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{{Short description|Arithmetic operation}}
{{Other uses}}
{{Redirect|Add||ADD (disambiguation)}}
{{Good article}}
[[File:Addition01.svg|right|thumb|upright=0.5|3 + 2 = 5 with [[apple]]s, a popular choice in textbooks<ref>{{harvtxt|Enderton|1977}}, p. [http://books.google.com/books?id=JlR-Ehk35XkC&pg=PA138 138]: "...select two sets ''K'' and ''L'' with card ''K'' = 2 and card ''L'' = 3. Sets of fingers are handy; sets of apples are preferred by textbooks."</ref>]]

'''Addition''' (usually signified by the [[Plus and minus signs#Plus sign|plus symbol]], +) is one of the four basic [[Operation (mathematics)|operations]] of [[arithmetic]], the other three being [[subtraction]], [[multiplication]], and [[Division (mathematics)|division]]. The addition of two [[Natural number|whole numbers]] results in the total or ''[[summation|sum]]'' of those values combined. For example, the adjacent image shows two columns of apples, one with three apples and the other with two apples, totaling to five apples. This observation is expressed as {{nowrap|1="3 + 2 = 5"}}, which is read as  "three plus two [[Equality (mathematics)|equal]]s five".

Besides [[counting]] items, addition can also be defined and executed without referring to [[concrete object]]s, using abstractions called [[number]]s instead, such as [[integer]]s, [[real number]]s, and [[complex number]]s. Addition belongs to arithmetic, a branch of [[mathematics]]. In [[algebra]], another area of mathematics, addition can also be performed on abstract objects such as [[Euclidean vector|vectors]], [[Matrix (mathematics)|matrices]], [[Linear subspace|subspaces]], and [[subgroup]]s.

Addition has several important properties. It is [[Commutative property|commutative]], meaning that the order of the [[operand|numbers being added]] does not matter, so {{nowrap|1=3 + 2 = 2 + 3}}, and it is [[associativity|associative]], meaning that when one adds more than two numbers, the order in which addition is performed does not matter. Repeated addition of {{num|1}} is the same as counting (see [[Successor function]]). Addition of {{num|0}} does not change a number. Addition also obeys rules concerning related operations such as subtraction and multiplication.

Performing addition is one of the simplest numerical tasks to perform. Addition of very small numbers is accessible to toddlers; the most basic task, {{nowrap|1 + 1}}, can be performed by infants as young as five months, and even some members of other animal species. In [[primary education]], students are taught to add numbers in the [[decimal]] system, beginning with single digits and progressively tackling more difficult problems. Mechanical aids range from the ancient [[abacus]] to the modern [[computer]], where research on the most efficient implementations of addition continues to this day.
{{Calculation results}}

== Notation and terminology ==
[[File:PlusCM128.svg|upright=0.4|thumb|The plus sign]]
Addition is written using the [[Plus and minus signs|plus sign]] "+" [[infix notation|between the terms]], and the result is expressed with an [[equals sign]]. For example, <math>1 + 2 = 3</math> reads "one plus two equals three".{{sfnp|Musser|Peterson|Burger|2013|p=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA87 87]}} Nonetheless, some situations where addition is "understood", even though no symbol appears: a whole number followed immediately by a [[fraction (mathematics)|fraction]] indicates the sum of the two, called a ''mixed number'', with an example,{{sfnp|Devine|Olson|Olson|1991|p=263}}<math display="block">3\frac{1}{2}=3+\frac{1}{2}=3.5.</math> This notation can cause confusion, since in most other contexts, [[Juxtaposition#Mathematics|juxtaposition]] denotes [[multiplication]] instead.{{sfnp|Mazur|2014|p=161}}

{{anchor|summand|addend}}
[[File:Addition1.png|thumb|upright=1|The terms of addends in the operation of an addition]]
The numbers or the objects to be added in general addition are collectively referred to as the '''terms''',{{sfnp|Department of the Army|1961|loc=[https://archive.org/details/TM11-684/page/16/mode/1up?view=theater Section 5.1]}} the '''addends''' or the '''summands'''.{{sfnp|Musser|Peterson|Burger|2013|p=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA87 87]}} This terminology carries over to the summation of multiple terms.
This is to be distinguished from ''factors'', which are [[multiplication|multiplied]].
Some authors call the first addend the ''augend''.{{sfnmp
 | 1a1 = Shmerko | 1a2 = Yanushkevich | 1a3 = Lyshevski | 1y = 2009 | 1p = 80
 | 2a1 = Schmid | 2y = 1974
 | 3a1 = Schmid | 3y = 1983
}} In fact, during the [[Renaissance]], many authors did not consider the first addend an "addend" at all. Today, due to the [[commutative property]] of addition, "augend" is rarely used, and both terms are generally called addends.{{sfnp|Schwartzman|1994|p=19}}

All of the above terminology derives from [[Latin]]. "[[wikt:addition|Addition]]" and "[[wikt:add|add]]" are [[English language|English]] words derived from the Latin [[verb]] {{lang|la|addere}}, which is in turn a [[compound (linguistics)|compound]] of {{lang|la|ad}} "to" and {{lang|la|dare}} "to give", from the [[Proto-Indo-European root]] {{lang|ine-x-proto|deh₃-}} "to give"; thus to ''add'' is to ''give to''.{{sfnp|Schwartzman|1994|p=19}} Using the [[gerundive]] [[Affix|suffix]] ''-nd'' results in "addend", "thing to be added".<ref group=lower-alpha>"Addend" is not a Latin word; in Latin it must be further conjugated, as in {{lang|la|numerus addendus}} "the number to be added".</ref> Likewise from {{lang|la|augere}} "to increase", one gets "augend", "thing to be increased".<ref>{{cite book |last=Schubert |first=Hermann |year=1903 |chapter=Monism in Arithmetic |title=Mathematical Essays and Recreations |place=Chicago |publisher=Open Court |page=10 |chapter-url=https://archive.org/details/cu31924062544915/page/n25/}}</ref>

[[File:AdditionNombryng.svg|thumb|upright=1|Redrawn illustration from ''The Art of Nombryng'', one of the first English arithmetic texts, in the 15th century.<ref>{{harvtxt|Karpinski|1925}}, pp. 56–57, reproduced on p. 104</ref>]]
"Sum" and "summand" derive from the Latin [[noun]] {{lang|la|summa}} "the highest" or "the top", used in Medieval Latin phrase {{lang|la|summa linea}} ("top line") meaning the sum of a column of numerical quantities, following the [[Ancient Greece|ancient Greek]] and [[Ancient Rome|Roman]] practice of putting the sum at the top of a column.{{sfnp|Schwartzman|1994|p=212}}
{{lang|la|Addere}} and {{lang|la|summare}} date back at least to [[Anicius Manlius Severinus Boethius|Boethius]], if not to earlier Roman writers such as [[Vitruvius]] and [[Sextus Julius Frontinus|Frontinus]]; Boethius also used several other terms for the addition operation. The later [[Middle English]] terms "adden" and "adding" were popularized by [[Geoffrey Chaucer|Chaucer]].{{sfnp|Karpinski|1925|pp=150–153}}

