rachkovsky]], as head of the Okhrana's Foreign Agency, had long ordered Okhrana agents to infiltrate and influence revolutionary movements abroad, Zubatov brought these tactics to a new level by setting up Okhrana-controlled trade unions, the foundation of police socialism.<ref>Richard J. Johnson, "Zagranichnaia Agentura: The Tsarist Political Police in Europe", ''Journal of Contemporary History'' 7 (1972): 226. Hingley, ''Russian Secret Police'', 87.</ref> Perhaps recognizing the same discontent among factory workers that the Bolsheviks sought to exploit to start a revolution, Zubatov hoped the unions would mollify factory workers with improvements in working conditions and thus prevent workers from joining revolutionary movements that threatened the monarchy. To this end, Zubatov set up the Moscow Mechanical Production Workers' Mutual Aid Society in May 1901. After Zubatov became head of the Special Section in 1902, he expanded his trade unions from Moscow to St. Petersburg and to Southern Russia.<ref>Hingley, ''Russian Secret Police'', 88–89.</ref>

Zubatovite trade unions achieved moderate success at channeling workers' political agitations away from revolutionary movements and toward labor improvements, especially in the cities of [[Minsk]] and [[Odessa]], with one high-ranking official noting that many revolutionaries and workers were joining the unions.<ref>Jonathan W. Daly, ''Autocracy Under Siege: Security Police and Opposition in Russia, 1866–1905'' (DeKalb: Northern Illinois University Press, 1998), 138.</ref> However, Zubatov, if not police socialism, became discredited in the summer of 1903 after the Okhrana officer in charge of the Odessa union allowed a strike to get out of hand, causing a mass movement which paralyzed the region.<ref>Hingley, Russian Secret Police, 89.</ref> Although the police-run unions continued to operate after Zubatov's ousting, without Okhrana funding, they proved more a liability than an asset. The Assembly of Working Men, a police-run union with about 6,000–8,000 members, formed by the alleged Okhrana agent [[Georgy Gapon|Father Georgy Gapon]], sparked the [[Bloody Sunday (1905)|Bloody Sunday massacre]] of January 1905, a milestone in the [[Revolution of 1905]] when union members marched peacefully on the [[Winter Palace]] in St. Petersburg and were fired upon by Imperial soldiers.<ref>Hingley, Russian Secret Police, 94–5.</ref> The Okhrana complemented police socialism and other projects to prevent the conditions in which revolutionary movements could take hold by pursuing initiatives to curtail the activities of existing organizations. [[Yevno Azef]], the notorious Okhrana [[Agent provocateur|provocateur]] who became the head of the [[Socialist-Revolutionary Party|Socialist Revolutionary]] Fighting Organization (SRFO), epitomized the Okhrana's inscrutable practice of revolutionary-group infiltration. While the Okhrana managed to imbed many of its agents in revolutionary organizations, the police preferred to slowly gather intelligence and to attempt to interfere with revolutionary work surreptitiously rather than to arrest known revolutionaries immediately. This policy led to numerous dubious acts on the part of police spies, who needed to participate in revolutionary activities to avoid suspicion, as when Yevno Azef, as head of the SRFO, ordered the assassination of [[Vyacheslav von Plehve|V. K. Plehve]] on July 15, 1904.<ref>Hingley, ''Russian Secret Police'', 92.</ref>

===The Revolution of 1905===
For over twenty years, the Okhrana had focused on interfering with the activities of relatively small, and distinct, revolutionary groups. The Revolution of 1905, characterized by seemingly spontaneous marches and strikes, exposed the Okhrana's inefficacy at controlling mass popular movements.<ref>Zuckerman, ''Political Police and Revolution'', 281.</ref> Not only did the Okhrana lack the capacity to prevent the mass movements of 1905, or even to contain them once they began, the Okhrana's misguided attempts may even have worsened the unrest. [[Dmitri Feodorovich Trepov|D. F. Trepov]], the Assistant Minister of the Interior in charge of police affairs, and P. I. Rachkovsky, now in charge of all domestic political-police operations, attempted to mount an aggressive offensive against those they believed to be responsible for the unrest, namely ''[[zemstvo]]'' employees, in May 1905, but backed down three months later.<ref>Zuckerman, ''Political Police and Revolution'', 282, 5.</ref> In October of that year, Trepov again attempted a violent repression of the revolution, only to call off the effort for lack of manpower. Since these attempts at repression never reached fruition, they only served to aggravate the already enraged Russian populace and to deepen their distrust of the Imperial government. Trepov's replacement by [[Pyotr Durnovo|P.N. Durnovo]] in late-October ushered in a period of even more vicious repression of the revolutionaries.<ref>Daly, ''Autocracy Under Siege'', 173.</ref> Indicative of this new period is the head of the St. Petersburg Special Section, A.V. Gerasimov's, strike on the St. Petersburg Soviet. To Emperor [[Nicholas II]]'s delight, Gerasimov arrested delegates of the Soviet ''en masse'' on December 3, 1905. Along with this repression and the end of the Revolution of 1905 came a shift in the political police's mentality; gone were the days of [[Nicholas I of Russia|Nicholas I]]'s white-gloved [[Third Section of His Imperial Majesty's Own Chancellery|moral police]]: post-1905 the political police feared that the Russian people were as eager to destroy them as to depose the Emperor.<ref>Zuckerman, “Political Police and Revolution,” 285, 287, 289–290.</ref>

Following the outbreak of the 1905 Revolution and assassination of Plehve, [[Pyotr Stolypin]], as the new MVD Minister and Chairman of the Council of Ministers, set up a nationwide net of Security Stations. By 1908 there were 31 Stations, and more than 60 by 1911. Two more Special Sections of the Department of Police were organized in 1906. The centralized Security Section of the Department of Police was created on February 9, 1907; it was located at 16, Fontanka, St. Petersburg.

The exposure of [[Yevno Azef]] (who had organized many assassinations, including that of [[Vyacheslav von Plehve|Plehve]]) and [[Dmitri Bogrov]] (who assassinated [[Pyotr Stolypin|Stolypin]] in 1911) as Okhrana [[double agent]]s put the agency's methods under great suspicion; the organisation was further compromised by the discovery of many similar [[double agent]]s-provocateur. In Autumn 1913, all of the Security Stations except the original Moscow, St Petersburg and Warsaw ones were dismissed. The start of [[World War I]] in 1914 marked a shift from anti-revolutionary activities of the Department of Police to [[counter-intelligence]]; however, the efforts of the department were poorly synchronised with counter-intelligence units of the General Staff and of the [[Imperial Russian Army|Army]].

===The 1917 Russian Revolution (February Revolution and October Revolution)===

Just as the Okhrana had once sponsored trade unions to divert activist energy from political causes, so too did the secret police attempt to promote the Bolshevik party, as the Bolsheviks seemed a relatively harmless alternative to more violent revolutionary groups. Indeed, to the Okhrana, [[Vladimir Lenin|Lenin]] seemed to actively hinder the revolutionary movement by denouncing other revolutionary groups and refusing to cooperate with them.<ref name="ReferenceA">Hingley, ''Russian Secret Police'', 105</ref>

To aid the Bolsheviks at the expense of other revolutionaries, the Okhrana helped [[Roman Malinovsky]] (a police spy who had managed to rise within the Bolshevik hierarchy and gain Lenin's trust) in his bid to become a Bolshevik delegate to the [[State Duma of the Russian Empire|Duma]] in 1912. To this end, the Okhrana sequestered Malinovsky's criminal record and arrested other candidates for the seat.<ref name="ReferenceA"/>

According to the transcribed recollections of Nikolay Vladimirovich Veselago, a former Okhrana officer and relative of the director of the Russian police department [[Stepan Petrovich Beletsky]], both Malinovsky and [[Joseph Stalin]] reported on Lenin as well as on each other although Stalin was unaware that Malinovosky was also a penetration agent.<ref>{{cite book |last1=Dziak |first1=John J. |title=Chekisty: A History of the KGB |date=1988 |publisher=Lexington Books |isbn=978-0-669-10258-1 |page=8 |url=https://books.google.com/books?id=yyIFAQAAIAAJ&q=Veselago+stalin+okhrana |language=en}}</ref><ref>{{cite book |last1=Hyde |first1=H. Montgomery |title=Stalin |date=21 March 1982 |publisher=Da Capo Press |isbn=978-0-306-80167-9 |page=618 |url=https://books.google.com/books?id=KGnvAAAAMAAJ&q=Stalin+okhrana+agent |language=en}}</ref><ref>{{cite book |last1=Brackman |first1=Roman |title=The Secret File of Joseph Stalin: A Hidden Life |date=23 November 2004 |publisher=Routledge |isbn=978-1-135-75840-0 |pages=369 |url=https://books.google.com/books?id=PY2RAgAAQBAJ&dq=Nikolay+Veselago+stalin&pg=PA369 |language=en}}</ref>

Malinovsky won the seat and led the Bolshevik delegation in the Fourth Duma until 1914, but even with the information Malinovsky and other informants provided to the Okhrana, the police were unprepared for the rise of Bolshevism in 1917. Although the secret police had agents within the Bolshevik organization, other factors contributed to the Okhrana's inefficacy at averting the events of 1917. Among these factors was the ban on police spies within the military promulgated by the Deputy Minister of the Interior [[Vladimir Dzhunkovsky]], who found the practice dishonorable and damaging to morale. While the beginning of World War I in 1914 moved the Okhrana's attentions initially from countering revolutionaries to countering German espionage, the focus quickly shifted back as it emerged that the Germans were heavily funding Russian revolutionary groups in order to destabilize the Russian Empire.<ref>Hingley, ''Russian Secret Police'', 106–9</ref>

Despite the renewed attention, the [[Russian Revolution]] of 1917 took the secret police, and the country, by surprise. Indeed, the Okhrana's persistent focus on revolutionary groups may have resulted in the secret police not fully appreciating the deep-seated popular unrest brewing in Russia.

The revolutionaries identified the Okhrana as one of the main symbols of [[Tsarist autocracy|Tsarist repression]], and its headquarters were sacked and burned on 27 February 1917. The newly formed [[Russian Provisional Government|Provisional Government]] then disbanded the whole organization and released most of the political prisoners held by the Tsarist regime. Revelations of the Okhrana's earlier abuses heightened public hostility towards the secret police after the 1917 [[February Revolution]] and made it very dangerous to be a political policeman. That fact, along with the [[Petrograd Soviet]]'s insistence on the dissolution of the regular Tsarist police force, as well as of the political police, meant that the Okhrana quickly and quietly disappeared.<ref>Hingley, ''Russian Secret Police'', 111.
</ref>

=== Successor organisations ===
Some Okhrana functionaries continued their activities in the [[Russian Civil War|civil-war period]] (1917–1923) within the [[White Armies]], notably in the {{ill|OSVAG|ru|ОСВАГ}} ({{langx |ru| ОСВАГ – ОСВедомительное АГентство | translit = OSVAG – OSVedomitel'noe AGentsvo | translation = Information Agency}}).<ref>
{{cite book
|last1 = Ehrenburg
|first1 = Ilya Grigoryevich
|author-link1 = Ilya Ehrenburg
|translator-last1 = Bostock
|translator-first1 = Anna
|translator-last2 = Kapp
|translator-first2 = Yvonne
|translator-link2 = Yvonne Kapp
|orig-date = 1960–1967
|title = Men, Years – life
|year = 1962
|url = https://books.google.com/books?id=mmUUAAAAIAAJ
|volume = 2
|location = London
|publisher = MacGibbon & Kee
|publication-date = 1962
|page = 93
|access-date = 21 January 2022
|quote = In the White Army there were men of the Black Hundreds, former members of the Okhrana (Tsarist secret police), gendarmes, hangmen. They occupied important posts in the administration, the counterintelligence and the Osvag.
}}
</ref>

After the 1917 [[October Revolution]], the government of the [[RSFSR]] under [[Vladimir Lenin]] replaced the Okhrana with a Soviet security organisation – the much larger and more efficient [[Cheka]] in December 1917, supplemented by the [[GRU (G.U.)|GRU]] (military intelligence) from October–November 1918. The Cheka and its [[Chronology of Soviet secret police agencies|successor organizations]] (notably the [[State Political Directorate|GPU]] and the [[OGPU]]) eventually became the [[KGB]] (1954–1991) after the death of the Soviet leader [[Joseph Stalin]] in March 1953. After the [[dissolution of the Soviet Union]] in December 1991 the [[KGB]] split into the [[Federal Counterintelligence Service|FSK]] (later reorganized into the [[Federal Security Service|FSB]] in 1995) and the [[Foreign Intelligence Service (Russia)|SVR]].

==See also==
* [[Chronology of Soviet secret police agencies]]
* [[Petrograd Military Revolutionary Committee]] (October–December 1917)
* [[Cheka]] (1917–1922)
* [[State Political Directorate|GPU]] (1922–1923)
* [[OGPU]] (1923–1934)
* [[NKVD]] (1934–1943)
* [[NKGB]] (1943–1946)
* [[Ministry for State Security (Soviet Union)|MGB]] (1946–1954)
* [[KGB]] (1954–1991)
* [[Federal Counterintelligence Service|FSK]] (1991–1995)
* [[Federal Security Service (Russia)|FSB]] (1995–present)
* [[Foreign Intelligence Service (Russia)|SVR]] (1991–present)

==Notes==
{{Reflist}}

==References==
* Jonathan Daly, [http://www.niupress.niu.edu/niupress/scripts/book/bookResults.asp?ID=46 Autocracy under Siege: Security Police and Opposition in Russia, 1866–1905] (DeKalb, Ill.: Northern Illinois University Press, 1998). {{ISBN|978-0-87580-243-5}}
* Jonathan Daly, [http://www.niupress.niu.edu/niupress/scripts/book/bookResults.asp?ID=219 The Watchful State: Security Police and Opposition in Russia, 1906–1917] (DeKalb, Ill.: Northern Illinois University Press, 2004). {{ISBN|978-0-87580-331-9}}
* Catherine Evtuhov, Stephen Kotkin; ''The Cultural Gradient: the transmission of ideas in Europe, 1789–1991''; Rowman & Littlefield, 2003 {{ISBN|0-7425-2063-3}}
* Charles A. Ruud, Sergei A. Stepanov; ''Fontanka 16: The Tsars' Secret Police''; McGill-Queen's University Press (paperback, 2002) {{ISBN|0-7735-2484-3}}
* [https://web.archive.org/web/20070613034033/https://www.cia.gov/library/center-for-the-study-of-intelligence/csi-publications/books-and-monographs/okhrana-the-paris-operations-of-the-russian-imperial-police/art1.pdf Paris Okhrana 1885–1905] [[CIA]] historical review program (Approved for release 22 September 1993)

==External links==
{{commons category|Okhrana}}
* [http://oac.cdlib.org/findaid/ark:/13030/kt538nf189/ Guide to the Okhrana Records] and [https://digitalcollections.hoover.org/objects/54048 photographs of activities and suspects online] at the [http://www.hoover.org/library-and-archives Hoover Institution Archives].
* Official history of the [[MVD]] of Russia: [https://web.archive.org/web/20050227171228/http://mvdinform.ru/index.php?docid=364 1857–1879] [https://web.archive.org/web/20050227171241/http://mvdinform.ru/index.php?docid=365 1880–1904] [https://web.archive.org/web/20050227171136/http://mvdinform.ru/index.php?docid=366 1905–1916] (in Russian)
* [http://www.spartacus-educational.com/RUSokhrana.htm "Okhrana"] in [[Spartacus Educational]] encyclopedia
*[https://www.cia.gov/library/center-for-the-study-of-intelligence/csi-publications/books-and-monographs/okhrana-the-paris-operations-of-the-russian-imperial-police/5474-1.html Okhrana: The Paris Operations of the Russian Imperial Police] {{Webarchive|url=https://web.archive.org/web/20121220144449/https://www.cia.gov/library/center-for-the-study-of-intelligence/csi-publications/books-and-monographs/okhrana-the-paris-operations-of-the-russian-imperial-police/5474-1.html |date=2012-12-20 }}

{{Authority control}}

[[Category:Okhrana personnel|*]]
[[Category:1881 establishments in the Russian Empire]]
[[Category:1917 disestablishments in Russia]]
[[Category:Branches of the secret services of the Russian Empire]]
[[Category:Government agencies established in 1881]]
[[Category:Organizations disestablished in 1917]]
[[Category:Secret police]]

Empty set

2025-05-26T02:12:52Z

173.79.19.248: /* Topology */ LQ

{{short description|Mathematical set containing no elements}}
{{Redirect|∅|similar symbols|Ø (disambiguation)}}
{{Other uses of|Empty}}

[[File:Nullset.svg|thumb|upright=0.6|The empty set is the set containing no elements.]]
In [[mathematics]], the '''empty set''' or '''void set''' is the unique [[Set (mathematics)|set]] having no [[Element (mathematics)|elements]]; its size or [[cardinality]] (count of elements in a set) is [[0|zero]].<ref name=":1">{{Cite web|last=Weisstein|first=Eric W.|title=Empty Set|url=https://mathworld.wolfram.com/EmptySet.html|access-date=2020-08-11|website=mathworld.wolfram.com|language=en}}</ref> Some [[axiomatic set theories]] ensure that the empty set exists by including an [[axiom of empty set]], while in other theories, its existence can be deduced. Many possible properties of sets are [[vacuously true]] for the empty set.