== Definition and interpretations ==
Addition is one of the four basic [[Operation (mathematics)|operations]] of [[arithmetic]], with the other three being [[subtraction]], [[multiplication]], and [[Division (mathematics)|division]]. This operation works by adding two or more terms.{{sfnp|Lewis|1974|p=1}} An arbitrary of many operation of additions is called the [[summation]].{{sfnp|Martin|2003|p=49}} An infinite summation is a delicate procedure known as a [[series (mathematics)|series]],{{sfnp|Stewart|1999|p=8}} and it can be expressed through [[capital sigma notation]] <math display="inline"> \sum </math>, which compactly denotes [[iteration]] of the operation of addition based on the given indexes.{{sfnp|Apostol|1967|p=37}} For example,
<math display="block">\sum_{k=1}^5 k^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 55.</math>

Addition is used to model many physical processes. Even for the simple case of adding [[natural number]]s, there are many possible interpretations and even more visual representations.

=== Combining sets ===
[[File:AdditionShapes.svg|right|thumb|upright=0.8|One set has three shapes while the other set has two. The total of shapes is five, which is a consequence of the addition of the objects from the two sets: <math> 3 + 2 = 5 </math>.]]
Possibly the most basic interpretation of addition lies in combining [[Set (mathematics)|sets]], that is:{{sfnp|Musser|Peterson|Burger|2013|p=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA87 87]}}
{{blockquote|When two or more disjoint collections are combined into a single collection, the number of objects in the single collection is the sum of the numbers of objects in the original collections.}}
This interpretation is easy to visualize, with little danger of ambiguity. It is also useful in higher mathematics (for the rigorous definition it inspires, see {{Section link||Natural numbers}} below). However, it is not obvious how one should extend this version of an addition's operation to include fractional or negative numbers.<ref>See {{harvtxt|Viro|2001}} for an example of the sophistication involved in adding with sets of "fractional cardinality".</ref>

One possible fix is to consider collections of objects that can be easily divided, such as pies or, still better, segmented rods. Rather than solely combining collections of segments, rods can be joined end-to-end, which illustrates another conception of addition: adding not the rods but the lengths of the rods.{{sfnp|National Research Council|2001|p=[http://books.google.com/books?id=pvI7uDPo0-YC&pg=PA74 74]}}
{{-}}

=== Extending a length ===
{{multiple image
 | image1 = AdditionLineAlgebraic.svg
 | caption1 = A number-line visualization of the algebraic addition <math> 2 + 4 = 6 </math>. A "jump" that has a distance of <math> 2 </math> followed by another that is as long as <math> 4 </math>, is the same as a translation by <math> 6 </math>.
 | image2 = AdditionLineUnary.svg
 | caption2 = A number-line visualization of the unary addition <math> 2 + 4 = 6 </math>. A translation by <math> 4 </math> is equivalent to four translations by <math> 1 </math>.
 | direction = vertical
 | total_width = 400
}}
A second interpretation of addition comes from extending an initial length by a given length:{{sfnp|Mosley|2001|p=[http://books.google.com/books?id=I-__WcWjemUC&pg=PA8 8]}}
{{blockquote|When an original length is extended by a given amount, the final length is the sum of the original length and the length of the extension.}}
The sum <math> a + b </math> can be interpreted as a [[binary operation]] that combines <math> a </math> and <math> b </math> algebraically, or it can be interpreted as the addition of <math> b </math> more units to <math> a </math>. Under the latter interpretation, the parts of a sum <math> a + b </math> play asymmetric roles, and the operation <math> a + b </math> is viewed as applying the [[unary operation]] <math> +b </math> to <math> a </math>.{{sfnp|Li|Lappan|2014|p=204}} Instead of calling both <math> a </math> and <math> b </math> addends, it is more appropriate to call <math> a </math> the "augend" in this case, since <math> a </math> plays a passive role. The unary view is also useful when discussing [[subtraction]], because each unary addition operation has an inverse unary subtraction operation, and vice versa.
{{-}}

== Properties ==
=== Commutativity ===
[[File:AdditionComm01.svg|right|upright=0.5|thumb|4 + 2 = 2 + 4 with blocks]]
Addition is [[commutative]], meaning that one can change the order of the terms in a sum, but still get the same result. Symbolically, if <math> a </math> and <math> b </math> are any two numbers, then:{{sfnp|Musser|Peterson|Burger|2013|p=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA89 89]}}
<math display="block"> a + b = b + a. </math>
The fact that addition is commutative is known as the "commutative law of addition"{{sfnp|Berg|1967|p=[https://books.google.com/books?id=aGXFCUaFCW0C&pg=PA14 14]}} or "commutative property of addition".{{sfnp|Behr|Jungst|1971|p=[https://books.google.com/books?id=GJXOBQAAQBAJ&pg=PA59 59]}} Some other [[binary operation]]s are commutative too as in [[multiplication]],{{sfnp|Rosen|2013|loc = See the [https://books.google.com/books?id=-oVvEAAAQBAJ&pg=SL1-PA1 Appendix I]}} but others are not as in [[subtraction]] and [[Division (mathematics)|division]].{{sfnp|Posamentier|Farber|Germain-Williams|Paris|2013|p=[https://books.google.com/books?id=VfCgAQAAQBAJ&pg=PA71 71]}}

=== Associativity ===
[[File:AdditionAsc.svg|upright=0.5|thumb|2 + (1 + 3) = (2 + 1) + 3 with segmented rods]]
Addition is [[associativity|associative]], which means that when three or more numbers are added together, the [[order of operations]] does not change the result. For any three numbers <math> a </math>, <math> b </math>, and <math> c </math>, it is true that:{{sfnp|Musser|Peterson|Burger|2013|p=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA90 90]}}
<math display="block"> (a + b) + c = a + (b + c). </math>
For example, <math> (1 + 2) + 3 = 1 + (2 + 3) </math>.