Any set other than the empty set is called ''non-empty''.

In some textbooks and popularizations, the empty set is referred to as the "null set".<ref name=":1" /> However, [[null set]] is a distinct notion within the context of [[measure theory]], in which it describes a set of measure zero (which is not necessarily empty).

==Notation==
{{Main|Null sign}}
[[Image:Empty set symbol.svg|thumb|upright=0.6|A symbol for the empty set]]
Common notations for the empty set include "{ }", "<math>\emptyset</math>", and "∅". The latter two symbols were introduced by the [[Bourbaki group]] (specifically [[André Weil]]) in 1939, inspired by the letter [[Ø]] ({{unichar|d8|LATIN CAPITAL LETTER O WITH STROKE}}) in the [[Danish orthography|Danish]] and [[Norwegian orthography|Norwegian]] alphabets.<ref>{{cite web| url = http://jeff560.tripod.com/set.html| title = Earliest Uses of Symbols of Set Theory and Logic.}}</ref> In the past, "0" (the numeral [[zero]]) was occasionally used as a symbol for the empty set, but this is now considered to be an improper use of notation.<ref>{{Cite book|url=https://archive.org/details/1979RudinW|title=Principles of Mathematical Analysis|last=Rudin|first=Walter|publisher=McGraw-Hill|year=1976|isbn=007054235X|edition=3rd|pages=300}}</ref>

The symbol ∅ is available at [[Unicode]] point {{unichar|2205|EMPTY SET}}.<ref>{{cite web| url = https://www.unicode.org/charts/PDF/U2200.pdf| title = Unicode Standard 5.2}}</ref> It can be coded in [[HTML]] as {{code|∅}} and as {{code|∅}} or as {{code|∅}}. It can be coded in [[LaTeX]] as {{code|\varnothing}}. The symbol <math>\emptyset</math> is coded in LaTeX as {{code|\emptyset}}.

When writing in languages such as Danish and Norwegian, where the empty set character may be confused with the alphabetic letter Ø (as when using the symbol in linguistics), the Unicode character U+29B0 REVERSED EMPTY SET ⦰ may be used instead.<ref>e.g. Nina Grønnum (2005, 2013) ''Fonetik og Fonologi: Almen og dansk.'' Akademisk forlag, Copenhagen.</ref>

== Properties ==
In standard [[axiomatic set theory]], by the [[axiom of extensionality|principle of extensionality]], two sets are equal if they have the same elements (that is, neither of them has an element not in the other). As a result, there can be only one set with no elements, hence the usage of "the empty set" rather than "an empty set".

The only subset of the empty set is the empty set itself; equivalently, the [[power set]] of the empty set is the set containing only the empty set. The number of elements of the empty set (i.e., its [[cardinality]]) is zero. The empty set is the only set with either of these properties.

[[For any]] set ''A'':
* The empty set is a [[subset]] of ''A''
* The [[union (set theory)|union]] of ''A'' with the empty set is ''A''
* The [[intersection (set theory)|intersection]] of ''A'' with the empty set is the empty set
* The [[Cartesian product]] of ''A'' and the empty set is the empty set

For any [[property (philosophy)|property]] ''P'':
* For every element of <math>\varnothing</math>, the property ''P'' holds ([[vacuous truth]]).
* There is no element of <math>\varnothing</math> for which the property ''P'' holds.

Conversely, if for some property ''P'' and some set ''V'', the following two statements hold:
* For every element of ''V'' the property ''P'' holds
* There is no element of ''V'' for which the property ''P'' holds
then <math>V = \varnothing.</math>

By the definition of [[subset]], the empty set is a subset of any set ''A''. That is, {{em|every}} element ''x'' of <math>\varnothing</math> belongs to ''A''. Indeed, if it were not true that every element of <math>\varnothing</math> is in ''A'', then there would be at least one element of <math>\varnothing</math> that is not present in ''A''. Since there are {{em|no}} elements of <math>\varnothing</math> at all, there is no element of <math>\varnothing</math> that is not in ''A''. Any statement that begins "for every element of <math>\varnothing</math>" is not making any substantive claim; it is a [[vacuous truth]]. This is often paraphrased as "everything is true of the elements of the empty set."

In the usual [[set-theoretic definition of natural numbers]], zero is modelled by the empty set.

=== Operations on the empty set ===

When speaking of the [[summation|sum]] of the elements of a finite set, one is inevitably led to the convention that the sum of the elements of the empty set (the [[empty sum]]) is zero. The reason for this is that zero is the [[identity element]] for addition. Similarly, the [[multiplication|product]] of the elements of the empty set (the [[empty product]]) should be considered to be [[1 (number)|one]], since one is the identity element for multiplication.<ref>{{cite book |author=David M. Bloom |title=Linear Algebra and Geometry |url=https://archive.org/details/linearalgebrageo0000bloo |url-access=registration |year=1979 |isbn=0521293243 |pages=[https://archive.org/details/linearalgebrageo0000bloo/page/45 45]}}</ref>

A [[derangement]] is a [[permutation]] of a set without [[fixed point (mathematics)|fixed point]]s. The empty set can be considered a derangement of itself, because it has only one permutation (<math>0!=1</math>), and it is vacuously true that no element (of the empty set) can be found that retains its original position.

==In other areas of mathematics==
=== Extended real numbers ===

Since the empty set has no member when it is considered as a subset of any [[ordered set]], every member of that set will be an upper bound and lower bound for the empty set. For example, when considered as a subset of the real numbers, with its usual ordering, represented by the [[real number line]], every real number is both an upper and lower bound for the empty set.<ref>Bruckner, A.N., Bruckner, J.B., and Thomson, B.S. (2008). ''[http://classicalrealanalysis.info/com/documents/TBB-AllChapters-Portrait.pdf Elementary Real Analysis]'', 2nd edition, p. 9.</ref> When considered as a subset of the [[extended reals]] formed by adding two "numbers" or "points" to the real numbers (namely [[negative infinity]], denoted <math>-\infty\!\,,</math> which is defined to be less than every other extended real number, and [[positive infinity]], denoted <math>+\infty\!\,,</math> which is defined to be greater than every other extended real number), we have that:
<math display=block>\sup\varnothing=\min(\{-\infty, +\infty \} \cup \mathbb{R})=-\infty,</math>
and
<math display=block>\inf\varnothing=\max(\{-\infty, +\infty \} \cup \mathbb{R})=+\infty.</math>

That is, the least upper bound (sup or [[supremum]]) of the empty set is negative infinity, while the greatest lower bound (inf or [[infimum]]) is positive infinity. By analogy with the above, in the domain of the extended reals, negative infinity is the identity element for the maximum and supremum operators, while positive infinity is the identity element for the minimum and infimum operators.

=== Topology ===
In any [[topological space]] <math>X</math>, the empty set is [[open set|open]] by definition, as is <math>X</math>. Since the [[Complement (set theory)|complement]] of an open set is [[Closed set|closed]] and the empty set and <math>X</math> are complements of each other, the empty set is also closed, making it a [[clopen set]]. Moreover, the empty set is [[Compact set|compact]] by the fact that every [[finite set]] is compact.

A topological space <math>X</math> is said to have the [[indiscrete topology]] if the only open sets are <math>\varnothing</math> and the entire space.

The [[Closure (mathematics)|closure]] of the empty set is empty. This is known as "preservation of [[nullary]] [[Union (set theory)|unions]]".<ref>{{cite book |last1=Munkres |first1=James Raymond |title=Topology |date=2018 |publisher=Pearson |location=New York, NY |isbn=978-0134689517 |edition=Second, reissue}}</ref>

=== Category theory ===
If <math>A</math> is a set, then there exists precisely one [[Function (mathematics)|function]] <math>f</math> from <math>\varnothing</math> to <math>A,</math> the [[empty function]]. As a result, the empty set is the unique [[initial object]] of the [[Category theory|category]] of sets and functions.

The empty set can be turned into a [[topological space]], called the empty space, in just one way: by defining the empty set to be [[Open set|open]]. This empty topological space is the unique initial object in the [[category of topological spaces]] with [[Continuous function (topology)|continuous maps]]. In fact, it is a [[strict initial object]]: only the empty set has a function to the empty set.

=== Set theory ===
In the [[von Neumann ordinal|von Neumann construction of the ordinals]], 0 is defined as the empty set, and the successor of an ordinal is defined as <math>S(\alpha)=\alpha\cup\{\alpha\}</math>. Thus, we have <math>0=\varnothing</math>, <math>1 = 0\cup\{0\}=\{\varnothing\}</math>, <math>2=1\cup\{1\}=\{\varnothing,\{\varnothing\}\}</math>, and so on. The von Neumann construction, along with the [[axiom of infinity]], which guarantees the existence of at least one infinite set, can be used to construct the set of natural numbers, <math>\N_0</math>, such that the [[Peano axioms]] of arithmetic are satisfied.

==Questioned existence==
=== Historical issues ===
In the context of sets of real numbers, Cantor used <math>P\equiv O</math> to denote "<math>P</math> contains no single point". This <math>\equiv O</math> notation was utilized in definitions; for example, Cantor defined two sets as being disjoint if their intersection has an absence of points; however, it is debatable whether Cantor viewed <math>O</math> as an existent set on its own, or if Cantor merely used <math>\equiv O</math> as an emptiness predicate. Zermelo accepted <math>O</math> itself as a set, but considered it an "improper set".<ref>A. Kanamori, "[https://math.bu.edu/people/aki/8.pdf The Empty Set, the Singleton, and the Ordered Pair]", p.275. Bulletin of Symbolic Logic vol. 9, no. 3, (2003). Accessed 21 August 2023.</ref>

=== Axiomatic set theory ===
In [[Zermelo set theory]], the existence of the empty set is assured by the [[axiom of empty set]], and its uniqueness follows from the [[axiom of extensionality]]. However, the axiom of empty set can be shown redundant in at least two ways:
*Standard [[first-order logic]] implies, merely from the [[logical axiom]]s, that {{em|something}} exists, and in the language of set theory, that thing must be a set. Now the existence of the empty set follows easily from the [[axiom of separation]].
*Even using [[free logic]] (which does not logically imply that something exists), there is already an axiom implying the existence of at least one set, namely the [[axiom of infinity]].

=== Philosophical issues ===
While the empty set is a standard and widely accepted mathematical concept, it remains an [[ontological]] curiosity, whose meaning and usefulness are debated by philosophers and logicians.

The empty set is not the same thing as {{em|[[nothing]]}}; rather, it is a set with nothing {{em|inside}} it and a set is always {{em|something}}. This issue can be overcome by viewing a set as a bag—an empty bag undoubtedly still exists. Darling (2004) explains that the empty set is not nothing, but rather "the set of all triangles with four sides, the set of all numbers that are bigger than nine but smaller than eight, and the set of all [[chess opening|opening moves]] in [[chess]] that involve a [[king (chess)|king]]."<ref name="Darling" />

The popular [[syllogism]]
:Nothing is better than eternal happiness; a ham sandwich is better than nothing; therefore, a ham sandwich is better than eternal happiness
is often used to demonstrate the philosophical relation between the concept of nothing and the empty set. Darling writes that the contrast can be seen by rewriting the statements "Nothing is better than eternal happiness" and "[A] ham sandwich is better than nothing" in a mathematical tone. According to Darling, the former is equivalent to "The set of all things that are better than eternal happiness is <math>\varnothing</math>" and the latter to "The set {ham sandwich} is better than the set <math>\varnothing</math>". The first compares elements of sets, while the second compares the sets themselves.<ref name="Darling">{{cite book|title=The Universal Book of Mathematics|author=D. J. Darling|publisher= [[John Wiley and Sons]]|year=2004 |isbn=0-471-27047-4|page=106}}</ref>

[[E. J. Lowe (philosopher)|Jonathan Lowe]] argues that while the empty set
:was undoubtedly an important landmark in the history of mathematics, … we should not assume that its utility in calculation is dependent upon its actually denoting some object.

it is also the case that:
:"All that we are ever informed about the empty set is that it (1) is a set, (2) has no members, and (3) is unique amongst sets in having no members. However, there are very many things that 'have no members', in the set-theoretical sense—namely, all non-sets. It is perfectly clear why these things have no members, for they are not sets. What is unclear is how there can be, uniquely amongst sets, a {{em|set}} which has no members. We cannot conjure such an entity into existence by mere stipulation."<ref name="Lowe">{{cite book|title=Locke|author=E. J. Lowe|publisher= [[Routledge]]|year=2005|page=87}}</ref>

[[George Boolos]] argued that much of what has been heretofore obtained by set theory can just as easily be obtained by [[plural quantification]] over individuals, without [[wikt:reification|reifying]] sets as singular entities having other entities as members.<ref>[[George Boolos]] (1984), "To be is to be the value of a variable", ''[[The Journal of Philosophy]]'' 91: 430–49. Reprinted in 1998, ''Logic, Logic and Logic'' ([[Richard Jeffrey]], and Burgess, J., eds.) [[Harvard University Press]], 54–72.</ref>

==See also==

* {{annotated link|0}}
* {{annotated link|Inhabited set}}
* {{annotated link|Nothing}}
* {{annotated link|Power set}}

==References==
{{reflist}}

==Further reading==
*[[Paul Halmos|Halmos, Paul]], ''[[Naive Set Theory (book)|Naive Set Theory]]''. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. {{ISBN|0-387-90092-6}} (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. {{ISBN|978-1-61427-131-4}} (paperback edition).
*{{cite book|last=Jech|first=Thomas|author-link=Thomas Jech|year=2002|title=Set Theory|edition=3rd millennium|series=Springer Monographs in Mathematics|publisher=Springer|isbn=3-540-44085-2}}
*{{cite book|last=Graham|first=Malcolm|title=Modern Elementary Mathematics|date=1975|publisher=[[Harcourt Brace Jovanovich]]|isbn=0155610392|edition=2nd}}

== External links ==
*{{MathWorld |title=Empty Set |id=EmptySet }}

{{Mathematical logic}}
{{Set theory}}

{{DEFAULTSORT:Empty Set}}
[[Category:Basic concepts in set theory]]
[[Category:0 (number)]]

Commutative ring

2025-05-25T11:41:55Z

173.79.19.248: Fix discussion location

{{short description|Algebraic structure}}
{{merge from|Composition ring| discuss=Talk:Composition ring#Merge?|date=May 2025}}
In [[mathematics]], a '''commutative ring''' is a [[Ring (mathematics)|ring]] in which the multiplication operation is [[commutative]]. The study of commutative rings is called [[commutative algebra]]. Complementarily, [[noncommutative algebra]] is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings.

Commutative rings appear in the following chain of [[subclass (set theory)|class inclusions]]:
{{Commutative ring classes}}

{{Ring theory sidebar}}
{{Algebraic structures |Ring}}

== Definition and first examples ==

=== Definition ===
{{details|topic=the definition of rings|Ring (mathematics)}}
A ''ring'' is a [[Set (mathematics)|set]] <math> R </math> equipped with two [[binary operation]]s, i.e. operations combining any two elements of the ring to a third. They are called ''addition'' and ''multiplication'' and commonly denoted by "<math>+</math>" and "<math>\cdot</math>"; e.g. <math>a+b</math> and <math>a \cdot b</math>. To form a ring these two operations have to satisfy a number of properties: the ring has to be an [[abelian group]] under addition as well as a [[monoid]] under multiplication, where multiplication [[distributive law|distributes]] over addition; i.e., <math>a \cdot \left(b + c\right) = \left(a \cdot b\right) + \left(a \cdot c\right)</math>. The identity elements for addition and multiplication are denoted <math> 0 </math> and <math> 1 </math>, respectively.

If the multiplication is commutative, i.e.
<math display="block">a \cdot b = b \cdot a,</math>
then the ring <math> R </math> is called ''commutative''. In the remainder of this article, all rings will be commutative, unless explicitly stated otherwise.

=== First examples ===
An important example, and in some sense crucial, is the [[integer|ring of integer]]s <math> \mathbb{Z} </math> with the two operations of addition and multiplication. As the multiplication of integers is a commutative operation, this is a commutative ring. It is usually denoted <math> \mathbb{Z} </math> as an abbreviation of the [[German language|German]] word ''Zahlen'' (numbers).

A [[field (mathematics)|field]] is a commutative ring where <math> 0 \neq 1 </math> and every [[0 (number)|non-zero]] element <math> a </math> is invertible; i.e., has a multiplicative inverse <math> b </math> such that <math> a \cdot b = 1 </math>. Therefore, by definition, any field is a commutative ring. The [[rational number|rational]], [[real number|real]] and [[complex number]]s form fields.

If <math> R </math> is a given commutative ring, then the set of all [[polynomial]]s in the variable <math> X </math> whose coefficients are in <math> R </math> forms the [[polynomial ring]], denoted <math> R \left[ X \right] </math>. The same holds true for several variables.

If <math> V </math> is some [[topological space]], for example a subset of some <math> \mathbb{R}^n </math>, real- or complex-valued [[continuous function]]s on <math> V </math> form a commutative ring. The same is true for [[differentiable function|differentiable]] or [[holomorphic function]]s, when the two concepts are defined, such as for <math> V </math> a [[complex manifold]].