When addition is used together with other operations, the [[order of operations]] becomes important. In the standard order of operations, addition is a lower priority than [[exponentiation]], [[nth root]]s, multiplication and division, but is given equal priority to subtraction.{{sfnp|Bronstein|Semendjajew|1987}}
{{-}}

=== Identity element ===
[[File:AdditionZero.svg|right|upright=0.33|thumb|5 + 0 = 5 with bags of dots]]
Adding [[0 (number)|zero]] to any number does not change the number. In other words, zero is the [[identity element]] for addition, and is also known as the [[additive identity]]. In symbols, for every <math> a </math>, one has:{{sfnp|Musser|Peterson|Burger|2013|p=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA90 90]}}
<math display="block"> a + 0 = 0 + a = a. </math>
This law was first identified in [[Brahmagupta]]'s ''[[Brahmasphutasiddhanta]]'' in 628&nbsp;AD, although he wrote it as three separate laws, depending on whether <math> a </math> is negative, positive, or zero itself, and he used words rather than algebraic symbols. Later [[Indian mathematicians]] refined the concept; around the year 830, [[Mahavira (mathematician)|Mahavira]] wrote, "zero becomes the same as what is added to it", corresponding to the unary statement <math> 0 + a = a </math>. In the 12th&nbsp;century, [[Bhāskara II|Bhaskara]] wrote, "In the addition of cipher, or subtraction of it, the quantity, positive or negative, remains the same", corresponding to the unary statement <math> a + 0 = a </math>.{{sfnp|Kaplan|2000|pp=69–71}}
{{-}}

=== Successor ===
{{main article|Successor function}}
Within the context of integers, addition of [[1 (number)|one]] also plays a special role: for any integer <math> a </math>, the integer <math> a + 1 </math> is the least integer greater than <math> a </math>, also known as the [[successor function|successor]] of <math> a </math>. For instance, 3 is the successor of 2, and 7 is the successor of 6. Because of this succession, the value of <math> a + b </math> can also be seen as the {{nowrap|1=<math> b </math>-}}th successor of <math> a </math>, making addition an iterated succession. For example, {{nowrap|6 + 2}} is 8, because 8 is the successor of 7, which is the successor of 6, making 8 the second successor of 6.{{sfnp|Hempel|2001|p=[http://books.google.com/books?id=yTY9La4P2n8C&pg=PA7 7]}}

=== Units ===
To numerically add physical quantities with [[units of measurement|units]], they must be expressed with common units.{{sfnp|Fierro|2012|loc=Section 2.3}} For example, adding 50&nbsp;milliliters to 150&nbsp;milliliters gives 200&nbsp;milliliters. However, if a measure of 5&nbsp;feet is extended by 2&nbsp;inches, the sum is 62&nbsp;inches, since 60&nbsp;inches is synonymous with 5&nbsp;feet. On the other hand, it is usually meaningless to try to add 3&nbsp;meters and 4&nbsp;square meters, since those units are incomparable; this sort of consideration is fundamental in [[dimensional analysis]].<ref>{{Cite book|last1=Moebs|first1=William|url=https://openstax.org/books/university-physics-volume-1/pages/1-4-dimensional-analysis|title=University Physics Volume 1|last2=Ling|first2=Samuel J.|publisher=[[OpenStax]]|year=2022|isbn=978-1-947172-20-3|chapter=1.4 Dimensional Analysis|display-authors=1}}</ref>

== Performing addition ==
=== Innate ability ===
Studies on mathematical development starting around the 1980s have exploited the phenomenon of [[habituation]]: [[infant]]s look longer at situations that are unexpected.{{sfnp|Wynn|1998|p=5}} A seminal experiment by [[Karen Wynn]] in 1992 involving [[Mickey Mouse]] dolls manipulated behind a screen demonstrated that five-month-old infants ''expect'' {{nowrap|1 + 1}} to be 2, and they are comparatively surprised when a physical situation seems to imply that {{nowrap|1 + 1}} is either 1 or 3. This finding has since been affirmed by a variety of laboratories using different methodologies.{{sfnp|Wynn|1998|p=15}} Another 1992 experiment with older [[toddler]]s, between 18 and 35&nbsp;months, exploited their development of motor control by allowing them to retrieve [[ping-pong]] balls from a box; the youngest responded well for small numbers, while older subjects were able to compute sums up to 5.{{sfnp|Wynn|1998|p=17}}

Even some nonhuman animals show a limited ability to add, particularly [[primate]]s. In a 1995 experiment imitating Wynn's 1992 result (but using [[eggplant]]s instead of dolls), [[rhesus macaque]] and [[cottontop tamarin]] monkeys performed similarly to human infants. More dramatically, after being taught the meanings of the [[Arabic numerals]] 0 through 4, one [[Common Chimpanzee|chimpanzee]] was able to compute the sum of two numerals without further training.{{sfnp|Wynn|1998|p=19}} More recently, [[Asian elephant]]s have demonstrated an ability to perform basic arithmetic.<ref>{{cite news |newspaper=The Guardian |last=Randerson |first=James |url=https://www.theguardian.com/science/2008/aug/21/elephants.arithmetic |title=Elephants have a head for figures |date=21 August 2008 |access-date=29 March 2015 |archive-url=https://web.archive.org/web/20150402103526/http://www.theguardian.com/science/2008/aug/21/elephants.arithmetic |archive-date=2 April 2015 |url-status=live }}</ref>

=== Childhood learning ===
Typically, children first master [[counting]]. When given a problem that requires that two items and three items be combined, young children model the situation with physical objects, often fingers or a drawing, and then count the total. As they gain experience, they learn or discover the strategy of "counting-on": asked to find two plus three, children count three past two, saying "three, four, ''five''" (usually ticking off fingers), and arriving at five. This strategy seems almost universal; children can easily pick it up from peers or teachers.{{sfnp|Smith|2002|p=130}} Most discover it independently. With additional experience, children learn to add more quickly by exploiting the commutativity of addition by counting up from the larger number, in this case, starting with three and counting "four, ''five''." Eventually children begin to recall certain addition facts ("[[number bond]]s"), either through experience or rote memorization. Once some facts are committed to memory, children begin to derive unknown facts from known ones. For example, a child asked to add six and seven may know that {{nowrap|1=6 + 6 = 12}} and then reason that {{nowrap|6 + 7}} is one more, or 13.<ref>{{Cite book |last=Carpenter |first=Thomas |author2=Fennema, Elizabeth|author2-link = Elizabeth Fennema |author3=Franke, Megan Loef |author4=Levi, Linda |author5=Empson, Susan|author5-link=Susan Empson |title=Children's mathematics: Cognitively guided instruction |publisher=Heinemann |year=1999 |location=Portsmouth, NH |isbn=978-0-325-00137-1 |url-access=registration |url=https://archive.org/details/childrensmathema0000unse_i5h7 }}</ref> Such derived facts can be found very quickly and most elementary school students eventually rely on a mixture of memorized and derived facts to add fluently.<ref name=Henry>{{Cite journal |last=Henry |first=Valerie J. |author2=Brown, Richard S. |title=First-grade basic facts: An investigation into teaching and learning of an accelerated, high-demand memorization standard |journal=Journal for Research in Mathematics Education |volume=39 |issue=2 |pages=153–183 |year=2008 |doi=10.2307/30034895|jstor=30034895 |doi-access=free }}</ref>