== Divisibility ==
In contrast to fields, where every nonzero element is multiplicatively invertible, the concept of [[divisibility (ring theory)|divisibility for rings]] is richer. An element <math> a </math> of ring <math> R </math> is called a [[unit (algebra)|unit]] if it possesses a multiplicative inverse. Another particular type of element is the [[zero divisor]]s, i.e. an element <math> a </math> such that there exists a non-zero element <math> b </math> of the ring such that <math> ab = 0 </math>. If <math> R </math> possesses no non-zero zero divisors, it is called an [[integral domain]] (or domain). An element <math> a </math> satisfying <math> a^n = 0 </math> for some positive integer <math> n </math> is called [[nilpotent element|nilpotent]].

=== Localizations ===
{{Main|Localization of a ring}}
The ''localization'' of a ring is a process in which some elements are rendered invertible, i.e. multiplicative inverses are added to the ring. Concretely, if <math> S </math> is a [[multiplicatively closed subset]] of <math> R </math> (i.e. whenever <math> s,t \in S </math> then so is <math> st </math>) then the ''localization'' of <math> R </math> at <math> S </math>, or ''ring of fractions'' with denominators in <math> S </math>, usually denoted <math> S^{-1}R </math> consists of symbols
{{block indent|1= <math>\frac{r}{s}</math> with <math> r \in R, s \in S </math> }}
subject to certain rules that mimic the cancellation familiar from rational numbers. Indeed, in this language <math> \mathbb{Q} </math> is the localization of <math> \mathbb{Z} </math> at all nonzero integers. This construction works for any integral domain <math> R </math> instead of <math> \mathbb{Z} </math>. The localization <math> \left(R\setminus \left\{0\right\}\right)^{-1}R </math> is a field, called the [[quotient field]] of <math> R </math>.

== Ideals and modules ==
{{hatnote|In the following, R denotes a commutative ring.}}

Many of the following notions also exist for not necessarily commutative rings, but the definitions and properties are usually more complicated. For example, all ideals in a commutative ring are automatically [[two-sided ideal|two-sided]], which simplifies the situation considerably.

=== Modules ===
{{Main|Module (mathematics)|l1=Module}}
For a ring <math> R </math>, an <math> R </math>-''module'' <math> M </math> is like what a vector space is to a field. That is, elements in a module can be added; they can be multiplied by elements of <math> R </math> subject to the same axioms as for a vector space.

The study of modules is significantly more involved than the one of [[vector space]]s, since there are modules that do not have any [[basis (linear algebra)|basis]], that is, do not contain a [[spanning set]] whose elements are [[linearly independent]]s. A module that has a basis is called a [[free module]], and a submodule of a free module needs not to be free.

A [[module of finite type]] is a module that has a finite spanning set. Modules of finite type play a fundamental role in the theory of commutative rings, similar to the role of the [[finite-dimensional vector space]]s in [[linear algebra]]. In particular, [[Noetherian rings]] (see also ''{{slink||Noetherian rings}}'', below) can be defined as the rings such that every submodule of a module of finite type is also of finite type.

=== Ideals ===
{{Main|Ideal (ring theory)|l1=Ideal|Factor ring}}
''Ideals'' of a ring <math> R </math> are the [[submodule]]s of <math> R </math>, i.e., the modules contained in <math> R </math>. In more detail, an ideal <math> I </math> is a non-empty subset of <math> R </math> such that for all <math> r </math> in <math> R </math>, <math> i </math> and <math> j </math> in <math> I </math>, both <math> ri </math> and <math> i+j </math> are in <math> I </math>. For various applications, understanding the ideals of a ring is of particular importance, but often one proceeds by studying modules in general.

Any ring has two ideals, namely the [[0 (number)|zero ideal]] <math> \left\{0\right\} </math> and <math> R </math>, the whole ring. These two ideals are the only ones precisely if <math> R </math> is a field. Given any subset <math> F=\left\{f_j\right\}_{j \in J} </math> of <math> R </math> (where <math> J </math> is some index set), the ideal ''generated by'' <math> F </math> is the smallest ideal that contains <math> F </math>. Equivalently, it is given by finite [[linear combination]]s
<math display="block"> r_1 f_1 + r_2 f_2 + \dots + r_n f_n .</math>

==== Principal ideal domains ====
If <math> F </math> consists of a single element <math> r </math>, the ideal generated by <math> F </math> consists of the multiples of <math> r </math>, i.e., the elements of the form <math> rs </math> for arbitrary elements <math> s </math>. Such an ideal is called a [[principal ideal]]. If every ideal is a principal ideal, <math> R </math> is called a [[principal ideal ring]]; two important cases are <math> \mathbb{Z} </math> and <math> k \left[X\right] </math>, the polynomial ring over a field <math> k </math>. These two are in addition domains, so they are called [[principal ideal domain]]s.

Unlike for general rings, for a principal ideal domain, the properties of individual elements are strongly tied to the properties of the ring as a whole. For example, any principal ideal domain <math> R </math> is a [[unique factorization domain]] (UFD) which means that any element is a product of irreducible elements, in a (up to reordering of factors) unique way. Here, an element <math> a </math> in a domain is called [[irreducible element|irreducible]] if the only way of expressing it as a product
<math display="block"> a=bc ,</math>
is by either <math> b </math> or <math> c </math> being a unit. An example, important in [[Field (mathematics)|field theory]], are [[irreducible polynomial]]s, i.e., irreducible elements in <math> k \left[X\right] </math>, for a field <math> k </math>. The fact that <math> \mathbb{Z} </math> is a UFD can be stated more elementarily by saying that any natural number can be uniquely decomposed as product of powers of prime numbers. It is also known as the [[fundamental theorem of arithmetic]].

An element <math> a </math> is a [[prime element]] if whenever <math> a </math> divides a product <math> bc </math>, <math> a </math> divides <math> b </math> or <math> c </math>. In a domain, being prime implies being irreducible. The converse is true in a unique factorization domain, but false in general.

==== Factor ring ====
The definition of ideals is such that "dividing" <math> I </math> "out" gives another ring, the ''factor ring'' <math> R / I </math>: it is the set of [[coset]]s of <math> I </math> together with the operations
<math display="block"> \left(a+I\right)+\left(b+I\right)=\left(a+b\right)+I </math> and <math> \left(a+I\right) \left(b+I\right)=ab+I </math>.
For example, the ring <math> \mathbb{Z}/n\mathbb{Z} </math> (also denoted <math> \mathbb{Z}_n </math>), where <math> n </math> is an integer, is the ring of integers modulo <math> n </math>. It is the basis of [[modular arithmetic]].

An ideal is ''proper'' if it is strictly smaller than the whole ring. An ideal that is not strictly contained in any proper ideal is called [[maximal ideal|maximal]]. An ideal <math> m </math> is maximal [[if and only if]] <math> R / m </math> is a field. Except for the [[zero ring]], any ring (with identity) possesses at least one maximal ideal; this follows from [[Zorn's lemma]].

=== Noetherian rings ===
{{Main|Noetherian ring}}
A ring is called ''Noetherian'' (in honor of [[Emmy Noether]], who developed this concept) if every [[ascending chain condition|ascending chain of ideals]]
<math display="block"> 0 \subseteq I_0 \subseteq I_1 \subseteq \dots \subseteq I_n \subseteq I_{n+1} \dots </math>
becomes stationary, i.e. becomes constant beyond some index <math> n </math>. Equivalently, any ideal is generated by finitely many elements, or, yet equivalent, [[submodule]]s of finitely generated modules are finitely generated.

Being Noetherian is a highly important finiteness condition, and the condition is preserved under many operations that occur frequently in geometry. For example, if <math> R </math> is Noetherian, then so is the polynomial ring <math> R \left[X_1,X_2,\dots,X_n\right] </math> (by [[Hilbert's basis theorem]]), any localization <math> S^{-1}R </math>, and also any factor ring <math> R / I </math>.

Any non-Noetherian ring <math> R </math> is the [[union (set theory)|union]] of its Noetherian subrings. This fact, known as [[Noetherian approximation]], allows the extension of certain theorems to non-Noetherian rings.

=== Artinian rings ===
A ring is called [[Artinian ring|Artinian]] (after [[Emil Artin]]), if every descending chain of ideals
<math display="block"> R \supseteq I_0 \supseteq I_1 \supseteq \dots \supseteq I_n \supseteq I_{n+1} \dots </math>
becomes stationary eventually. Despite the two conditions appearing symmetric, Noetherian rings are much more general than Artinian rings. For example, <math> \mathbb{Z} </math> is Noetherian, since every ideal can be generated by one element, but is not Artinian, as the chain
<math display="block"> \mathbb{Z} \supsetneq 2\mathbb{Z} \supsetneq 4\mathbb{Z} \supsetneq 8\mathbb{Z} \dots </math>
shows. In fact, by the [[Hopkins–Levitzki theorem]], every Artinian ring is Noetherian. More precisely, Artinian rings can be characterized as the Noetherian rings whose Krull dimension is zero.

== Spectrum of a commutative ring ==

=== Prime ideals ===
{{Main|Prime ideal}}
As was mentioned above, <math> \mathbb{Z} </math> is a [[unique factorization domain]]. This is not true for more general rings, as algebraists realized in the 19th century. For example, in
<math display="block">\mathbb{Z}\left[\sqrt{-5}\right]</math>
there are two genuinely distinct ways of writing 6 as a product:
<math display="block">6 = 2 \cdot 3 = \left(1 + \sqrt{-5}\right)\left(1 - \sqrt{-5}\right).</math>
Prime ideals, as opposed to prime elements, provide a way to circumvent this problem. A prime ideal is a proper (i.e., strictly contained in <math> R </math>) ideal <math> p </math> such that, whenever the product <math> ab </math> of any two ring elements <math> a </math> and <math> b </math> is in <math> p, </math> at least one of the two elements is already in <math> p .</math> (The opposite conclusion holds for any ideal, by definition.) Thus, if a prime ideal is principal, it is equivalently generated by a prime element. However, in rings such as <math>\mathbb{Z}\left[\sqrt{-5}\right],</math> prime ideals need not be principal. This limits the usage of prime elements in ring theory. A cornerstone of algebraic number theory is, however, the fact that in any [[Dedekind ring]] (which includes <math>\mathbb{Z}\left[\sqrt{-5}\right]</math> and more generally the [[algebraic integers|ring of integers in a number field]]) any ideal (such as the one generated by 6) decomposes uniquely as a product of prime ideals.

Any maximal ideal is a prime ideal or, more briefly, is prime. Moreover, an ideal <math>I</math> is prime if and only if the factor ring <math>R/I</math> is an integral domain. Proving that an ideal is prime, or equivalently that a ring has no zero-divisors can be very difficult. Yet another way of expressing the same is to say that the [[Complement (set theory)|complement]] <math>R \setminus p</math> is multiplicatively closed. The localisation <math>\left(R \setminus p\right)^{-1}R</math> is important enough to have its own notation: <math>R_p</math>. This ring has only one maximal ideal, namely <math>pR_p</math>. Such rings are called [[local ring|local]].

=== Spectrum ===
{{Main|Spectrum of a ring}}
[[Image:Spec Z.png|right|400px|thumb|Spec ('''Z''') contains a point for the zero ideal. The closure of this point is the entire space. The remaining points are the ones corresponding to ideals (''p''), where ''p'' is a prime number. These points are closed.]]
The ''spectrum of a ring'' <math>R</math>,{{efn|This notion can be related to the [[Spectrum of an operator|spectrum]] of a linear operator; see ''[[Spectrum of a C*-algebra]]'' and ''[[Gelfand representation]]''.}} denoted by <math>\text{Spec}\ R</math>, is the set of all prime ideals of <math>R</math>. It is equipped with a topology, the [[Zariski topology]], which reflects the algebraic properties of <math>R</math>: a basis of open subsets is given by
<math display="block">D\left(f\right) = \left\{p \in \text{Spec} \ R,f \not\in p\right\},</math> where <math>f</math> is any ring element.
Interpreting <math>f</math> as a function that takes the value ''f'' mod ''p'' (i.e., the image of ''f'' in the residue field ''R''/''p''), this subset is the locus where ''f'' is non-zero. The spectrum also makes precise the intuition that localisation and factor rings are complementary: the natural maps {{nowrap|''R'' → ''R''''f''}} and {{nowrap|''R'' → ''R'' / ''fR''}} correspond, after endowing the spectra of the rings in question with their Zariski topology, to complementary [[open immersion|open]] and [[closed immersion]]s respectively. Even for basic rings, such as illustrated for {{nowrap|1=''R'' = '''Z'''}} at the right, the Zariski topology is quite different from the one on the set of real numbers.

The spectrum contains the set of maximal ideals, which is occasionally denoted mSpec (''R''). For an [[algebraically closed field]] ''k'', mSpec (k[''T''1, ..., ''T''''n''] / (''f''1, ..., ''f''''m'')) is in bijection with the set
{{block indent|1= {''x'' =(''x''1, ..., ''x''''n'') ∊ ''k''''n'' | ''f''1(''x'') = ... = ''f''''m''(''x'') = 0.} }}
Thus, maximal ideals reflect the geometric properties of solution sets of polynomials, which is an initial motivation for the study of commutative rings. However, the consideration of non-maximal ideals as part of the geometric properties of a ring is useful for several reasons. For example, the minimal prime ideals (i.e., the ones not strictly containing smaller ones) correspond to the [[irreducible component]]s of Spec ''R''. For a Noetherian ring ''R'', Spec ''R'' has only finitely many irreducible components. This is a geometric restatement of [[primary decomposition]], according to which any ideal can be decomposed as a product of finitely many [[primary ideal]]s. This fact is the ultimate generalization of the decomposition into prime ideals in Dedekind rings.

=== Affine schemes ===
The notion of a spectrum is the common basis of commutative algebra and [[algebraic geometry]]. Algebraic geometry proceeds by endowing Spec ''R'' with a [[sheaf (mathematics)|sheaf]] <math>\mathcal O</math> (an entity that collects functions defined locally, i.e. on varying open subsets). The datum of the space and the sheaf is called an [[affine scheme]]. Given an affine scheme, the underlying ring ''R'' can be recovered as the [[global section]]s of <math>\mathcal O</math>. Moreover, this one-to-one correspondence between rings and affine schemes is also compatible with ring homomorphisms: any ''f'' : ''R'' → ''S'' gives rise to a [[continuous map]] in the opposite direction
{{block indent|1= Spec ''S'' → Spec ''R'', ''q'' ↦ ''f''−1(''q''), i.e. any prime ideal of ''S'' is mapped to its [[preimage]] under ''f'', which is a prime ideal of ''R''. }}
The resulting [[equivalence of categories|equivalence]] of the two said categories aptly reflects algebraic properties of rings in a geometrical manner.

Similar to the fact that [[manifold (mathematics)|manifolds]] are locally given by open subsets of '''R'''''n'', affine schemes are local models for [[scheme (mathematics)|schemes]], which are the object of study in algebraic geometry. Therefore, several notions concerning commutative rings stem from geometric intuition.

=== Dimension ===
{{Main|Krull dimension}}
The ''Krull dimension'' (or dimension) dim ''R'' of a ring ''R'' measures the "size" of a ring by, roughly speaking, counting independent elements in ''R''. The dimension of algebras over a field ''k'' can be axiomatized by four properties:
* The dimension is a local property: {{nowrap|1=dim ''R'' = supp ∊ Spec ''R'' dim ''R''''p''}}.
* The dimension is independent of nilpotent elements: if {{nowrap|''I'' ⊆ ''R''}} is nilpotent then {{nowrap|1=dim ''R'' = dim ''R'' / ''I''}}.
* The dimension remains constant under a finite extension: if ''S'' is an ''R''-algebra which is finitely generated as an ''R''-module, then dim ''S'' = dim ''R''.
* The dimension is calibrated by dim {{nowrap|1=''k''[''X''1, ..., ''X''''n''] = ''n''}}. This axiom is motivated by regarding the polynomial ring in ''n'' variables as an algebraic analogue of [[affine space|''n''-dimensional space]].

The dimension is defined, for any ring ''R'', as the supremum of lengths ''n'' of chains of prime ideals
{{block indent|1= ''p''0 ⊊ ''p''1 ⊊ ... ⊊ ''p''''n''. }}
For example, a field is zero-dimensional, since the only prime ideal is the zero ideal. The integers are one-dimensional, since chains are of the form (0) ⊊ (''p''), where ''p'' is a [[prime number]]. For non-Noetherian rings, and also non-local rings, the dimension may be infinite, but Noetherian local rings have finite dimension. Among the four axioms above, the first two are elementary consequences of the definition, whereas the remaining two hinge on important facts in [[commutative algebra]], the [[going-up theorem]] and [[Krull's principal ideal theorem]].

== Ring homomorphisms ==
{{Main|Ring homomorphism}}
A ''ring homomorphism'' or, more colloquially, simply a ''map'', is a map {{nowrap|''f'' : ''R'' → ''S''}} such that
{{block indent|1= ''f''(''a'' + ''b'') = ''f''(''a'') + ''f''(''b''), ''f''(''ab'') = ''f''(''a'')''f''(''b'') and ''f''(1) = 1. }}
These conditions ensure {{nowrap|1=''f''(0) = 0}}. Similarly as for other algebraic structures, a ring homomorphism is thus a map that is compatible with the structure of the algebraic objects in question. In such a situation ''S'' is also called an ''R''-algebra, by understanding that ''s'' in ''S'' may be multiplied by some ''r'' of ''R'', by setting
{{block indent|1= ''r'' · ''s'' := ''f''(''r'') · ''s''. }}

The ''kernel'' and ''image'' of ''f'' are defined by {{nowrap|1=ker(''f'') = {{mset|1=''r'' ∈ ''R'', ''f''(''r'') = 0}}}} and {{nowrap|1=im(''f'') = ''f''(''R'') = {{mset|''f''(''r''), ''r'' ∈ ''R''}}}}. The kernel is an [[ring ideal|ideal]] of ''R'', and the image is a [[subring]] of ''S''.

A ring homomorphism is called an isomorphism if it is bijective. An example of a ring isomorphism, known as the [[Chinese remainder theorem]], is
<math display="block">\mathbf Z/n = \bigoplus_{i=0}^k \mathbf Z/p_i ,</math>
where {{nowrap|1=''n'' = ''p''1''p''2...''p''''k''}} is a product of pairwise distinct [[prime number]]s.

Commutative rings, together with ring homomorphisms, form a [[category (mathematics)|category]]. The ring '''Z''' is the [[initial object]] in this category, which means that for any commutative ring ''R'', there is a unique ring homomorphism '''Z''' → ''R''. By means of this map, an integer ''n'' can be regarded as an element of ''R''. For example, the [[binomial formula]]
<math display="block">(a+b)^n = \sum_{k=0}^n \binom n k a^k b^{n-k}</math>
which is valid for any two elements ''a'' and ''b'' in any commutative ring ''R'' is understood in this sense by interpreting the binomial coefficients as elements of ''R'' using this map.

[[File:Tensor product of algebras.png|thumb|The [[universal property]] of {{nowrap|''S'' ⊗''R'' ''T''}} states that for any two maps {{nowrap|''S'' → ''W''}} and {{nowrap|''T'' → ''W''}} which make the outer quadrangle commute, there is a unique map {{nowrap|''S'' ⊗''R'' ''T'' → ''W''}} that makes the entire diagram commute.]]
Given two ''R''-algebras ''S'' and ''T'', their [[tensor product of algebras|tensor product]]
{{block indent|1= ''S'' ⊗''R'' ''T'' }}
is again a commutative ''R''-algebra. In some cases, the tensor product can serve to find a ''T''-algebra which relates to ''Z'' as ''S'' relates to ''R''. For example,
{{block indent|1= ''R''[''X''] ⊗''R'' ''T'' = ''T''[''X'']. }}

=== Finite generation ===
An ''R''-algebra ''S'' is called [[finitely generated algebra|finitely generated (as an algebra)]] if there are finitely many elements ''s''1, ..., ''s''''n'' such that any element of ''s'' is expressible as a polynomial in the ''s''''i''. Equivalently, ''S'' is isomorphic to
{{block indent|1= ''R''[''T''1, ..., ''T''''n''] / ''I''. }}

A much stronger condition is that ''S'' is [[finitely generated module|finitely generated as an ''R''-module]], which means that any ''s'' can be expressed as a ''R''-linear combination of some finite set ''s''1, ..., ''s''''n''.

== Local rings ==
A ring is called [[local ring|local]] if it has only a single maximal ideal, denoted by ''m''. For any (not necessarily local) ring ''R'', the localization
{{block indent|1= ''R''''p'' }}
at a prime ideal ''p'' is local. This localization reflects the geometric properties of Spec ''R'' "around ''p''". Several notions and problems in commutative algebra can be reduced to the case when ''R'' is local, making local rings a particularly deeply studied class of rings. The [[residue field]] of ''R'' is defined as
{{block indent|1= ''k'' = ''R'' / ''m''. }}
Any ''R''-module ''M'' yields a ''k''-vector space given by {{nowrap|''M'' / ''mM''}}. [[Nakayama's lemma]] shows this passage is preserving important information: a finitely generated module ''M'' is zero if and only if {{nowrap|''M'' / ''mM''}} is zero.

=== Regular local rings ===
[[File:Node_(algebraic_geometry).png|thumb|left|The [[cubic plane curve]] (red) defined by the equation ''y''2 = ''x''2(''x'' + ''1'') is [[singularity (mathematics)|singular]] at the origin, i.e., the ring ''k''[''x'', ''y''] / ''y''2 − ''x''2(''x'' + ''1''), is not a regular ring. The tangent cone (blue) is a union of two lines, which also reflects the singularity.]]
The ''k''-vector space ''m''/''m''2 is an algebraic incarnation of the [[cotangent space]]. Informally, the elements of ''m'' can be thought of as functions which vanish at the point ''p'', whereas ''m''2 contains the ones which vanish with order at least 2. For any Noetherian local ring ''R'', the inequality
{{block indent|1= dim''k'' ''m''/''m''2 ≥ dim ''R'' }}
holds true, reflecting the idea that the cotangent (or equivalently the tangent) space has at least the dimension of the space Spec ''R''. If equality holds true in this estimate, ''R'' is called a [[regular local ring]]. A Noetherian local ring is regular if and only if the ring (which is the ring of functions on the [[tangent cone]])
<math display="block">\bigoplus_n m^n / m^{n+1}</math>
is isomorphic to a polynomial ring over ''k''. Broadly speaking, regular local rings are somewhat similar to polynomial rings.{{sfn|Matsumura|1989|p=143|loc=§7, Remarks|ps=}} Regular local rings are UFD's.{{sfn|Matsumura|1989|loc=§19, Theorem 48|ps=}}

[[Discrete valuation ring]]s are equipped with a function which assign an integer to any element ''r''. This number, called the valuation of ''r'' can be informally thought of as a zero or pole order of ''r''. Discrete valuation rings are precisely the one-dimensional regular local rings. For example, the ring of germs of holomorphic functions on a [[Riemann surface]] is a discrete valuation ring.

=== Complete intersections ===
[[File:Twisted_cubic_curve.png|thumb|The [[twisted cubic]] (green) is a set-theoretic complete intersection, but not a complete intersection.]]
By [[Krull's principal ideal theorem]], a foundational result in the [[dimension theory (algebra)|dimension theory of rings]], the dimension of
{{block indent|1= ''R'' = ''k''[''T''1, ..., ''T''''r''] / (''f''1, ..., ''f''''n'') }}
is at least ''r'' − ''n''. A ring ''R'' is called a [[complete intersection ring]] if it can be presented in a way that attains this minimal bound. This notion is also mostly studied for local rings. Any regular local ring is a complete intersection ring, but not conversely.

A ring ''R'' is a ''set-theoretic'' complete intersection if the reduced ring associated to ''R'', i.e., the one obtained by dividing out all nilpotent elements, is a complete intersection. As of 2017, it is in general unknown, whether curves in three-dimensional space are set-theoretic complete intersections.{{sfn|Lyubeznik|1989|ps=}}

=== Cohen–Macaulay rings ===
The [[depth (ring theory)|depth]] of a local ring ''R'' is the number of elements in some (or, as can be shown, any) maximal regular sequence, i.e., a sequence ''a''1, ...,
''a''''n'' ∈ ''m'' such that all ''a''''i'' are non-zero divisors in
{{block indent|1= ''R'' / (''a''1, ..., ''a''''i''−1). }}
For any local Noetherian ring, the inequality
{{block indent|1= depth(''R'') ≤ dim(''R'') }}
holds. A local ring in which equality takes place is called a [[Cohen–Macaulay ring]]. Local complete intersection rings, and a fortiori, regular local rings are Cohen–Macaulay, but not conversely. Cohen–Macaulay combine desirable properties of regular rings (such as the property of being [[universally catenary ring]]s, which means that the (co)dimension of primes is well-behaved), but are also more robust under taking quotients than regular local rings.{{sfn|Eisenbud|1995|loc=Corollary 18.10, Proposition 18.13|ps=}}

== Constructing commutative rings ==
There are several ways to construct new rings out of given ones. The aim of such constructions is often to improve certain properties of the ring so as to make it more readily understandable. For example, an integral domain that is [[integral element#Equivalent definitions|integrally closed]] in its [[field of fractions]] is called [[normal ring|normal]]. This is a desirable property, for example any normal one-dimensional ring is necessarily [[Regular local ring|regular]]. Rendering{{clarify|date=March 2012}} a ring normal is known as ''normalization''.

=== Completions ===
If ''I'' is an ideal in a commutative ring ''R'', the powers of ''I'' form [[neighborhood (topology)|topological neighborhoods]] of ''0'' which allow ''R'' to be viewed as a [[topological ring]]. This topology is called the [[I-adic topology|''I''-adic topology]]. ''R'' can then be completed with respect to this topology. Formally, the ''I''-adic completion is the [[inverse limit]] of the rings ''R''/''In''. For example, if ''k'' is a field, ''k''<nowiki>[[</nowiki>''X''<nowiki>]]</nowiki>, the [[formal power series]] ring in one variable over ''k'', is the ''I''-adic completion of ''k''[''X''] where ''I'' is the principal ideal generated by ''X''. This ring serves as an algebraic analogue of the disk. Analogously, the ring of [[p-adic number|''p''-adic integers]] is the completion of '''Z''' with respect to the principal ideal (''p''). Any ring that is isomorphic to its own completion, is called [[complete ring|complete]].

Complete local rings satisfy [[Hensel's lemma]], which roughly speaking allows extending solutions (of various problems) over the residue field ''k'' to ''R''.

== Homological notions ==
Several deeper aspects of commutative rings have been studied using methods from [[homological algebra]]. {{harvtxt|Hochster|2007}} lists some open questions in this area of active research.

=== Projective modules and Ext functors ===
Projective modules can be defined to be the [[direct summand]]s of free modules. If ''R'' is local, any finitely generated projective module is actually free, which gives content to an analogy between projective modules and [[vector bundle]]s.{{refn|See also [[Serre–Swan theorem]]}} The [[Quillen–Suslin theorem]] asserts that any finitely generated projective module over ''k''[''T''1, ..., ''T''''n''] (''k'' a field) is free, but in general these two concepts differ.
A local Noetherian ring is regular if and only if its [[global dimension]] is finite, say ''n'', which means that any finitely generated ''R''-module has a [[resolution (homological algebra)|resolution]] by projective modules of length at most ''n''.

The proof of this and other related statements relies on the usage of homological methods, such as the
[[Ext functor]]. This functor is the [[derived functor]] of the functor
{{block indent|1= Hom''R''(''M'', −). }}
The latter functor is exact if ''M'' is projective, but not otherwise: for a surjective map {{nowrap|''E'' → ''F''}} of ''R''-modules, a map {{nowrap|''M'' → ''F''}} need not extend to a map {{nowrap|''M'' → ''E''}}. The higher Ext functors measure the non-exactness of the Hom-functor. The importance of this standard construction in homological algebra stems can be seen from the fact that a local Noetherian ring ''R'' with residue field ''k'' is regular if and only if
{{block indent|1= Ext''n''(''k'', ''k'') }}
vanishes for all large enough ''n''. Moreover, the dimensions of these Ext-groups, known as [[Betti number]]s, grow polynomially in ''n'' if and only if ''R'' is a [[local complete intersection]] ring.{{sfn|Christensen|Striuli|Veliche|2010|ps=}} A key argument in such considerations is the [[Koszul complex]], which provides an explicit free resolution of the residue field ''k'' of a local ring ''R'' in terms of a regular sequence.

=== Flatness ===
The [[tensor product]] is another non-exact functor relevant in the context of commutative rings: for a general ''R''-module ''M'', the functor
{{block indent|1= ''M'' ⊗''R'' − }}
is only right exact. If it is exact, ''M'' is called [[flat module|flat]]. If ''R'' is local, any finitely presented flat module is free of finite rank, thus projective. Despite being defined in terms of homological algebra, flatness has profound geometric implications. For example, if an ''R''-algebra ''S'' is flat, the dimensions of the fibers
{{block indent|1= ''S'' / ''pS'' = ''S'' ⊗''R'' ''R'' / ''p'' }}
(for prime ideals ''p'' in ''R'') have the "expected" dimension, namely {{nowrap|dim ''S'' − dim ''R'' + dim(''R'' / ''p'')}}.

== Properties ==
By [[Wedderburn's little theorem|Wedderburn's theorem]], every finite [[division ring]] is commutative, and therefore a [[finite field]]. Another condition ensuring commutativity of a ring, due to [[Nathan Jacobson|Jacobson]], is the following: for every element ''r'' of ''R'' there exists an integer {{nowrap|''n'' > 1}} such that {{nowrap|1=''r''''n'' = ''r''}}.{{sfn|Jacobson|1945|ps=}} If, {{nowrap|1=''r''2 = ''r''}} for every ''r'', the ring is called [[Boolean ring]]. More general conditions which guarantee commutativity of a ring are also known.{{sfn|Pinter-Lucke|2007|ps=}}

== Generalizations ==

=== Graded-commutative rings ===
[[File:Pair_of_pants.png|thumb|A [[pair of pants (mathematics)|pair of pants]] is a [[cobordism]] between a circle and two disjoint circles. Cobordism classes, with the [[cartesian product]] as multiplication and [[disjoint union]] as the sum, form the [[cobordism ring]].]]
A [[graded ring]] {{nowrap|1=''R'' = ⨁''i''∊'''Z''' ''R''''i''}} is called [[graded-commutative ring|graded-commutative]] if, for all homogeneous elements ''a'' and ''b'',
{{block indent|1= ''ab'' = (−1)deg ''a'' ⋅ deg ''b'' ''ba''. }}
If the ''R''''i'' are connected by differentials ∂ such that an abstract form of the [[product rule]] holds, i.e.,
{{block indent|1= ∂(''ab'') = ∂(''a'')''b'' + (−1)deg ''a''a∂(''b''), }}
''R'' is called a [[differential graded algebra|commutative differential graded algebra]] (cdga). An example is the complex of [[differential form]]s on a [[manifold (mathematics)|manifold]], with the multiplication given by the [[exterior product]], is a cdga. The cohomology of a cdga is a graded-commutative ring, sometimes referred to as the [[cohomology ring]]. A broad range examples of graded rings arises in this way. For example, the [[Lazard's universal ring|Lazard ring]] is the ring of cobordism classes of complex manifolds.

A graded-commutative ring with respect to a grading by '''Z'''/2 (as opposed to '''Z''') is called a [[superalgebra]].

A related notion is an [[almost commutative ring]], which means that ''R'' is [[filtration (mathematics)|filtered]] in such a way that the associated graded ring
{{block indent|1= gr ''R'' := ⨁ ''F''''i''''R'' / ⨁ ''F''''i''−1''R'' }}
is commutative. An example is the [[Weyl algebra]] and more general rings of [[differential operator]]s.

=== Simplicial commutative rings ===
A [[simplicial commutative ring]] is a [[simplicial object]] in the category of commutative rings. They are building blocks for (connective) [[derived algebraic geometry]]. A closely related but more general notion is that of [[E-infinity ring|E∞-ring]].

== Applications of the commutative rings ==
* [[Holomorphic function]]s
* [[Algebraic K-theory]]
* [[Topological K-theory]]
* [[Divided power structure]]s
* [[Witt vector]]s
* [[Hecke algebra]] (used in [[Wiles's proof of Fermat's Last Theorem]])
* [[Fontaine's period rings]]
* [[Cluster algebra]]
* [[Convolution algebra]] (of a commutative group)
* [[Fréchet algebra]]

== See also ==
* [[Almost ring]], a certain generalization of a commutative ring
* [[Divisibility (ring theory)]]: [[nilpotent element]], (ex. [[dual number]]s)
* Ideals and modules: [[Radical of an ideal]], [[Morita equivalence]]
* [[Ring homomorphism]]s: [[integral element]]: [[Cayley–Hamilton theorem]], [[Integrally closed domain]], [[Krull ring]], [[Krull–Akizuki theorem]], [[Mori–Nagata theorem]]
* Primes: [[Prime avoidance lemma]], [[Jacobson radical]], [[Nilradical of a ring]], Spectrum: [[Compact space]], [[Connected ring]], [[Differential calculus over commutative algebras]], [[Banach–Stone theorem]]
* [[Local ring]]s: [[Gorenstein local ring]] (also used in [[Wiles's proof of Fermat's Last Theorem]]): [[Duality (mathematics)]], [[Eben Matlis]]; [[Dualizing module]], [[Popescu's theorem]], [[Artin approximation theorem]].