Different nations introduce whole numbers and arithmetic at different ages, with many countries teaching addition in pre-school.<ref>
Beckmann, S. (2014). The twenty-third ICMI study: primary mathematics study on whole numbers. International Journal of STEM Education, 1(1), 1–8.
Chicago
</ref> However, throughout the world, addition is taught by the end of the first year of elementary school.<ref>Schmidt, W., Houang, R., & Cogan, L. (2002). "A coherent curriculum". ''American Educator'', 26(2), 1–18.</ref>

=== Decimal system ===
The prerequisite to addition in the [[decimal]] system is the fluent recall or derivation of the 100 single-digit "addition facts". One could [[memorize]] all the facts by [[rote learning|rote]], but pattern-based strategies are more enlightening and, for most people, more efficient:{{sfnp|Fosnot|Dolk|2001|p=99}}
* ''Commutative property'': Mentioned above, using the pattern <math> a + b = b + a </math> reduces the number of "addition facts" from 100 to 55.
* ''One or two more'': Adding 1 or 2 is a basic task, and it can be accomplished through counting on or, ultimately, [[intuition (knowledge)|intuition]].{{sfnp|Fosnot|Dolk|2001|p=99}}
* ''Zero'': Since zero is the additive identity, adding zero is trivial. Nonetheless, in the teaching of arithmetic, some students are introduced to addition as a process that always increases the addends; [[word problem (mathematics education)|word problems]] may help rationalize the "exception" of zero.{{sfnp|Fosnot|Dolk|2001|p=99}}
* ''Doubles'': Adding a number to itself is related to counting by two and to [[multiplication]]. Doubles facts form a backbone for many related facts, and students find them relatively easy to grasp.{{sfnp|Fosnot|Dolk|2001|p=99}}
* ''Near-doubles'': Sums such as 6 + 7 = 13 can be quickly derived from the doubles fact {{nowrap|1=6 + 6 = 12}} by adding one more, or from {{nowrap|1=7 + 7 = 14}} but subtracting one.{{sfnp|Fosnot|Dolk|2001|p=99}}
* ''Five and ten'': Sums of the form 5 + {{mvar|x}} and 10 + {{mvar|x}} are usually memorized early and can be used for deriving other facts. For example, {{nowrap|1=6 + 7 = 13}} can be derived from {{nowrap|1=5 + 7 = 12}} by adding one more.{{sfnp|Fosnot|Dolk|2001|p=99}}
* ''Making ten'': An advanced strategy uses 10 as an intermediate for sums involving 8 or 9; for example, {{nowrap|1=8 + 6 = 8 + 2 + 4 =}} {{nowrap|1=10 + 4 = 14}}.{{sfnp|Fosnot|Dolk|2001|p=99}}
As students grow older, they commit more facts to memory and learn to derive other facts rapidly and fluently. Many students never commit all the facts to memory, but can still find any basic fact quickly.<ref name=Henry/>

==== Carry ====
{{main|Carry (arithmetic)}}
[[File:Addition with carry.png|thumb|upright=1|An addition with [[Carry (mathematics)|carry]]]]
The standard algorithm for adding multidigit numbers is to align the addends vertically and add the columns, starting from the ones column on the right. If a column exceeds nine, the extra digit is "[[carry (arithmetic)|carried]]" into the next column. For example, in the following image, the ones in the addition of {{nowrap|59 + 27}} is 9 + 7 = 16, and the digit 1 is the carry.<ref>Some authors think that "carry" may be inappropriate for education; {{harvtxt|van de Walle|2004}}, p. 211 calls it "obsolete and conceptually misleading", preferring the word "trade". However, "carry" remains the standard term.</ref> An alternate strategy starts adding from the most significant digit on the left; this route makes carrying a little clumsier, but it is faster at getting a rough estimate of the sum.<ref>For example, [[al-Khwarizmi]] performed multi-digit addition in this way from left to right. {{pb}} {{cite journal |last1=Crossley |first1=J. N. |last2=Henry |first2=A. S. |year=1990 |title=Thus spake al-Khwārizmī: A translation of the text of Cambridge University Library Ms. Ii.vi.5. |journal=Historia Mathematica |volume=17 |number=2 |pages=103–131 |doi=10.1016/0315-0860(90)90048-i}}</ref>

==== Decimal fractions ====
[[Decimal fractions]] can be added by a simple modification of the above process.<ref>Rebecca Wingard-Nelson (2014) ''Decimals and Fractions: It's Easy'' Enslow Publishers, Inc.</ref> One aligns two decimal fractions above each other, with the decimal point in the same location. If necessary, one can add trailing zeros to a shorter decimal to make it the same length as the longer decimal. Finally, one performs the same addition process as above, except the decimal point is placed in the answer, exactly where it was placed in the summands.

As an example, 45.1 + 4.34 can be solved as follows:
    4 5 . 1 0
 +  0 4 . 3 4
 ————————————
    4 9 . 4 4

==== Scientific notation ====
{{main|Scientific notation#Basic operations}}
In [[scientific notation]], numbers are written in the form <math>x=a\times10^{b}</math>, where <math>a</math> is the significand and <math>10^{b}</math> is the exponential part. To add numbers in scientific notation, they should be expressed with the same exponent, so that the two significands can simply be added.<ref>{{cite book |last1=Cassidy |first1=David |last2=Holton |first2=Gerald |last3=Rutherford |first3=James |year=2002 |chapter=Reviewing Units, Mathematics, and Scientific Notation |title=Understanding Physics |publisher=Springer |place=New York |doi=10.1007/0-387-21660-X_3 |page=11}}</ref>

For example:
: <math>\begin{align}
&2.34\times10^{-5} + 5.67\times10^{-6} \\
&\quad = 2.34\times10^{-5} + 0.567\times10^{-5} \\
&\quad = 2.907\times10^{-5}.
\end{align}</math>

=== Non-decimal ===
{{main|Binary addition}}
Addition in other bases is very similar to decimal addition. As an example, one can consider addition in binary.<ref>Dale R. Patrick, Stephen W. Fardo, Vigyan Chandra (2008) ''Electronic Digital System Fundamentals'' The Fairmont Press, Inc. p. 155</ref> Adding two single-digit binary numbers is relatively simple, using a form of carrying:
: 0 + 0 → 0
: 0 + 1 → 1
: 1 + 0 → 1
: 1 + 1 → 0, carry 1 (since 1 + 1 = 2 = 0 + (1 × 2<sup>1</sup>))
Adding two "1" digits produces a digit "0", while 1 must be added to the next column. This is similar to what happens in decimal when certain single-digit numbers are added together; if the result equals or exceeds the value of the radix (10), the digit to the left is incremented:
: 5 + 5 → 0, carry 1 (since 5 + 5 = 10 = 0 + (1 × 10<sup>1</sup>))
: 7 + 9 → 6, carry 1 (since 7 + 9 = 16 = 6 + (1 × 10<sup>1</sup>))

This is known as ''carrying''.<ref>P.E. Bates Bothman (1837) ''The common school arithmetic''. Henry Benton. p. 31</ref> When the result of an addition exceeds the value of a digit, the procedure is to "carry" the excess amount divided by the radix (that is, 10/10) to the left, adding it to the next positional value. This is correct since the next position has a weight that is higher by a factor equal to the radix. Carrying works the same way in binary:

   {{brown|1 1 1 1 1    (carried digits)}}
     0 1 1 0 1
 +   1 0 1 1 1
 —————————————
   1 0 0 1 0 0 = 36

In this example, two numerals are being added together: 01101<sub>2</sub> (13<sub>10</sub>) and 10111<sub>2</sub> (23<sub>10</sub>). The top row shows the carry bits used. Starting in the rightmost column, {{nowrap|1=1 + 1 = 10<sub>2</sub>}}. The 1 is carried to the left, and the 0 is written at the bottom of the rightmost column. The second column from the right is added: {{nowrap|1=1 + 0 + 1 = 10<sub>2</sub>}} again; the 1 is carried, and 0 is written at the bottom. The third column: {{nowrap|1=1 + 1 + 1 = 11<sub>2</sub>}}. This time, a 1 is carried, and a 1 is written in the bottom row. Proceeding like this gives the final answer 100100<sub>2</sub> (36<sub>10</sub>).

=== Computers ===
[[File:Opampsumming2.svg|right|frame|Addition with an op-amp. See [[Operational amplifier applications#Summing amplifier|Summing amplifier]] for details.]]
[[Analog computer]]s work directly with physical quantities, so their addition mechanisms depend on the form of the addends. A mechanical adder might represent two addends as the positions of sliding blocks, in which case they can be added with an [[arithmetic mean|averaging]] [[lever]]. If the addends are the rotation speeds of two [[axle|shafts]], they can be added with a [[differential (mechanics)|differential]]. A hydraulic adder can add the [[pressure]]s in two chambers by exploiting [[Newton's laws of motion|Newton's second law]] to balance forces on an assembly of [[piston]]s. The most common situation for a general-purpose analog computer is to add two [[voltage]]s (referenced to [[ground (electricity)|ground]]); this can be accomplished roughly with a [[resistor]] [[Electronic circuit|network]], but a better design exploits an [[operational amplifier]].{{sfnp|Truitt|Rogers|1960|pp=1;44–49, 2;77–78}}

Addition is also fundamental to the operation of [[computer|digital computers]], where the efficiency of addition, in particular the [[carry (arithmetic)|carry]] mechanism, is an important limitation to overall performance.{{sfnp|Gschwind|McCluskey|1975|p=[http://books.google.com/books?id=VLmrCAAAQBAJ&pg=PA233 233]}}

[[File:BabbageDifferenceEngine.jpg|left|thumb|Part of Charles Babbage's [[Difference Engine]] including the addition and carry mechanisms]]
The [[abacus]], also called a counting frame, is a calculating tool that was in use centuries before the adoption of the written modern numeral system and is still widely used by merchants, traders and clerks in [[Asia]], [[Africa]], and elsewhere; it dates back to at least 2700–2300&nbsp;BC, when it was used in [[Sumer]].<ref>{{cite book |last=Ifrah |first=Georges |year=2001 |title=The Universal History of Computing: From the Abacus to the Quantum Computer |publisher=John Wiley & Sons, Inc. |location=New York |isbn=978-0-471-39671-0 |url=https://archive.org/details/unset0000unse_w3q2 }} p. 11</ref>

[[Blaise Pascal]] invented the mechanical calculator in 1642;<ref>{{harvtxt|Marguin|1994}}, p. 48. Quoting {{harvtxt|Taton|1963}}.</ref> it was the first operational [[adding machine]]. It made use of a gravity-assisted carry mechanism. It was the only operational mechanical calculator in the 17th&nbsp;century<ref>See [[Pascal's calculator#Competing designs|Competing designs]] in Pascal's calculator article</ref> and the earliest automatic, digital computer. [[Pascal's calculator]] was limited by its carry mechanism, which forced its wheels to only turn one way so it could add. To subtract, the operator had to use the [[method of complements|Pascal's calculator's complement]], which required as many steps as an addition.<ref>{{cite journal |last=Kistermann |first=F. W. |year=1998 |title=Blaise Pascal's adding machine: new findings and conclusions |journal=IEEE Annals of the History of Computing |volume=20 |number=1 |pages=69–76 |doi=10.1109/85.646211}}</ref> [[Gottfried Leibniz]] built [[Stepped reckoner|another mechanical calculator]], finished in 1694, and [[Giovanni Poleni]] improved on the design in 1709 with a calculating clock made of wood that could perform all four arithmetical operations. These early attempts were not commercially successful but inspired later mechanical calculators of the 19th century.<ref>{{cite book |last=Campanile |first=Benedetta |chapter=La girandola di Poleni: un progetto destinato a scomparire |language=it |title=Atti del XLIII Convegno annuale SISFA |year=2024 |pages=151–158 |doi=10.6093/978-88-6887-278-6 |doi-access=free |editor1-last=Di Mauro |editor1-first=Marco |editor2-last=Romano |editor2-first=Luigi |editor3-last=Zanini |editor3-first=Valeria}}</ref>

[[File:Full-adder.svg|thumb|"[[Adder (electronics)|Full adder]]" logic circuit that adds two binary digits, ''A'' and ''B'', along with a carry input ''C<sub>in</sub>'', producing the sum bit, ''S'', and a carry output, ''C<sub>out</sub>''.]]
[[Adder (electronics)|Adders]] execute integer addition in electronic digital computers, usually using [[binary arithmetic]]. The simplest architecture is the ripple carry adder, which follows the standard multi-digit algorithm. One slight improvement is the [[Carry bypass adder|carry skip]] design, again following human intuition; one does not perform all the carries in computing {{nowrap|999 + 1}}, but one bypasses the group of 9s and skips to the answer.{{sfnp|Flynn|Oberman|2001|pp=2, 8}}