== Notes ==
{{notelist}}

== Citations ==
{{reflist}}

== References ==
{{refbegin}}
* {{Citation | last1=Christensen | first1=Lars Winther | first2=Janet | last2=Striuli | first3=Oana | last3=Veliche | title=Growth in the minimal injective resolution of a local ring | journal=Journal of the London Mathematical Society | series=Second Series | volume=81 | issue=1 | pages=24–44 | year=2010 | doi=10.1112/jlms/jdp058 | arxiv=0812.4672| s2cid=14764965 }}
* {{Citation | last1=Eisenbud | first1=David | author1-link=David Eisenbud | title=Commutative algebra. With a view toward algebraic geometry. | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Graduate Texts in Mathematics]] | isbn=978-0-387-94268-1 | mr=1322960 | year=1995 | volume=150}}
* {{Citation | last=Hochster | first=Melvin | title=Homological conjectures, old and new | journal=Illinois J. Math. | volume=51 | issue=1 | year=2007 | pages=151–169 | doi=10.1215/ijm/1258735330| doi-access=free }}
* {{Citation | doi=10.2307/1969205 | last1=Jacobson | first1=Nathan | author1-link=Nathan Jacobson | title=Structure theory of algebraic algebras of bounded degree | year=1945 | journal=[[Annals of Mathematics]] | issn=0003-486X | volume=46 | issue=4 | pages=695–707 | jstor=1969205}}
* {{Citation | last1=Lyubeznik |first1=Gennady | chapter=A survey of problems and results on the number of defining equations | title=Representations, resolutions and intertwining numbers | pages=375–390 | year=1989 | zbl=0753.14001}}
* {{Citation | last1=Matsumura | first1=Hideyuki | title=Commutative Ring Theory | publisher=[[Cambridge University Press]] | edition=2nd | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-36764-6 | year=1989}}
* {{Citation | last1=Pinter-Lucke | first1=James | title=Commutativity conditions for rings: 1950–2005 | doi=10.1016/j.exmath.2006.07.001 | year=2007 | journal=Expositiones Mathematicae | issn=0723-0869 | volume=25 | issue=2 | pages=165–174| doi-access=free }}
{{refend}}

== Further reading ==
{{refbegin}}
* {{Citation | last1=Atiyah | first1=Michael | author1-link=Michael Atiyah | last2=Macdonald | first2=I. G. | author2-link=Ian G. Macdonald | title=Introduction to commutative algebra | publisher=Addison-Wesley Publishing Co. | year=1969 }}
* {{Citation | last1=Balcerzyk | first1=Stanisław | last2=Józefiak | first2=Tadeusz | title=Commutative Noetherian and Krull rings | publisher=Ellis Horwood Ltd. | location=Chichester | series=Ellis Horwood Series: Mathematics and its Applications | isbn=978-0-13-155615-7 | year=1989}}
* {{Citation | last1=Balcerzyk | first1=Stanisław | last2=Józefiak | first2=Tadeusz | title=Dimension, multiplicity and homological methods | publisher=Ellis Horwood Ltd. | location=Chichester | series=Ellis Horwood Series: Mathematics and its Applications. | isbn=978-0-13-155623-2 | year=1989}}
* {{Citation | last1=Kaplansky | first1=Irving | author1-link=Irving Kaplansky | title=Commutative rings | publisher=[[University of Chicago Press]] | edition=Revised | mr=0345945 | year=1974}}
* {{Citation | last1=Nagata | first1=Masayoshi | author1-link=Masayoshi Nagata | title=Local rings | publisher=Interscience Publishers | series=Interscience Tracts in Pure and Applied Mathematics | isbn=978-0-88275-228-0 |year=1975 | mr=0155856 | orig-year=1962 | volume=13 | pages=xiii+234}}
* {{Citation | last1=Zariski | first1=Oscar | author1-link=Oscar Zariski | last2=Samuel | first2=Pierre | author2-link=Pierre Samuel | title=Commutative Algebra I, II | publisher= D. van Nostrand, Inc. | location=Princeton, N.J. | series=University series in Higher Mathematics | year=1958–60}} ''(Reprinted 1975–76 by Springer as volumes 28–29 of Graduate Texts in Mathematics.)''
{{refend}}

{{Authority control}}

[[Category:Commutative algebra]]
[[Category:Ring theory]]
[[Category:Algebraic structures]]

Colorado hairstreak

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{{Short description|Species of butterfly}}
{{Speciesbox
| image = Colorado Hairstreak, MM.JPG
| image_caption = Upperside
| status = G5
| status_system = TNC
| status_ref = <ref>{{Cite web |title=NatureServe Explorer 2.0 |url=https://explorer.natureserve.org/Taxon/ELEMENT_GLOBAL.2.116716/Hypaurotis_crysalus |access-date=2024-12-08 |website=explorer.natureserve.org}}</ref>
| display_parents = 3
| genus = Hypaurotis
| parent_authority = [[Samuel Hubbard Scudder|Scudder]], 1876
| species = crysalus
| authority = [[William Henry Edwards|W. H. Edwards]], 1873
| range_map = Hypaurotis crysalus range map.JPG
}}

The '''Colorado hairstreak''' ('''''Hypaurotis crysalus''''') is a montane [[butterfly]] native to oak [[scrubland]] in the southwestern United States and northern [[Mexico]]. It was designated the [[state insect]] of [[Colorado]] in 1996.<ref name=stateinsect>{{cite web |url= https://archives.colorado.gov/collections/symbols-emblems |title= Colorado State Archives Symbols & Emblems: State Insect |work= Colorado Department of Personnel and Administration |access-date= 1 December 2024}}</ref> It is the [[Monotypic taxon|only species]] in the [[genus]] '''''Hypaurotis'''''.<ref>[http://ftp.funet.fi/pub/sci/bio/life/insecta/lepidoptera/ditrysia/papilionoidea/lycaenidae/theclinae/hypaurotis/ "''Hypaurotis'' Scudder, 1876"] at Markku Savela's ''Lepidoptera and Some Other Life Forms''</ref>

[[File:Colorado Hairstreak 2, MM.JPG|thumb|left|Underside of the wings]]
[[File:Gambel oak 1.jpg|left|thumb|The Colorado hairstreak is found associated with the [[Gambel oak]].]]

The upperside of the Colorado hairstreak's wings is dark purple with a broad black or dark border. Each wing has orange spots at the lower outer edge, and each hindwing has a thin hairlike tail. The lower side of the wings is pale to dark gray with white and dark markings, an orange patch on the margin of each forewing, and an orange spot with a black center on the hindwing near the tail. It has a wingspan of {{convert|3.1|-|3.8|cm|in|abbr=on}}. The thin hair-like tails that extend from its rear wings is likely the source of the "hairstreak" part of its name.<ref>{{cite web|url=https://butterflies.org/thecoloradohairstreak/|title=Colorado’s State Insect - The Colorado Hairstreak (Hypaurotis crysalus)|publisher=Butterflies.org|last=Hokamp|first=Kathryn|date=|access-date=May 16, 2025}}</ref>

This species depends upon the [[Gambel oak]] (''Quercus gambelii''), which is both the favorite roost of adults and usual food source for [[caterpillar]]s. Eggs are laid singly in late summer on twigs of Gambel oaks or another oak species. The caterpillars emerge in the spring and eat young leaves. Adults feed on [[sap (plant)|tree sap]] and probably [[honeydew (secretion)|honeydew]] secreted by other [[insect]]s rather than on [[flower]]s.{{Citation needed|date=December 2024}} The species has one flight, usually from mid-June to August. It is found mainly in [[Utah]], [[Colorado]], [[Arizona]], and [[New Mexico]], the [[Colorado Plateau]] states, although it can also be found in small portions of [[Nevada]], [[Wyoming]], and [[Durango]], [[Mexico]].{{Citation needed|date=December 2024}}

==References==
{{Reflist}}
* {{cite web | url = http://www.nearctica.com/butter/plate6/Hcrysal.htm | title = Colorado Hairstreak (''Hypaurotis crysalus'') | work = Nearctica | accessdate = 18 June 2006 | url-status = dead | archiveurl = https://web.archive.org/web/20060319072513/http://www.nearctica.com/butter/plate6/Hcrysal.htm | archivedate = 19 March 2006 }}
* {{cite book | title = Field Guide to Western Butterflies, A | edition = 2nd | last = Opler | first = Paul | others = Amy Bartlett Wright (ill.) | publisher = Houghton Mifflin Company | location = Boston; New York | year = 1999 }}

==External links==
*{{commons-inline|italic=1}}

{{Taxonbar|from=Q3815713}}

[[Category:Theclini]]
[[Category:Butterflies of North America]]
[[Category:Fauna of the Western United States]]
[[Category:Colorado Plateau]]
[[Category:Symbols of Colorado]]
[[Category:Butterflies described in 1873]]
[[Category:Taxa named by William Henry Edwards]]


{{Lycaenidae-stub}}

Lake Kivijärvi (South Karelia)

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{{Short description|Lake in the country of Finland}}
{{other uses|Kivijärvi (disambiguation)}}
{{Infobox lake
| name = Lake Kivijärvi
| image = Луумяки 150805.jpg
| alt =
| caption =
| image_bathymetry =
| alt_bathymetry =
| caption_bathymetry =
| location =
| coords = {{coord|60.975|N|27.6734|E|type:waterbody|display=inline,title}}
| lake_type = Natural
| inflow =
| outflow =
| catchment = Kymijoki
| basin_countries = [[Finland]]
| date-built =
| date-flooded =
| length =
| width =
| area = {{convert|76.365|km2|abbr=on}}
| depth = {{convert|5.26|m|ft|abbr=on}}
| max-depth = {{convert|27|m|ft|abbr=on}}
| volume = {{convert|0.402|km3|abbr=on}}
| residence_time =
| shore = {{convert|365.27|km|abbr=on}}
| elevation = {{convert|75.2|m|ft|abbr=on}}
| frozen = December–April
| islands = Haapasalo, Parkkisaari, Eteissaari, Muuraissaari
| cities = [[Luumäki]]

| pushpin_map = Finland
| pushpin_label_position =
| pushpin_map_alt = Location of Lake Kivijärvi in Finland.
| pushpin_map_caption =

| website =
| reference =
}}

'''Lake Kivijärvi''' is a medium-sized [[lake]] in [[South Karelia]], Eastern [[Finland]], with a total area of {{convert|76.4|km2|sqmi|0}}.<ref name=jw>[http://www.jarviwiki.fi/w/index.php?title=Kivij%C3%A4rvi_%2803.021.1.001%29&setlang=en Järviwiki Web Service.] [[Finnish Environment Institute]]. Retrieved 2014-02-26. {{in lang|en}}</ref> The name Kivijärvi is rather common name of the lake. In Finland there are 121 lakes with the same name. This one is the second biggest.<ref>[http://www.jarviwiki.fi/wiki/Lake_statistics Lake statistics]. Järviwiki Web Service. Retrieved 2014-02-25. {{in lang|en}}</ref>

==See also==
*[[List of lakes in Finland]]

==References==
{{Reflist}}

{{DEFAULTSORT:Kivijarvi (South Karelia)}}
[[Category:Lakes of Finland]]

{{SouthernFinland-geo-stub}}

Eustace Jotham

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{{Short description|Recipient of the Victoria Cross}}
{{Use dmy dates|date=April 2014}}
{{Use British English|date=April 2014}}
{{Infobox military person
|name=Eustace Jotham
|honorific_suffix=
|birth_date={{birth date|df=y|1883|11|28}}
|death_date={{death date and age|df=y|1915|1|7|1883|11|28}}
|birth_place= [[Kidderminster]], [[Worcestershire]]
|death_place= [[Tochi Valley]], [[North-West Frontier Province (1901–1955)|North West Frontier]], [[British India]]
|placeofburial= Miranshah Cemetery, [[North Waziristan]]
|image= File:Jotham card.jpg
|image_size=
|caption= Jotham as pictured on a cigarette card
|nickname=
|allegiance={{flag|United Kingdom}}
|serviceyears=1903–1915
|rank=[[Captain (British Army and Royal Marines)|Captain]]
|branch=[[File:British Raj Red Ensign.svg|23px]] [[British Indian Army]]
|commands=
|unit=
{{plainlist|
*[[North Staffordshire Regiment]]
*[[102nd Prince of Wales's Own Grenadiers]]
*[[51st Sikhs (Frontier Force)|51st Sikhs]]
}}
|battles= [[World War I]]{{KIA}}
|awards= [[Victoria Cross]]
|laterwork=
}}
'''Eustace Jotham''' [[Victoria Cross|VC]] (28 November 1883 – 7 January 1915) was an [[England|English]] recipient of the [[Victoria Cross]], the highest and most prestigious award for gallantry in the face of the enemy that can be awarded to [[United Kingdom|British]] and [[Commonwealth of Nations|Commonwealth]] forces.

==Early life and career==
Born in [[Kidderminster]] 28 November 1883, Jotham attended [[Lucton School]] in [[Herefordshire]] for five years and at the age of 16, in 1899, he transferred to [[Bromsgrove School]]. In 1901, at the age of 18, he left to attend the [[Royal Military College, Sandhurst]]. He was commissioned into the [[North Staffordshire Regiment|Prince of Wales's (North Staffordshire) Regiment]] as a [[second lieutenant]] on 22 April 1903<ref>{{London Gazette|issue=27545|page=2530|date=21 April 1903}}</ref> and joined the 2nd Battalion of his regiment, embarking for a tour of duty in India.{{sfn|Gliddon|2005|p=21}}

In 1903 Jotham sailed to India with the North Staffords and served with them until 1905. Under the Indian Army regulations for admission paragraph 13<ref>{{cite book |title=The India List and India Office List 1905 |url=https://archive.org/details/bub_gb_3VQTAAAAYAAJ |access-date=28 May 2009 |year=1905 |publisher=India Office |location=London |pages=[https://archive.org/details/bub_gb_3VQTAAAAYAAJ/page/n247 221]–222}}</ref> British army officers could transfer to the Indian army. Jotham transferred to the [[British Indian Army|Indian Army]] on 23 June 1905 (backdated to his commissioning date in 1903),<ref name=Gazette27864>{{London Gazette|issue=27864|page=9011|date=15 December 1905}}</ref> and promoted to [[first lieutenant|lieutenant]] in the [[102nd Prince of Wales's Own Grenadiers]] on 22 July 1905.<ref name=Gazette27864/> In October 1906 he transferred to the [[51st Sikhs (Frontier Force)|51st Sikhs]] and is listed in the records as a 'double company officer'.<ref>January 1908 Indian Army List</ref> He was promoted [[captain (British Army and Royal Marines)|captain]] on 22 April 1912.<ref>{{London Gazette|issue=28611|page=3797|date=24 May 1912}}</ref>{{sfn|Gliddon|2005|p=21}}

In 1913 he returned to England for what would be his last period of home leave. Just before he returned to India he was a passenger on a train involved in the [[1913 Ais Gill rail accident|accident at Ais Gill]] and participated in rescue operations. He appeared as a witness at the inquiry held immediately after the accident.<ref>{{cite report|author= Major J.W. Pringle |url=http://www.railwaysarchive.co.uk/documents/BoT_AisGill1913.pdf |title=Report by Major J.W. Pringle R.E on the fatal collision that occurred on the 2nd September, 1913, between two passenger trains near Ais Gill, on the Midland Railway |page=41 |date=10 October 1913}}</ref>

On his return to India he was attached to the [[North Waziristan Militia]] during 1914–1915 and it was during this attachment on operations in the [[Tochi Valley]] area of the [[North-West Frontier Province (1901–1955)|North West Frontier]] that he was [[killed in action]] on 7 January 1915. His bravery during these operations earned him a posthumous Victoria Cross.

==Citation==
{{quote|For most conspicuous bravery on 7th January, 1915, at Spina Khaisora (Tochi Valley).

During operations against the Khostwal tribesmen, Captain Jotham, who was commanding a party of about a dozen of the North Waziristan Militia, was attacked in a nullah and almost surrounded by an overwhelming force of some 1,500 tribesmen. He gave the order to retire, and could have himself escaped, but most gallantly sacrificed his own life by attempting to effect the rescue of one of his men who had lost his horse.|The London Gazette, 23 July 1915<ref>{{London Gazette|issue=29240|supp=y|page=7279|date=23 July 1915}}</ref>}}

==Burial and memorials==
He was buried in the [[Miranshah]] Cemetery, [[North Waziristan]], and is commemorated on the [[India Gate|Delhi Memorial (India Gate)]].<ref>{{cite web|url=http://www.cwgc.org/find-war-dead/casualty/1437380|title=Commonwealth War Graves Commission — casualty details|publisher=[[CWGC]]|access-date=26 November 2007}}</ref>{{sfn|Buzzell|1997|p=172}}

On 7 January 2015, the 100th anniversary of his death, a memorial paving stone to Jotham was unveiled at St Mary's and All Saints Church in his home town of Kidderminster.<ref>{{cite web |url=https://www.bbc.co.uk/news/uk-england-hereford-worcester-30710745 |title=Kidderminster VC memorial to Capt Eustace Jotham unveiled |work=BBC News |date=7 January 2015 |access-date=14 April 2015}}</ref> Jotham's VC is held at [[Bromsgrove School]].<ref>{{cite web |url=http://www.memorialstovalour.co.uk/vc568.html |title=VC568 Eustace Jotham |work=Memorials to Valour |access-date=14 April 2015}}</ref>

==References==
{{Reflist}}

==Bibliography==
*{{cite book|year=1997|title=The Register of the Victoria Cross|editor-last=Buzzell|editor-first=Nora|location=[[Cheltenham]], Gloucestershire|publisher=This England Alma House|isbn=0-906324-27-0}}
*{{cite book|last=Gliddon|first=Gerald|year=2005|title=The Sideshows|series=[[VCs of the First World War]]|location=[[Gloucestershire]], [[England]]|publisher=Sutton Publishing|isbn=978-0-7509-2084-1}}

==External links==
*{{Find a Grave|7996537}}

{{Authority control}}

{{DEFAULTSORT:Jotham, Eustace}}
[[Category:1883 births]]
[[Category:1915 deaths]]
[[Category:Military personnel from Worcestershire]]
[[Category:British Indian Army officers]]
[[Category:British World War I recipients of the Victoria Cross]]
[[Category:Graduates of the Royal Military College, Sandhurst]]
[[Category:Indian Army personnel of World War I]]
[[Category:British Indian Army personnel killed in action]]
[[Category:People educated at Lucton School]]
[[Category:People educated at Bromsgrove School]]
[[Category:People from Kidderminster]]
[[Category:North Staffordshire Regiment officers]]
[[Category:20th-century British Army personnel]]

Cakaudrove East (Fijian Communal Constituency, Fiji)

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{{short description|Former electoral division of Fiji}}
{{Politics of Fiji}}

'''Cakaudrove East Fijian Provincial Communal''' is a former [[electoral division]] of [[Fiji]], one of 23 [[communal constituencies]] reserved for [[Fijians|indigenous Fijians]]. Established by the [[1997 Constitution of Fiji|1997 Constitution]], it came into being in 1999 and was used for the [[Elections in Fiji|parliamentary elections]] of [[1999 Fijian general election|1999]], [[2001 Fijian general election|2001]], and [[2006 Fijian general election|2006]]. (Of the remaining 48 seats, 23 were reserved for other ethnic communities, and 25, called [[Open constituencies|Open Constituencies]], were elected by [[universal suffrage]]). The electorate covered eastern areas of [[Cakaudrove Province|Cakaudrove]] [[Local government in Fiji|Province]]; it included south-eastern [[Vanua Levu]] and the nearby islands of [[Taveuni]], [[Rabi Island|Rabi]], and [[Kioa]].