In practice, computational addition may be achieved via [[Exclusive or|XOR]] and [[Bitwise operation#AND|AND]] bitwise logical operations in conjunction with bitshift operations as shown in the [[pseudocode]] below. Both XOR and AND gates are straightforward to realize in digital logic, allowing the realization of [[Adder (electronics)|full adder]] circuits, which in turn may be combined into more complex logical operations. In modern digital computers, integer addition is typically the fastest arithmetic instruction, yet it has the largest impact on performance, since it underlies all [[floating-point arithmetic|floating-point operations]] as well as such basic tasks as [[memory address|address]] generation during [[memory (computers)|memory]] access and fetching [[instruction (computer science)|instructions]] during [[control flow|branching]]. To increase speed, modern designs calculate digits in [[parallel algorithm|parallel]]; these schemes go by such names as carry select, [[carry lookahead adder|carry lookahead]], and the [[Ling adder|Ling]] pseudocarry. Many implementations are, in fact, hybrids of these last three designs.{{sfnmp
 | 1a1 = Flynn | 1a2 = Oberman | 1y = 2001 | 1pp = 1–9
 | 2a1 = Liu | 2a2 = Tan | 2a3 = Song | 2a4 = Chen | 2y = 2010 | 2p = 194
}} Unlike addition on paper, addition on a computer often changes the addends. Both addends are destroyed on the ancient [[abacus]] and adding board, leaving only the sum. The influence of the abacus on mathematical thinking was strong enough that early [[Latin]] texts often claimed that in the process of adding "a number to a number", both numbers vanish.{{sfnp|Karpinski|1925|pp=102–103}} In modern times, the ADD instruction of a [[microprocessor]] often replaces the augend with the sum but preserves the addend.<ref>The identity of the augend and addend varies with architecture. For ADD in [[x86]] see {{harvtxt|Horowitz|Hill|2001}}, p. 679. For ADD in [[68k]] see {{harvtxt|Horowitz|Hill|2001}}, p. 767.</ref> In a [[high-level programming language]], evaluating <math> a + b </math> does not change either <math> a </math> or <math> b </math>; if the goal is to replace <math> a </math> with the sum this must be explicitly requested, typically with the statement <math> a = a + b </math>. Some languages like [[C (programming language)|C]] or [[C++]] allow this to be abbreviated as {{nowrap|1=<code>''a'' += ''b''</code>}}.

<syntaxhighlight lang="c">
// Iterative algorithm
int add(int x, int y) {
    int carry = 0;
    while (y != 0) {      
        carry = AND(x, y);   // Logical AND
        x     = XOR(x, y);   // Logical XOR
        y     = carry << 1;  // left bitshift carry by one
    }
    return x; 
}

// Recursive algorithm
int add(int x, int y) {
    return x if (y == 0) else add(XOR(x, y), AND(x, y) << 1);
}
</syntaxhighlight>

On a computer, if the result of an addition is too large to store, an [[arithmetic overflow]] occurs, resulting in an incorrect answer. Unanticipated arithmetic overflow is a fairly common cause of [[software bug|program errors]]. Such overflow bugs may be hard to discover and diagnose because they may manifest themselves only for very large input data sets, which are less likely to be used in validation tests.<ref>Joshua Bloch, [http://googleresearch.blogspot.com/2006/06/extra-extra-read-all-about-it-nearly.html "Extra, Extra – Read All About It: Nearly All Binary Searches and Mergesorts are Broken"] {{Webarchive|url=https://web.archive.org/web/20160401140544/http://googleresearch.blogspot.com/2006/06/extra-extra-read-all-about-it-nearly.html |date=2016-04-01 }}. Official Google Research Blog, June 2, 2006.</ref> The [[Year 2000 problem]] was a series of bugs where overflow errors occurred due to the use of a 2-digit format for years.{{sfnp|Neumann|1987}}

Computers have another way of representing numbers, called ''[[floating-point arithmetic]]'', which is similar to scientific notation described above and which reduces the overflow problem. Each floating point number has two parts, an exponent and a mantissa. To add two floating-point numbers, the exponents must match, which typically means shifting the mantissa of the smaller number. If the disparity between the larger and smaller numbers is too great, a loss of precision may result. If many smaller numbers are to be added to a large number, it is best to add the smaller numbers together first and then add the total to the larger number, rather than adding small numbers to the large number one at a time. This makes floating point addition non-associative in general. See [[floating-point arithmetic#Accuracy problems]].{{cn|date=June 2025}}

== Addition of numbers ==
To prove the usual properties of addition, one must first define addition for the context in question. Addition is first defined on the [[natural number]]s. In [[set theory]], addition is then extended to progressively larger sets that include the natural numbers: the [[integer]]s, the [[rational number]]s, and the [[real number]]s.<ref>[[Herbert Enderton|Enderton]] chapters 4 and 5, for example, follow this development.</ref> In [[mathematics education]],<ref>According to a survey of the nations with highest TIMSS mathematics test scores; see {{harvtxt|Schmidt|Houang|Cogan|2002}}, p. 4.</ref> positive fractions are added before negative numbers are even considered; this is also the historical route.<ref>{{harvtxt|Baez|Dolan|2001}}, p. 37 explains the historical development, in "stark contrast" with the set theory presentation: "Apparently, half an apple is easier to understand than a negative apple!"</ref>

=== Natural numbers ===
{{Further|Natural number}}
There are two popular ways to define the sum of two natural numbers <math> a </math> and <math> b </math>. If one defines natural numbers to be the [[Cardinal number|cardinalities]] of finite sets (the cardinality of a set is the number of elements in the set), then it is appropriate to define their sum as follows:{{sfnmp
 | 1a1 = Begle | 1y = 1975 | 1p = 49
 | 2a1 = Johnson | 2y = 1975 | 2p = 120
 | 3a1 = Devine | 3a2 = Olson | 3a3 = Olson | 3y = 1991 | 3p = 75
}}
{{blockquote|Let <math> N(S) </math> be the cardinality of a set <math> S </math>. Take two disjoint sets <math> A </math> and <math> B </math>, with <math> N(A) = a </math> and <math> N(B) = b </math>. Then <math> a + b </math> is defined as <math> N(A \cup B)</math>.
}}
Here <math> A \cup B </math> means the [[union (set theory)|union]] of <math> A </math> and <math> B </math>. An alternate version of this definition allows <math> A </math> and <math> B </math> to possibly overlap and then takes their [[disjoint union]], a mechanism that allows common elements to be separated out and therefore counted twice.