The [[2013 Constitution of Fiji|2013 Constitution]] promulgated by the [[Republic of Fiji Military Forces|Military]]-backed interim government abolished all constituencies and established a form of [[proportional representation]], with the entire country voting as a single electorate.

== Election results ==
In the following tables, the ''primary vote'' refers to [[Electoral system of Fiji|first-preference votes]] cast. The ''final vote'' refers to the final tally after votes for low-polling candidates have been progressively redistributed to other candidates according to pre-arranged electoral agreements (see ''[[electoral fusion]]''), which may be customized by the voters (see ''[[Instant-runoff voting|instant run-off voting]]'').

=== 1999 ===
{| table width="75%" border="1" align="center" cellpadding=3 cellspacing=0 style="margin:5px; border:3px solid;"
|td width="30%" style="border-bottom:3px solid; background:#efefef;"|'''Candidate'''||td width="40%" style="border-bottom:3px solid; background:#efefef;"|'''Political party'''||td width="20%" style="border-bottom:3px solid; background:#efefef;"|'''Votes'''||td width="10%" style="border-bottom:3px solid; background:#efefef;"|'''%'''
|-
|[[Ratu]] [[Inoke Kubuabola]]||[[Soqosoqo ni Vakavulewa ni Taukei]] (SVT)||5,135||78.66
|-
|[[Ratu Inoke Tabualevu]]||[[Christian Democratic Alliance (Fiji)|Christian Democratic Alliance]] (VLV)||1,099||16.84
|-
|[[Josua Gabirieli]]||(UNLP)||294||4.50
|-
|colspan=2|'''Total'''||'''6,528'''||'''100.00'''
|-
|}

=== 2001 ===
{| table width="75%" border="1" align="center" cellpadding=3 cellspacing=0 style="margin:5px; border:3px solid;"
|td width="30%" style="border-bottom:3px solid; background:#efefef;"|'''Candidate'''||td width="40%" style="border-bottom:3px solid; background:#efefef;"|'''Political party'''||td width="20%" style="border-bottom:3px solid; background:#efefef;"|'''Votes'''||td width="10%" style="border-bottom:3px solid; background:#efefef;"|'''%'''
|-
|[[Ratu]] [[Naiqama Lalabalavu]]||[[Conservative Alliance-Matanitu Vanua|Conservative Alliance]] (CAMV)||3,278||56.09
|-
|[[Ratu]] [[Inoke Kubuabola]]||[[Soqosoqo ni Vakavulewa ni Taukei]] (SVT)||1,742||29.81
|-
|[[Ratu]] [[Josefa Lului]]||[[Soqosoqo Duavata ni Lewenivanua]] (SDL)||824||14.10
|-
|colspan=2|'''Total'''||'''5,844'''||'''100.00'''
|-
|}

=== 2006 ===
{| table width="75%" border="1" align="center" cellpadding=3 cellspacing=0 style="margin:5px; border:3px solid;"
|td width="30%" style="border-bottom:3px solid; background:#efefef;"|'''Candidate'''||td width="40%" style="border-bottom:3px solid; background:#efefef;"|'''Political party'''||td width="20%" style="border-bottom:3px solid; background:#efefef;"|'''Votes'''||td width="10%" style="border-bottom:3px solid; background:#efefef;"|'''%'''
|-
|[[Ratu]] [[Naiqama Lalabalavu]]||[[Soqosoqo Duavata ni Lewenivanua]] (SDL)||6,120||88.86
|-
|[[Melania Lutuvakula]]||[[National Alliance Party of Fiji|National Alliance Party]] (NAPF)||480||6.97
|-
|[[Vilimone Vosarogo]]||Independent||287||4.17
|-
|colspan=2|'''Total'''||'''6,887'''||'''100.00'''
|-
|}

== Sources ==
* [http://psephos.adam-carr.net/countries/f/fiji/fiji2001index.shtml Psephos - Adam Carr's electoral archive]
* [http://www.fijihosting.com/pcgov/fiji_facts.htm Fiji Facts]

{{Electoral divisions of Fiji}}

Hans Dahl

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{{Short description|Norwegian painter (1849–1937)}}
{{distinguish|Hans Fredrik Dahl|Hans Normann Dahl|Hans Dahl (footballer)}}
{{ Infobox artist
| name = Hans Dahl
| image = Dahl Hans Selbstportrait.jpg
| caption =
| birth_date = 19 February 1849
| birth_place = [[Granvin]], [[Hardangerfjord]], [[Sweden-Norway]]
| death_date = 27 July 1937 (aged 88)
| death_place = [[Balestrand]], [[Sogn og Fjordane]], Norway
| nationality = Norwegian
| field = [[Painting]]
| training = [[Kunstakademie Düsseldorf]]
| movement = Norwegian romanticist
| works = Landscape paintings
}}

'''Hans Dahl''' (19 February 1849 – 27 July 1937) was a Norwegian painter. He was best known for his paintings of Norwegian fjords and surrounding landscapes.<ref>{{cite web|url= https://nkl.snl.no/Hans_Dahl |title= Hans Dahl (1849–1937) |website= Norsk Kunstnerleksikon
|date= 20 February 2017 |access-date=December 1, 2020}}</ref>

==Background==
Hans Dahl was born in the village of [[Granvin]], on the [[Hardangerfjord]] in [[Hordaland]], Norway. His talent was already evident when Dahl was 16 years old. However, it was only after service in the army that Dahl received artistic education. Hans Dahl was educated first to become an officer and became a lieutenant in 1871. He served in the Bergenske Brigade until 1874. After leaving the army, he apprenticed with [[Johan Fredrik Eckersberg]] and [[Knud Bergslien]]. He went to [[Karlsruhe]], where he studied under [[Hans Fredrik Gude]] and {{ill|Wilhelm Riefstahl|de}} and then to [[Düsseldorf]], where his teachers included [[Eduard von Gebhardt]] and [[Wilhelm Sohn]]. His art became associated with the [[Düsseldorf school of painting]], which was characterized by finely detailed yet still fanciful landscapes.
<ref>{{cite web|url=https://lokalhistoriewiki.no/wiki/Hans_Dahl_%281849–1937%29 |title= Hans Dahl (1849–1937) |website= lokalhistoriewiki.no |access-date=December 1, 2020}}</ref>

==Career==
Dahl had his first exhibition in Düsseldorf in 1876. Dahl lived in [[Düsseldorf]] until 1888, when he moved to Berlin. Almost every summer, he was back to Norway. In 1893, he commissioned the firm of [[Jacob Digre]] in [[Trondheim]] to build his summer residence, {{ill|Villa Strandheim|nn}}. It is located on the banks of the [[Sognefjord]] at [[Balestrand]] in the county of [[Sogn og Fjordane]]. Norwegian painter [[Adelsteen Normann]] had settled in Balestrand during 1891. Dahl's villa was of a similar design to the villa built for Normann.<ref>[http://www.nrk.no/nyheter/distrikt/nrk_sogn_og_fjordane/fylkesleksikon/1910231.html ''Villa Strandheim'' (NRK Sogn og Fjordane)] {{webarchive|url=https://web.archive.org/web/20121102035946/http://www.nrk.no/nyheter/distrikt/nrk_sogn_og_fjordane/fylkesleksikon/1910231.html |date=2012-11-02 }}</ref>

Between 1888 and 1919, Dahl lived mostly in Berlin-Wilmersdorf, and spent only the summer in Balestrand. After 1919, Dahl no longer traveled to Berlin. Balestrand became a much visited tourist destination in western Norway. German emperor, [[Kaiser Wilhelm II]], became one of Dahl's patrons, conferring a professorship upon him in 1910, and visiting Dahl in Balestrand several summers in a row.<ref>[http://www.sffarkiv.no/sffbasar/default.asp?p=result&db=dbatlas_leks&art_id=828&spraak_id=2&ptype=single&show_kart=no&lang=eng The artist Hans Dahl ] ''Kulturhistorisk Leksikon''</ref>
[[File:Bundesarchiv N 1572 Bild-1925-127, Polarfahrt Dampfer "München", Maler Hans Dahl.jpg|thumb|250px| Hans Dahl in front of his villa in Balholmen]]
Dahl resisted the transition in art from [[romanticism]] to [[modernism]]. In the 1890s a new school of art arose, and artists like Dahl were not very popular in the leading circles in the capital. He was particularly criticised by the art historian [[Jens Thiis]]. He was severely criticized by fellow artists especially by [[Christian Krohg]], who was one of the leading figures in the transition from romanticism to [[Naturalism (visual arts)|naturalism]] which characterized Norwegian art in this period. Throughout his life, he increasingly narrowed his range of topics. Dahl often described the scenery of the western part of Norway in brilliant sunshine with smiling people in national costumes. His vibrant colors and charming portrayals of young Norwegian girls in their national costume have always been very popular.

==Personal life==
Hans Dahl was married to Helene Bewer, the daughter of the German painter [[Clemens Bewer]] (1820–1884). Their son was the Norwegian painter {{ill|Hans Andreas Dahl|de}} (1881–1919), who died of tuberculosis at the age of 37.

In 1902 Dahl was appointed knight of the [[Royal Norwegian Order of St. Olav]], 1st class. Hans Dahl died in [[Balestrand]] in [[Sogn og Fjordane]] during 1937. His grave is located in the churchyard of [[Tjugum Church]].<ref>[https://runeberg.org/salmonsen/2/5/0435.html Hans Dahl] ''Salmonsens konversationsleksikon''</ref>

<gallery mode=packed heights="135">
File:Dahl Hans (Norwegian) 1849 to 1937 By The FJord O C 49.5 by 67.3 cm.jpg|''By the Fjord''
File:Hans Dahl - By the Water's Edge.jpg|''By the Water's Edge''
File:Hans Dahl - Fiord with sailing boat.jpg|''Fjord with sailing boat''
File:Hans Dahl - Upon sunny waves.jpg|''Upon Sunny Waves''
File:Dahl Hans - Beim Heutransport 52x36cm.jpg|''Transporting Hay ''
File:Dahl Hans 1901 41x68cm.jpg|''Boating in Balestrand''
File:On the Mountaintop.jpg|''On the Mountaintop''
File:Dahl Hans - Sognefjord von seinem Grundstück in Balestrand 1920 21x30cm.jpg|''Sognefjord Near Summer Home in Balestrand''
File:Hans-dahl peasant-girl.jpg|''Peasant Girl''
File:Hans Dahl - On the banks of the Fjord.jpg|''On the banks of the Fjord''
File:Hans-Dahl sommerwind.jpg|''Sommerwind''
File:On the Fjord by Hans Dahl.jpg|''On the Fjord''
</gallery>

==References==
{{Commons category|Hans Dahl}}
{{Reflist}}

==External links==
{{Spoken Wikipedia|En-hansdahl.ogg|date=2005-12-10}}
*[https://web.archive.org/web/20121102035953/http://www.nrk.no/sf/leksikon/index.php/Villa_Strandheim Villa Strandheim]
*[http://www.himmeljord.no/kirke.aspx?id=3 Tjugum kyrkje]
* {{Cite EB1911|wstitle=Dahl, Hans|volume=7|page=731}}

{{Düsseldorf school of painting}}
{{Authority control (arts)}}

{{DEFAULTSORT:Dahl, Hans}}
[[Category:1849 births]]
[[Category:1937 deaths]]
[[Category:People from Granvin]]
[[Category:Kunstakademie Düsseldorf alumni]]
[[Category:Düsseldorf school of painting]]
[[Category:Norwegian landscape painters]]
[[Category:19th-century Norwegian painters]]
[[Category:20th-century Norwegian painters]]
[[Category:Norwegian male painters]]
[[Category:Norwegian expatriates in Germany]]
[[Category:Order of Saint Olav]]
[[Category:19th-century Norwegian male artists]]
[[Category:20th-century Norwegian male artists]]

Nicholas Grimald

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{{short description|16th-century English poet and dramatist}}
{{EngvarB|date=July 2017}}
{{Use dmy dates|date=July 2017}}
'''Nicholas Grimald''' (or '''Grimoald''') (1519–1562) was an English poet and dramatist.

==Life==
{{Marshalsea}}
Nicholas Grimald was born to a modest yeoman family of farmers in 1519–20. His parents are unknown, despite the popular belief that his father was Giovanni Baptista Grimaldi. The poem ''A funeral song, upon the death of Annes his {{not a typo|moother}}'', accounts for his mother's death. Grimald's mother has been speculated to be Agnes Gyrmbold, who dies in 1555. The poem mentions his hardworking father, but focuses upon the fondness that he had for his mother. Grimald saw his parents' love and devotion for each other and expressed that in the poem. He was the only boy out of many girls.

Maps found of Huntingdonshire in the 16th century do not show a place that could be Brownshold. What was found was an estate named [[Leighton Bromswold]]. This is the closest match to how Grimald describes his home in the poem ''A funeral song, upon the death of Annes his {{not a typo|moother}}''.

The Grimald family was seen to be living in Leighton Bromswold for close to four hundred years. Their last name had gone through variations as shown by legal documents kept by the village. Variations include: Grymbaud, Grymbold, Grymbolde, Grimbold, Grimald, Grimbald, and Grymbalde.<ref name=Grimald-Merrill-1925 >{{harvnb|Grimald|Merrill|1925|p=}}{{page needed|date=January 2014}}.</ref>

There has been no evidence of Grimald ever being married or having children. Some of his contributed verses in Tottle's ''Songes and Sonnettes'' refer to two women Grimald my have admired, Carie Day and Mistress Damascene Awdley. These verses show a debate on if marriage has the desired result that is wanted.{{sfn|Brennan|2004}}

Grimald's connection to [[Nicholas Ridley (martyr)]], bishop of London, brought him under suspicion, and he was imprisoned in the [[Marshalsea]]. It is said that he escaped the penalties of [[heresy]] by recanting his errors, and was despised accordingly by his [[Protestant]] contemporaries.{{sfn|Chisholm|1911|p=598}}

The exact date, cause, and location of his death are unknown. [[Barnabe Googe]], in his ''Eglogs, Epythaphes, and Sonettes'' (1563), included ''An Epytaphes on the Death of Nicholas Grimaold'', which was written before Googe's departure abroad in 1562.{{sfn|Brennan|2004}}{{sfn|Merrill|1926|pp=483–485}}

==Education and early career==
When Nicholas Grimald was fifteen years old, he started showing signs of poetic talent. His mother, who placed an emphasis on education, sent him to continue his education at Cambridge. This separation was hard for Grimald's mother, as he was the only boy in the family.

In 1539–40, Grimald graduated from [[Christ's College, Cambridge|Christ's College]] with a B.A.. Grimald travelled to [[Oxford]] a year later, when the prebendary of [[Leighton Bromswold]] Gilbert Smith, a family friend, was impressed by his work. In his first few years in Oxford, Grimald attended [[Brasenose College]]. When Grimald was unable to continue his school work because he lacked his books, his first drama was written. Grimald received encouragement from the Matthew Smith, the president of the school, along with other teachers and students who were eager to participate in the arts. The play was the [[Latin]] resurrection play ''Christus Redivivus''. Grimald dedicated his work to Smith. It was written in 1541, but was published two years later in Germany in 1543.

At twenty-three years old in 1542, Grimald was able to acquire his B.A. at Oxford. This degree allowed his admission into [[Merton College]], where he received his M.A. in 1544. In that same year, Grimald travelled back to Cambridge to get his M.A. at Christ's College.<ref name=Grimald-Merrill-1925/> Grimald chose to stay in Oxford after his degrees in 1544. His next work was another Latin tragedy based on the life and death of [[John the Baptist]] called ''Archipropheta'' in 1548. Grimald was licensed as a lecturer in 1552 by Richard Sampson, this allowed him to preach at Eccles. The next year he was appointed chaplain to [[Nicholas Ridley (martyr)|Nicholas Ridley]], bishop of London. Ridely's high opinion of Grimald was shown when the bishop chose him to deliver a Latin address in April 1553, ''Oratio ad pontifices''.{{sfn|Brennan|2004}}

==Involvement with Nicholas Ridley==

Nicholas Grimald was appointed chaplain to Nicholas Ridley, bishop of London, in 1552. Previously, in a letter written to Sir John Gates and Sir William Cecil, Ridley praised Grimald for his preferment and commended him for his "eloquence in both English and Latin" (Matthew; Harrison 13). Ridley's high opinion was later confirmed when the Bishop appointed Grimald to deliver the Latin address in April 1553, at an assembled bishops' meeting on absentee clergy.