The other popular definition is recursive:{{sfnp|Enderton|1977|p=[http://books.google.com/books?id=JlR-Ehk35XkC&pg=PA79 79]}}
{{blockquote|Let <math> n^+ </math> be the successor of <math> n </math>, that is the number following <math> n </math> in the natural numbers, so <math> 0^+ = 1 </math>, <math> 1^+ = 2 </math>. Define <math> a + 0 = a </math>. Define the general sum recursively by <math> a + b^+ = (a + b)^+ </math>. Hence <math> 1 + 1 = 1 + 0^+ = (1 + 0)^+ = 1^+ = 2 </math>.
}}
Again, there are minor variations upon this definition in the literature. Taken literally, the above definition is an application of the [[Recursion#The recursion theorem|recursion theorem]] on the [[partially ordered set]] <math> \mathbb{N}^2 </math>.<ref>For a version that applies to any poset with the [[descending chain condition]], see {{harvtxt|Bergman|2005}}, p. 100</ref> On the other hand, some sources prefer to use a restricted recursion theorem that applies only to the set of natural numbers. One then considers <math> a </math> to be temporarily "fixed", applies recursion on <math> b </math> to define a function "<math> a + </math>", and pastes these unary operations for all <math> a </math> together to form the full binary operation.<ref>{{harvtxt|Enderton|1977}}, p. [http://books.google.com/books?id=JlR-Ehk35XkC&pg=PA79 79] observes, "But we want one binary operation <math> + </math>, not all these little one-place functions."</ref>

This recursive formulation of addition was developed by Dedekind as early as 1854, and he would expand upon it in the following decades. He proved the associative and commutative properties, among others, through [[mathematical induction]].{{sfnp|Ferreirós|1999|p=223}}

=== Integers ===
{{Further|Integer}}
The simplest conception of an integer is that it consists of an [[absolute value]] (which is a natural number) and a [[sign (mathematics)|sign]] (generally either [[positive number|positive]] or [[negative numbers|negative]]). The integer zero is a special third case, being neither positive nor negative. The corresponding definition of addition must proceed by cases:{{sfnmp
 | 1a1 = Smith | 1y = 1980 | 1p = 234
 | 2a1 = Sparks | 2a2 = Rees | 2y = 1979 | 2p = 66
}}
{{blockquote|For an integer <math> n </math>, let <math> |n| </math> be its absolute value. Let <math> a </math> and <math> b </math> be integers. If either <math> a </math> or <math> b </math> is zero, treat it as an identity. If <math> a </math> and <math> b </math> are both positive, define <math> a + b = |a| + |b| </math>. If <math> a </math> and <math> b </math> are both negative, define <math> a + b = -(|a| + |b|) </math>. If <math> a </math> and <math> b </math> have different signs, define <math> a + b </math> to be the difference between <math> |a| </math> and <math> |b| </math>, with the sign of the term whose absolute value is larger.
}}
As an example, {{nowrap|1=−6 + 4 = −2}}; because −6 and 4 have different signs, their absolute values are subtracted, and since the absolute value of the negative term is larger, the answer is negative.

Although this definition can be useful for concrete problems, the number of cases to consider complicates proofs unnecessarily. So the following method is commonly used for defining integers. It is based on the remark that every integer is the difference of two natural integers and that two such differences, <math> a - b </math> and <math> c - d </math> are equal if and only if <math> a + d = b + c </math>. So, one can define formally the integers as the [[equivalence class]]es of [[ordered pair]]s of natural numbers under the [[equivalence relation]] <math> (a,b) \sim (c,d) </math> if and only if <math> a + d = b + c </math>.{{sfnp|Campbell|1970|p=[https://archive.org/details/structureofarith00camp/page/83 83]}} The equivalence class of <math> (a,b) </math> contains either <math> (a-b,0) </math> if <math> a \ge b </math>, or <math> (0,b-a) </math> if otherwise. Given that <math> n </math> is a natural number, then one can denote <math> +n </math> the equivalence class of <math> (n,0) </math>, and by <math> -n </math> the equivalence class of <math> (0,n) </math>. This allows identifying the natural number <math> n </math> with the equivalence class <math> +n </math>.

The addition of ordered pairs is done component-wise:{{sfnp|Campbell|1970|p=[https://archive.org/details/structureofarith00camp/page/84 84]}}
<math display="block"> (a,b) + (c,d) = (a+c, b+d).</math>
A straightforward computation shows that the equivalence class of the result depends only on the equivalence classes of the summands, and thus that this defines an addition of equivalence classes, that is, integers.{{sfnp|Enderton|1977|p=[http://books.google.com/books?id=JlR-Ehk35XkC&pg=PA92 92]}} Another straightforward computation shows that this addition is the same as the above case definition.

This way of defining integers as equivalence classes of pairs of natural numbers can be used to embed into a [[group (mathematics)|group]] any commutative [[semigroup]] with [[cancellation property]]. Here, the semigroup is formed by the natural numbers, and the group is the additive group of integers.{{sfnp|Cohn|2003|p=[http://books.google.com/books?id=jlLmBwAAQBAJ&pg=PA29 29]}} The rational numbers are constructed similarly, by taking as a semigroup the nonzero integers with multiplication.<ref>{{cite encyclopedia |encyclopedia=Encyclopaedia of Mathematics |volume=7 |title=Rational Number |publisher=Kluwer |page=498–499 |url=https://archive.org/details/encyclopaediaofm0007unse/page/498/mode/1up?q=%22number+expressible+as+a+fraction%22 |url-access=limited |last=Kudryavtsev |first=L.D. |editor-last=Hazewinkel |editor-first=Michiel |isbn=1-55608-006-9 |year=1991 }}</ref>

This construction has also been generalized under the name of [[Grothendieck group]] to the case of any commutative semigroup. Without the cancellation property, the [[semigroup homomorphism]] from the semigroup into the group may be non-injective. Originally, the Grothendieck group was the result of this construction applied to the equivalence classes under isomorphisms of the objects of an [[abelian category]], with the [[direct sum]] as semigroup operation.{{cn|date=June 2025}}

=== Rational numbers (fractions) ===
{{main article|Field of fractions}}
Addition of [[rational number]]s involves the [[fraction]]s. The computation can be done by using the [[least common denominator]], but a conceptually simpler definition involves only integer addition and multiplication:
<math display="block"> \frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}.</math>
As an example, the sum <math display="inline">\frac 34 + \frac 18 = \frac{3 \,\times\, 8 \,+\, 4 \,\times\, 1}{4 \times 8} = \frac{24 \,+\, 4}{32} = \frac{28}{32} = \frac78</math>.

Addition of fractions is much simpler when the [[denominator]]s are the same; in this case, one can simply add the numerators while leaving the denominator the same:
<math display="block"> \frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}, </math>
so <math display="inline">\frac 14 + \frac 24 = \frac{1\,+\,2}{4} = \frac 34</math>.{{sfnp|Cameron|Craig|2013|p=29}}

The commutativity and associativity of rational addition are easy consequences of the laws of integer arithmetic.<ref>The verifications are carried out in {{harvtxt|Enderton|1977|p=[http://books.google.com/books?id=JlR-Ehk35XkC&pg=PA104 104]}} and sketched for a general field of fractions over a commutative ring in {{harvtxt|Dummit|Foote|1999}}, p. 263.</ref>