The possible apostasy under Mary I put considerable strife on their relationship, and ultimately lead to its demise. Following Ridley's successful recommendation for Grimald to be appointed chantership of [[Old St Paul's Cathedral|St Paul's Cathedral]], Queen Mary acceded to the throne. Queen Mary's brutal prosecution of Protestants led to Ridley's imprisonment. While in prison, Ridley wrote to Grimald and potentially sent him [[Lorenzo Valla]]'s [[Lorenzo Valla#On the Donation of Constantine|''De falso credita et ementita Constantini Donatione declamatio'']], a text denouncing the pope's claim to dominion (13). The correspondence caught the attention of the Catholic authority and consequently led to Grimald's imprisonment in the [[Marshalsea]] prison in 1555 (Stephen;Lee 1917).

Grimald's imprisonment was brief, and many speculated the poet's apostasy to Catholicism. Originally, Grimald quelled the accusations. In response to rumours of Grimald's collusion with the Catholic Church, Ridley said, "it will not sink into my head to think that Grimbol would ever play me such a Judas's part" (Matthew;Harrison 13). Ridley later confirmed Grimald's betrayal, and said, in response Grimald's imprisonment and apostasy to Catholicism, that "he (Grimald) escaped not without some becking and bowing (alas) of is knee unto [[Baal]]" (13). It is unclear whether or not there was reconciliation prior to Ridley's execution.{{sfn|Brennan|2004}}{{sfn|Lee|1886|pp=249–250}}

==Works==

Upon arriving at Oxford, Grimald began his first major work; the Latin resurrection and tragicomedy ''Christus Redivus'' or ''The Resurrection of Christ''. His motivation for starting the play was to redeem not being able to perform his studies because his books had come late. When proposing the idea of writing a play, Grimald received support from his peers, teachers, and even the college's president. The story of the play consisted of the resurrection of Christ, and is thought to have been made for an Eastertime performance. It is believed Grimald wrote a companion play later in his life called ''Christus nascens'', a nativity play made to be performed for Christmas. ''Christus Redivus'' was dedicated to the Archdeacon of Peterborough Gilbert Smith, and was eventually published in 1543 at Cologne.{{sfn|Brennan|2004}}<ref name=Grimald-Merrill-1925/>

It cannot be determined whether Grimald was familiar with [[George Buchanan (humanist)|George Buchanan]]'s ''Baptistes'' (1543), or with Jakob Schöpper's ''Johannes decollatus vel Ectrachelistes'' (1546). Grimald provides a purely romantic motive for the catastrophe in the passionate attachment of [[Herodias]] to [[Herod Antipas]], and constantly resorts to lyrical methods. As a poet Grimald is memorable as the earliest follower of [[Henry Howard, Earl of Surrey]] in the production of [[blank verse]]. He writes sometimes simply enough, as in the lines on his own childhood addressed to his mother, but in general his style is more artificial, and his metaphors more studied than is the case with the other contributors to the ''Miscellany''. His classical reading shows itself in the comparative terseness and smartness of his verses.{{sfn|Chisholm|1911|p=598}}

''Archipropheta'' or ''The Archprophet'', published in 1548, was a Latin drama that depicted the life of John the Baptist. Grimald submitted the play with his application for a fellowship with the new Dean of [[Christ Church, Oxford|Christ Church]], Dr Richard Cox. The application got him a job at the college as "a senior or theologist" where he would give lectures on rhetoric. ''Archipropheta'' is one of the first tragedies to be penned by an Englishman.<ref name=Grimald-Merrill-1925/>

Grimald's name is attributed to the work ''Vox Populi'', or ''The People's Complaint'', published in 1549. The piece of writing criticised Church officials for failing to carry out their jobs correctly.{{sfn|Lee|1886|pp=249–250}} According to Merrill, the work revolves around "those rectors, vicars, archdeacons, deans, prebendaries, etc., who spend their lives far from their flocks, or do not perform their sacred duties."<ref name=Grimald-Merrill-1925/>

He translated [[Cicero]]'s ''[[De Officiis]]'' as ''Marcus Tullius Ciceroes thre bokes of duties'' (1556); a [[Latin]] paraphrase of [[Virgil]]'s ''[[Georgics]]'' (printed 1591) is attributed to him, but most of the works assigned to him by [[John Bale]] are lost. Two Latin tragedies are extant: ''Archipropheta sive Johannes Baptista'', printed at Cologne in 1548, probably performed at Oxford the year before, and ''Christus redivivus'' (Cologne, 1543), edited by JM Hart (for the Modern Language Association of America, 1886, separately issued 1899).{{sfn|Chisholm|1911|p=598}}

Grimald contributed forty poems to the original edition (June 1557) of ''[[Songes and Sonettes]]'' (commonly known as Tottel's ''Miscellany'').{{sfn|Chisholm|1911|p=598}} Two of Grimald's poems printed in ''Miscellany'', ''The Death of Zoroas'' and ''Marcus Tullius Ciceroes Death'', are regarded as some of the first examples of English blank verse ever published. It is also possible that Grimald was an editor of the first edition of ''Miscellany''. It is speculated that most of Grimald's work was removed from the second edition due to his recantation of Protestantism. Tottel feared that Grimald's recantation would cause book sales to decline, and thus decided to omit many of Grimald's original works from the updated version. All of Grimald's more personal works were left out of the second edition, and only his initials were used to signify his authorship of the nine remaining poems.{{sfn|Brennan|2004}}

==Bibliography==

* ''Christus Redivivus'' (1543)
* ''Archipropheta'' (1548)
* ''Vox Popli'' (1549)
* ''Marcus Tullius Ciceroes thre bokes of duties'' (Translation of Cicero's ''De Officiis'') (1556)
* Contributed poems to the collection ''Songes & Sonettes'' (1557)

==See also==
{{portal|Poetry}}
*[[Canons of Elizabethan poetry]]

==Notes==
{{reflist|30em}}

==References==
*{{Cite ODNB |last=Brennan |first=Michael G. |title=Grimald, Nicholas (''b.'' 1519/20, ''d.'' in or before 1562) |id=11629 |year=2004 }}
*{{cite book|last1=Grimald |first1=Nicholas |last2=Merrill |first2=LeRoy |year=1925 |title=The Life and Poems of Nicholas Grimald |location=New Haven |publisher=Yale University Press}}
*{{cite journal|last=Merrill |first=L. R. |title=Nicholas Grimald The Life and Poems of Nicholas Grimald |editor-first=A. W |editor-last=Reed |journal=The Review of English Studies |volume=2 |number=8 |date=October 1926 |pages=483–485 |publisher=Oxford University Press |jstor=507970}}
*{{Cite DNB|last=Lee |first=Sidney |wstitle=Grimald, Nicholas |volume=8 |pages=249–250}}

'''Attribution:'''
*{{EB1911|wstitle=Grimald, Nicholas|volume=12 |page=598}} Endnotes:
** [[C. H. Herford]], ''Studies in the Literary Relations of England and Germany'' (pp. 113–119, 1886).
** ''A Catalogue of printed books ... by writers bearing the name of Grimaldi'' (ed. A. B. Grimaldi), printed 1883
** [[Edward Arber]]'s reprint of [[Tottel's Miscellany]].

{{Authority control}}

{{DEFAULTSORT:Grimald, Nicholas}}
[[Category:1519 births]]
[[Category:1562 deaths]]
[[Category:People from Huntingdonshire]]
[[Category:Alumni of Christ's College, Cambridge]]
[[Category:English Renaissance dramatists]]
[[Category:16th-century English poets]]
[[Category:Inmates of the Marshalsea]]
[[Category:16th-century English dramatists and playwrights]]
[[Category:16th-century English male writers]]
[[Category:English male dramatists and playwrights]]
[[Category:English male poets]]

Jesse Carleton

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{{short description|American amateur golfer}}
{{Use mdy dates|date=January 2020}}
{{Infobox golfer
| name = Jesse Carleton
| image = Jesse Lee Carleton.png
| alt =
| caption =
| fullname = Jesse Lee Carleton
| birth_date = {{Birth date|1862|08|20}}
| birth_place = [[Cumberland, Maryland]], U.S.
| death_date = {{Death date and age|1921|12|06|1862|08|20}}
| death_place = [[St. Louis|St. Louis, Missouri]], U.S.
| spouse =
| children =
| status = Amateur
| show-medals = yes
| medaltemplates =
{{Medal|Competition|[[Olympic Games]]}}
{{MedalSport | Men's [[Golf at the Summer Olympics|Golf]]}}
{{MedalBronze| [[1904 Summer Olympics|1904 St. Louis]]|[[Golf at the 1904 Summer Olympics – Men's team|Team]]}}
}}
'''Jesse Lee Carleton''' (August 20, 1862 – December 6, 1921) was an American [[golf]]er who competed in the [[1904 Summer Olympics]].<ref>{{cite web|url=https://www.olympedia.org/athletes/18151 |title=Jesse Carleton |work=Olympedia |accessdate=July 4, 2020}}</ref>

==Biography==
Jesse Carleton was born in [[Cumberland, Maryland]].<ref name=Dies>{{Cite news |url=https://www.newspapers.com/clip/41711314/jesse_l_carleton_dies_suddenly_at_his/ |title=Jesse L. Carleton Dies Suddenly At His Home |work=[[St. Louis Post-Dispatch]] |page=31 |date=1921-12-07 |access-date=2020-01-04 |via=Newspapers.com}}</ref>

In 1904 he was part of the American team which won the bronze medal. He finished twelfth in this competition. In the [[Golf at the 1904 Summer Olympics – Men's individual|individual competition]] he finished 16th in the qualification and was eliminated in the first round of the match play.

Outside of golf, he owned the Carleton Dry Goods Company. He was president of the Missouri State Golf Association for three years, and president of the Sunset Hill Country Club for three years.<ref name=Dies/>

Jesse Carleton died from a cerebral hemorrhage at his home in St. Louis on December 6, 1921.<ref name=Dies/>

==References==
{{Reflist}}

==External links==
* [https://web.archive.org/web/20070214063525/http://www.databaseolympics.com/players/playerpage.htm?ilkid=CARLEJES01 Profile]
{{Commons category}}
*{{sports links}}

{{DEFAULTSORT:Carleton, Jesse}}
[[Category:American male golfers]]
[[Category:Amateur golfers]]
[[Category:Golfers at the 1904 Summer Olympics]]
[[Category:Olympic bronze medalists for the United States in golf]]
[[Category:Medalists at the 1904 Summer Olympics]]
[[Category:Golfers from Maryland]]
[[Category:Sportspeople from Cumberland, Maryland]]
[[Category:1862 births]]
[[Category:1921 deaths]]
[[Category:20th-century American sportsmen]]

{{US-golf-bio-stub}}

Puri, Angola

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{{Infobox settlement


|name = Puri
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|population_urban_footnotes = <ref>[https://www.citypopulation.de/en/angola/cities/ Citypopulation.de] Population of cities & urban localities in Angola</ref>
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''' Puri ''' is a town and [[Municipalities of Angola|municipality]] in [[Uíge Province]] in [[Angola]].<ref>{{cite web|url=http://www.statoids.com/yao.html|title=City councils of Angola|publisher=Statoids|access-date=April 9, 2009}}</ref> The municipality had a population of 37,910 in 2014.<ref name =Censo2014>{{cite web|url=https://www.ine.gov.ao/images/Censo_2014/Publicacao__Uige__14_11_2016.pdf |title=Resultados Definitivos Recenseamento Geral da População e Habitação – 2014 Província do Uíge |access-date=3 May 2020 |website=Instituto Nacional de Estatística, República de Angola| df=dmy }}</ref>

==References==
{{Reflist}}

{{Municipalities of Angola}}

[[Category:Populated places in Uíge Province]]
[[Category:Municipalities of Angola]]

{{Angola-geo-stub}}

Melakadambur

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{{Use Indian English|date=January 2023}}
{{Use British English|date=October 2011}}
{{Use dmy dates|date=February 2021}}
{{more citations needed|date=September 2014}}
{{cleanup rewrite|date=November 2018}}
{{Infobox settlement
| name = Melakadambur
| native_name =
| native_name_lang =
| other_name = Kadambur
| nickname =
| settlement_type = village
| image_skyline = Profile view, Amrithakadeswarar temple Melakadambur.jpg
| image_alt =
| image_caption = [[Mela Kadambur Amirthakadeswarar Temple|Amirthakadeswarar Temple]]
| pushpin_map = India Tamil Nadu#India
| pushpin_label_position = left
| pushpin_map_alt =
| pushpin_map_caption = Location in Tamil Nadu, India
| coordinates = {{coord|11.14|N|79.31|E|display=inline,title}}
| subdivision_type = Country
| subdivision_name = {{flag|India}}
| subdivision_type1 = [[States and union territories of India|State]]
| subdivision_name1 = [[Tamil Nadu]]
| subdivision_type2 = [[List of districts in India|District]]
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| area_footnotes =
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| area_total_km2 = 5
| elevation_footnotes =
| elevation_m = 5
| population_total = 2000 approx.
| population_as_of = 2008
| population_rank =
| population_density_km2 = 160
| population_demonym =
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| demographics_type1 = Languages
| demographics1_title1 = Official
| demographics1_info1 = [[Tamil language|Tamil]]
| timezone1 = [[Indian Standard Time|IST]]
| utc_offset1 = +5:30
| postal_code_type = [[Postal Index Number|PIN]]
| postal_code = 608304
| area_code_type = Telephone code
| area_code = 04144 std
| registration_plate = TN-31
| blank1_name_sec1 = Nearest city
| blank1_info_sec1 = Chidambaram
| blank2_name_sec1 = [[Human sex ratio|Sex ratio]]
| blank2_info_sec1 = 1:1 [[male|♂]]/[[female|♀]]
| blank3_name_sec1 = Literacy
| blank3_info_sec1 = 50%
| blank4_name_sec1 = [[Lok Sabha]] constituency
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| blank5_name_sec1 = [[Vidhan Sabha]] constituency
| blank5_info_sec1 = Kattumannarkoil
| blank1_name_sec2 = [[Climate of India|Climate]]
| blank1_info_sec2 = [[Climatic regions of India|as on plain]] ([[Köppen climate classification|Köppen]])
| website =
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}}
'''Melakadambur''' is a village located {{convert|31|km|mi}} from [[Chidambaram]], and can reach at 6 km from Kattumannarkoil, [[Cuddalore district]], [[Tamil Nadu]], India. It is known for [[Mela Kadambur Amirthakadeswarar Temple|Arulmigu Amirthakateshwarar Thirukkoil]] temple, constructed during the reign of [[Chola]]s by [[Kulottunga I]]. The temple and the village are well known in Ponniyin Selvan history, as they are more than 1,000 years old. The [[Shiva|Sivan]] temple at Melakadambur is a [[Paadal Petra Sthalam]]. This Sivalinga is self-manifested or [[Swayambhu]] Linga.

==''Ponniyin Selvan''==
''[[Ponniyin Selvan]]'' is a historical novel written by Tamil writer [[Kalki|Kalki Krishnamurthy]], from Tamil Nadu, who is also known as Krishnamurthy. This historical novel appeared in serial in the Tamil weekly ''Kalki'', in the name of the author, many times and attracted many readers.<ref>{{cite news|title=Age hardly withers charm of Ponniyin Selvan|url=http://www.thehindu.com/books/age-hardly-withers-charm-of-ponniyin-selvan/article2550847.ece|work=The Hindu|date=10 July 2011|accessdate=2014-09-19}}</ref>

==Devara shrine==
It is a Devara shrine of north shored [[Kaveri]] river. There are 63 shrines along the North bank of River [[Kaveri]], including Melakadambur that lists as the 34th. The well known Nalvar praised the Lord of this shrine and made three pathigams on him. Also, Arunagirinathar, Vannacharabamn Thandapani Swamigal, and Vallalar came here to sing in praise of this Lord. In every ''[[yuga]]'', some Devas visited this place to worship this Lord Amirthakadeswara. In ''[[Treta Yuga]]'', Surya, Chandra, Indira, Romarishi, Ashtaparvathas and Parvatharajan came to worship. In ''[[Kali Yuga]]'', Pathanjali came to worship this sthala.

==Karakkoil==
There are nine types of Hindu temple structures. Amongst them, Karakkoil also called [[Mela Kadambur Amirthakadeswarar Temple]] is a structure that the vimana is in chariot like structure. The structure is pulled by horses looks an enchanting appearance.

[[File:Karakkoil.jpg|330px|thumb|right|Karakkoil]]

This Karakkoil was built by emperor [[Kulottunga Chola I]] in the year 1113 AD, the king's 43rd governing year.<ref>{{cite book|title=Faiths Across Time: 5,000 Years of Religious History [4 Volumes]: 5,000 Years of Religious History|author=J. Gordon Melton|publisher=ABC-CLIO, 15-Jan-2014 - Reference - 2023 pages|page=814}}</ref> This temple is like a chariot like structure with two horses pulling the car or Ratha. The temple is finely sculptured with many purana stories like Ramayan, Krishna Leela, Nayanmar stories and other stories of siva.

The basement is called Adhittanam this temples adhittanam is laid of 18 layered adittanam called pathmaga pandham, and manjapathram.

==Sun worship==
It is rare scene to view. During the 3rd, 4th, 5th day of the Tamil month of Panguni, twelfth month of the [[Tamil calendar]], early during sunrise at 6:15 am, the Sun's rays will fall on Lord Amirthakateshwara Lingam for about fifteen minutes. At that time special poojas are offered to the Lord. The shrine Goddess Jothiminnammai seems as Kalaimagal in morning and Thirumagal in evening and Malaimagal in night.

==Vinayaka==
When Narasimhavarma Pallavan won the war on Chalukya, a Ganapati statue was brought from the Chalukyan capital of Vatapi — by Sirutondar. Like that, cholas made a war towards Ganga region by the King Rajendra I at that time and they took the Ganapathy statue and Dvarabalaka statue from vengi region. While the Ganapathy statue was positioned in Kadambur temple, the Davarabalaka statue was kept in the entrance of Gangai konda cholapuram temple.

==Dasa Bhuja Rishaba Thandavamoorthy==
[[File:Dasabuja rishaba thandava moorthy.jpg|thumb|[[Pala Empire|Pala dynasty]] image of [[Nataraja]]]]
A special idol kept in this temple, called Dasa Bhuja Rishaba Thandavamoorthy, depicts Lord Shiva dancing on a bull with all ten hands spread and holding weapons, with all Devas surrounding his feet. This is believed to be a [[Pala Empire|Pala dynasty]] idol that came to Tamil Nadu-Melakadambur along with victorious King [[Rajendra Chola I]] of the Bengal war. After Rajendra this idol has been kept in this Melakadambur temple built by the king [[Kulothunga I]], who donated this idol and some Vidangars in 1110 AD. Devotees can see this idol only on Pradosha days.

'''Kadamba tree'''

The Kadambu tree is the sthala vriksha of the Siva temple situated here. This village is named as Kadambur because of the Kadambu tree, named in botany as [[Neolamarckia cadamba]] of [[Rubiaceae]] family. This Kadambur then bifurcated into Melakadambur and Keezhakadambur.

==Festivals of this village==

This village was called as Thirukkadambur in 6th–19th century. We can know this by means of the padhigams of Thirunavukkarasar and Thirugnanasambandar nayanmars. Every month festivals being celebrated in good manner.

1. Chithirai—Tamil varuda pirappu, Nataraja abishegam Thirunavukarasar guru pooja

2. Vaikasi - Visaga uthsavam for Murugan, Thirugnanasambandar gurupooja

3. Aani - Aani Thirumanjanam, Manickavasaga gurupooja

4. Aadi - Amman Abishegam in pooram, Sundarar gurupooj

5. Avani - Vinayagar Chathurthi

6. Purattasi - Navaraththiri

7. Aippasi - Kandhasashti, Thirukkalyanam

8. Karthigai - 108 sangabishegam in {{lang|te-Latn|somavaram}}

9. Margazhi - Thirupalliyezhuchi in early morning

10. Thai - Pongal pooja magarasangranthi

11. Maasi - Mahasivarathri

12. Panguni—Sooriya pooja on 3,4,5 at early sunrise

==Dakshinamurthy==
[[File:dakshanamoorthy.jpg|thumb|right|Dakshinamurthy]]

In Kadambur temple the moolasthanam is made up as a chariot-like structure. On that structure's south side is a stone statue of [[Dakshinamurthy]] who is sitting on a bull below a banyan tree. The beauty of this sculpture is a hole that extends from the left ear to the right ear.

== Notes ==
{{Reflist}}

[[Category:Villages in Cuddalore district]]

Bill Allcorn

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{{Short description|American politician}}
{{Use mdy dates|date=April 2025}}
{{More citations needed|date=January 2021}}
'''Bill Allcorn''' (died August 24, 1962) was an American politician. He was Texas Land Commissioner in the late 1950s.

Allcorn was from [[Brownwood, Texas]]. In February 1958, Governor [[Price Daniel]] appointed Allcorn as Texas Land Commissioner when [[James Earl Rudder]] was appointed vice president of [[Texas A&M University]]. Allcorn was elected to one two-year term in 1958 and was succeeded by [[Jerry Sadler]] in 1961. Allcorn died on August 24, 1962, at the age of 38.<ref>{{cite web|url=https://www.wacotrib.com/obituaries/leath-maritia-allcorn/article_d027e5d3-64a2-520c-b2c3-e103811e0758.html|title=Leath, Maritia Allcorn|publisher=Waco Tribune-Herald|accessdate=November 13, 2019}}</ref>

{{s-start}}
{{s-ppo}}
{{s-bef|before=[[James Earl Rudder]]}}
{{s-ttl|title=[[Democratic Party (United States)|Democratic]] nominee for [[Texas General Land Office|Land Commissioner of Texas]]|years=1958}}
{{s-aft|after=[[Jerry Sadler]]}}
{{s-off}}
{{succession box
| before=[[James Earl Rudder]]
| title=[[Commissioner of the Texas General Land Office]]
| after=[[Jerry Sadler]]
| years=1958–1961
}}
{{s-end}}

==References==
{{Reflist}}

{{DEFAULTSORT:Allcorn, Bill}}
[[Category:Texas land commissioners]]
[[Category:Texas Democrats]]
[[Category:Year of birth missing]]
[[Category:1962 deaths]]

{{Texas-politician-stub}}

Wick-A-Te-Wah, Michigan

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{{Use mdy dates|date=July 2023}}
{{Infobox settlement
|name = Wick-A-Te-Wah, Michigan
|settlement_type = [[Unincorporated area|Unincorporated community]]
|nickname =
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|image_skyline =
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|pushpin_map = Michigan
|pushpin_map_caption =Location within the state of Michigan
|pushpin_label = Wick-A-Te-Wah
|pushpin_label_position = right 
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|coordinates = {{coord|44|21|21|N|86|13|26|W|region:US-MI_type:city_source:GNIS|display=inline,title}}
|coordinates_footnotes =
|subdivision_type = [[List of sovereign states|Country]]
|subdivision_name = United States
|subdivision_type1 = [[U.S. state|State]]
|subdivision_name1 = [[Michigan]]
|subdivision_type2 = [[List of counties in Michigan|County]]
|subdivision_name2 = [[Manistee County, Michigan|Manistee]]
|subdivision_type3 = [[Civil township|Township]]
|subdivision_name3 = [[Onekama Township, Michigan|Onekama]]


|government_footnotes =
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'''Wick-A-Te-Wah''' (or '''Wic-a-te-wah''') is an unincorporated summer resort area of [[Onekama Township, Michigan|Onekama Township]], [[Manistee County, Michigan|Manistee County]] in the [[U.S. state]] of [[Michigan]]. It is located on the south shore of [[Portage Lake (Michigan)|Portage Lake]] at {{coord|44|21|21|N|86|13|26|W|}},<ref name="gnis">{{gnis|1622152|Wic-a-te-wah, Michigan}}</ref> between the Portage Lake Bible Camp Camp Delight on the east and [[Red Park, Michigan|Red Park]] on the west.

==History==
Mr. H. Ward Leonard, a developer from [[Manistee, Michigan|Manistee]], developed this area as a summer resort in the early 1900s. In 1917, Leonard issued a brochure describing the property and the availability of lots for sale.

==Festivities==
The annual Fourth of July Wick-A-Te-Wah Parade is considered by many to be one of the best Fourth of July parades in Michigan, drawing considerable crowds and extensive neighborhood participation.<ref>{{cite web|url=http://news.pioneergroup.com/manisteenews/2014/07/04/wick-te-wah-parade-brings-neighbors-together/|title=Wick-a-te-wah parade brings neighbors together - Manistee News|date=4 July 2014|website=News.pioneergroup.com|accessdate=8 July 2018}}</ref>

==References==
{{Reflist}}

==Further reading==
* Heidi Berg, "Wick-A-Te-Wah," in ''Wellspring: Interesting interviews... The way things were...'' (Manistee: J.B. Publications, 1992), vol. 2, pp. 23–26.

{{Manistee County, Michigan}}

{{authority control}}

[[Category:Unincorporated communities in Manistee County, Michigan]]
[[Category:Unincorporated communities in Michigan]]

Frank Owen (baseball)

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{{Short description|American baseball player (1879–1942)}}
{{distinguish|text=Frank [[Yip Owens]], a Major League Baseball catcher}}
{{Use mdy dates|date=June 2024}}
{{Infobox baseball biography
|name=Frank Owen
|position=[[Pitcher]]
|image=Frank Owen baseball card.jpg
|caption=Frank Owen baseball card
|bats=Switch
|throws=Right
|birth_date={{Birth date|1879|12|23|mf=y}}
|birth_place=[[Ypsilanti, Michigan]], U.S.
|death_date={{death date and age|1942|11|24|1879|12|23}}
|death_place=[[Dearborn, Michigan]], U.S.
|debutleague = MLB
|debutdate=April 29
|debutyear=1901
|debutteam=Detroit Tigers
|finalleague = MLB
|finaldate=May 12
|finalyear=1909
|finalteam=Chicago White Sox
|statleague = MLB
|stat1label=[[Win–loss record (pitching)|Win–loss record]]
|stat1value=82–67
|stat3label=[[Strikeout]]s
|stat3value=443
|stat2label=[[Earned run average]]
|stat2value=2.55
|teams=
*[[Detroit Tigers]] (1901)
*[[Chicago White Sox]] (1903–1909)
|awards=
*[[World Series champion]] ({{wsy|1906}})
}}
'''Frank Malcolm Owen''' (December 23, 1879 – November 24, 1942) was a [[pitcher]] in [[Major League Baseball]] who played eight seasons with the [[Detroit Tigers]] and [[Chicago White Sox]].

Born in [[Ypsilanti, Michigan]] (and nicknamed "Yip" for it), he pitched the final six innings of Game 2 of the [[1906 World Series]], replacing [[Doc White]]. In 194 career games, Owen had an 82–67 won-loss record with a 2.55 [[Earned run average|ERA]].

Owen was the first [[American League]] pitcher to pitch complete game wins in both games of a doubleheader, winning against the [[St. Louis Browns]] on July 1, 1905.<ref>{{cite book|last1=Nemec|first1=David|title=This Day in Baseball: A Day-by-day Record of the Events that Shaped the Game|date=2008|publisher=Rowman & Littlefield|location=Lanham, Maryland|isbn=978-1589793804|pages=330}}</ref> Owen was mistakenly referred to as "Billy Owen" in the 1906 version of the "Fan Craze" board game, released by the Fan Craze Co of [[Cincinnati]].<ref>{{cite web|title=The Mystery of Billy Owen|url=http://baseballgames.dreamhosters.com/BillyOwen.htm|website=Baseball Games|access-date=April 28, 2015}}</ref>

In 1904, as a member of the White Sox, in 315 innings of work, he handled 151 chances (21 PO, 130 A) without an error and also executed 8 double plays.

==References==
<references />

==External links==
{{Baseballstats|br=o/owenfr01|brm=owen--001fra}}

{{1906 Chicago White Sox}}
{{Chicago White Sox Opening Day starting pitchers}}

{{DEFAULTSORT:Owen, Frank}}
[[Category:1879 births]]
[[Category:1942 deaths]]
[[Category:Chicago White Sox players]]
[[Category:Detroit Tigers players]]
[[Category:Major League Baseball pitchers]]
[[Category:Baseball players from Washtenaw County, Michigan]]
[[Category:Sportspeople from Ypsilanti, Michigan]]
[[Category:Detroit Tigers (Western League) players]]
[[Category:Wilkes-Barre Coal Barons players]]
[[Category:Omaha Indians players]]
[[Category:Toledo Mud Hens players]]
[[Category:Kansas City Blues (baseball) players]]

{{US-baseball-pitcher-1870s-stub}}

Sadie Grant Pack

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{{no footnotes|date=October 2018}}
{{Infobox Latter Day Saint biography
| name = Sadie Grant Pack
| image = Sadie Grant Pack.jpg
| alt = Photo of Sadie Grant Pack
| caption = Pack (ca. 1936)
| birth_name = Sadie Grant
| birth_date = {{Birth date|1877|12|20}}
| birth_place = [[West Bountiful, Utah|West Bountiful]], [[Utah Territory]]
| death_date = {{Death date and age|1960|08|24|1877|12|20}}
| death_place = [[Salt Lake City]], Utah, US
| death_cause =
| resting_place = [[Salt Lake City Cemetery]]
| resting_place_coordinates = {{Coord|40.777|-111.858|type:landmark|display=inline|name=Salt Lake City Cemetery}}
| monuments =
| residence =
| education =
| alma_mater = [[University of Utah]]
| occupation =
| employer =
| organization =
| notable_works =
| title =
| spouse = [[Frederick J. Pack]]
| children = 4
| parents = Joseph H. Grant Evaletta Eldredge
| relatives = [[Jedediah M. Grant]] (grandfather)
| awards =
| signature =
| website = 

| position_or_quorum1 = [[Primary (LDS Church)#Chronology of the general presidency of the Primary|First Counselor]] in the general presidency of the [[Primary (LDS Church)|Primary]]
| called_by1 = [[May Anderson]]
| ordination_reason1 =
| predecessor1 = May Anderson
| successor1 = [[Isabelle S. Ross]]
| start_date1 = {{start date|1925|10|06}}
| end_date1 = {{end date|1929|09|11}}
| end_reason1 = }}
'''Sadie Grant Pack''' (December 20, 1877 – August 24, 1960) was the first counselor to [[May Anderson]] in the general presidency of the [[Primary (LDS Church)|Primary]] of [[the Church of Jesus Christ of Latter-day Saints]] from 1925 to 1929.

Sadie Grant was born in [[West Bountiful, Utah|West Bountiful]], [[Utah Territory]] to Joseph Hyrum Grant (a son of [[Jedediah M. Grant]]) and Evaletta Eldredge. She was educated at the [[University of Utah]], where she met her future husband, [[Frederick J. Pack]], who was also a native of West Bountiful. Grant and Pack married in November 1896 and later that year began service as an LDS Church [[Mormon missionary|missionaries]] in the [[Colorado]] [[Mission (LDS Church)|Mission]] of the church.

From 1904 to 1906 while her husband was a student at [[Columbia University]], Sadie was president of the LDS Relief Society in New York and Brooklyn.

Pack held a variety of [[Calling (LDS Church)|callings]] in the [[Relief Society]], the [[Young Women's Mutual Improvement Association]] and the [[Primary (LDS Church)|Primary]] prior to her call as first counselor in the general presidency of the Primary. In 1925, when [[May Anderson]] was called as the general president of the Primary, she selected Pack as her first counselor. Pack was released in 1929 and was succeeded by [[Isabelle S. Ross]].

Pack and her husband were the parents of four children. Pack died in [[Salt Lake City, Utah]].

Saide Grant Pack was a granddaughter of prominent church leader [[Jedediah M. Grant]] and was a niece to [[President of the Church (LDS Church)|LDS Church president]] [[Heber J. Grant]].

==References==
*{{cite book
| last = Jenson
| first = Andrew
| authorlink = Andrew Jenson
| title = Latter-day Saint biographical encyclopedia: A compilation of biographical sketches of prominent men and women in the Church of Jesus Christ of Latter-Day Saints
| url = http://contentdm.lib.byu.edu/u?/BYUIBooks,5238
| accessdate = December 20, 2011
| volume = 4
| year = 1936
| publisher = The Andrew Jenson Memorial Association (Printed by The Deseret News Press)
| location = Salt Lake City, Utah
| isbn = 1-58958-026-5
| pages = [http://contentdm.lib.byu.edu/u?/BYUIBooks,5204 292–293]}}
*''[[The Children's Friend (LDS magazine)|Children's Friend]]'', vol. 25, p. 23
{|style="margin:1em auto;"
|-
|{{S-start}}
{{s-rel | mo}}
{{s-bef | before = [[May Anderson]]}}
{{s-ttl | title = [[Primary (LDS Church)#Chronology of the general presidency of the Primary|First Counselor]] in the general presidency of the [[Primary (LDS Church)|Primary]] | years = {{start date|1925|10|06}} – {{end date|1929|09|11}}}}
{{s-aft | after = [[Isabelle S. Ross]]}}
{{s-end}}
|-
|{{LDSprimary}}
|}

{{DEFAULTSORT:Pack, Sadie Grant}}
[[Category:1877 births]]
[[Category:1960 deaths]]
[[Category:19th-century Mormon missionaries]]
[[Category:American Mormon missionaries in the United States]]
[[Category:American leaders of the Church of Jesus Christ of Latter-day Saints]]
[[Category:Burials at Salt Lake City Cemetery]]
[[Category:Counselors in the General Presidency of the Primary (LDS Church)]]
[[Category:Female Mormon missionaries]]
[[Category:20th-century Mormon missionaries]]
[[Category:Latter Day Saints from Utah]]
[[Category:People from Davis County, Utah]]
[[Category:University of Utah alumni]]