=== Real numbers ===
{{Further|Construction of the real numbers}}
A common construction of the set of real numbers is the Dedekind completion of the set of rational numbers. A real number is defined to be a [[Dedekind cut]] of rationals: a [[non-empty set]] of rationals that is closed downward and has no [[greatest element]]. The sum of real numbers ''a'' and ''b'' is defined element by element:{{sfnp|Enderton|1977|p=[http://books.google.com/books?id=JlR-Ehk35XkC&pg=PA114 114]}}
<math display="block"> a+b = \{q+r \mid q\in a, r\in b\}.</math>
This definition was first published, in a slightly modified form, by [[Richard Dedekind]] in 1872.<ref>{{harvtxt|Ferreirós|1999}}, p. 135; see section 6 of ''[http://www.ru.nl/w-en-s/gmfw/bronnen/dedekind2.html Stetigkeit und irrationale Zahlen] {{webarchive |url=https://web.archive.org/web/20051031071536/http://www.ru.nl/w-en-s/gmfw/bronnen/dedekind2.html |date=2005-10-31 }}''.</ref>
The commutativity and associativity of real addition are immediate; defining the real number 0 as the set of negative rationals, it is easily seen as the additive identity. Probably the trickiest part of this construction pertaining to addition is the definition of additive inverses.<ref>The intuitive approach, inverting every element of a cut and taking its complement, works only for irrational numbers; see {{harvtxt|Enderton|1977}}, p. [http://books.google.com/books?id=JlR-Ehk35XkC&pg=PA117 117] for details.</ref>

[[File:AdditionRealCauchy.svg|right|thumb|Adding <math> \pi^2/6 </math> and <math> e </math> using Cauchy sequences of rationals.]]
Unfortunately, dealing with the multiplication of Dedekind cuts is a time-consuming case-by-case process similar to the addition of signed integers.<ref>Schubert, E. Thomas, Phillip J. Windley, and James Alves-Foss. "Higher Order Logic Theorem Proving and Its Applications: Proceedings of the 8th International Workshop, volume 971 of." ''Lecture Notes in Computer Science'' (1995).</ref> Another approach is the metric completion of the rational numbers. A real number is essentially defined to be the limit of a [[Cauchy sequence]] of rationals, lim&nbsp;''a''<sub>''n''</sub>. Addition is defined term by term:<ref>Textbook constructions are usually not so cavalier with the "lim" symbol; see {{harvtxt|Burrill|1967}}, p. 138 for a more careful, drawn-out development of addition with Cauchy sequences.</ref>
<math display="block">\lim_n a_n + \lim_n b_n = \lim_n (a_n + b_n).</math>
This definition was first published by [[Georg Cantor]], also in 1872, although his formalism was slightly different.{{sfnp|Ferreirós|1999|p=128}}
One must prove that this operation is well-defined, dealing with co-Cauchy sequences. Once that task is done, all the properties of real addition follow immediately from the properties of rational numbers. Furthermore, the other arithmetic operations, including multiplication, have straightforward, analogous definitions.{{sfnp|Burrill|1967|p=140}}

=== Complex numbers ===
[[File:Complex Number Addition Visualization svg.svg|right|thumb|Addition of two complex numbers can be done geometrically by constructing a parallelogram.]]
[[Complex number]]s are added by adding the real and imaginary parts of the summands.<ref>{{Citation |last=Conway |first=John B. |title=Functions of One Complex Variable I |year=1986 |publisher=Springer |isbn=978-0-387-90328-6}}</ref><ref>{{Citation |last1=Joshi |first1=Kapil D
|title=Foundations of Discrete Mathematics |publisher=[[John Wiley & Sons]] |location=New York |isbn=978-0-470-21152-6|year=1989}}</ref> That is to say:
:<math>(a+bi) + (c+di) = (a+c) + (b+d)i.</math>
Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers ''A'' and ''B'', interpreted as points of the complex plane, is the point ''X'' obtained by building a [[parallelogram]] three of whose vertices are ''O'', ''A'' and ''B''.{{sfnp|Özhan|2022|p=[http://books.google.com/books?id=IS51EAAAQBAJ&pg=PA10 10]}} Equivalently, ''X'' is the point such that the [[triangle]]s with vertices ''O'', ''A'', ''B'', and ''X'', ''B'', ''A'', are [[Congruence (geometry)|congruent]].

== Generalizations ==
Many binary operations can be viewed as generalizations of the addition operation on the real numbers. The field of algebra is centrally concerned with such generalized operations, and they also appear in [[set theory]] and [[category theory]].

=== Algebra ===
{{main|Vector addition|Matrix addition|Modular arithmetic|Linear combination}}
In [[linear algebra]], a [[vector space]] is an algebraic structure that allows for adding any two [[coordinate vector|vectors]] and for scaling vectors. A familiar vector space is the set of all ordered pairs of real numbers; the ordered pair <math> (a,b) </math> is interpreted as a vector from the origin in the Euclidean plane to the point <math> (a,b) </math> in the plane. The sum of two vectors is obtained by adding their individual coordinates:
<math display="block"> (a,b) + (c,d) = (a+c,b+d). </math>
This addition operation is central to [[classical mechanics]], in which [[velocity|velocities]], [[acceleration]]s and [[force]]s are all represented by vectors.{{sfnp|Gbur|2011|p=1}}

Matrix addition is defined for two matrices of the same dimensions. The sum of two ''m'' × ''n'' (pronounced "m by n") matrices '''A''' and '''B''', denoted by {{nowrap|'''A''' + '''B'''}}, is again an {{nowrap|''m'' × ''n''}} matrix computed by adding corresponding elements:<ref>Lipschutz, S., & Lipson, M. (2001). Schaum's outline of theory and problems of linear algebra. Erlangga.</ref><ref>{{cite book |title=Mathematical methods for physics and engineering |url=https://archive.org/details/mathematicalmeth00rile |url-access=registration |first1=K.F. |last1=Riley |first2=M.P.|last2=Hobson |first3=S.J. |last3=Bence |publisher=Cambridge University Press |year=2010 |isbn=978-0-521-86153-3}}</ref>
<math display=block>\begin{align}
\mathbf{A}+\mathbf{B} &= 
\begin{bmatrix}
 a_{11} & a_{12} & \cdots & a_{1n} \\
 a_{21} & a_{22} & \cdots & a_{2n} \\
 \vdots & \vdots & \ddots & \vdots \\
 a_{m1} & a_{m2} & \cdots & a_{mn} \\
\end{bmatrix} +
\begin{bmatrix}
 b_{11} & b_{12} & \cdots & b_{1n} \\
 b_{21} & b_{22} & \cdots & b_{2n} \\
 \vdots & \vdots & \ddots & \vdots \\
 b_{m1} & b_{m2} & \cdots & b_{mn} \\
\end{bmatrix}\\[8mu]
 &= \begin{bmatrix}
 a_{11} + b_{11} & a_{12} + b_{12} & \cdots & a_{1n} + b_{1n} \\
 a_{21} + b_{21} & a_{22} + b_{22} & \cdots & a_{2n} + b_{2n} \\
 \vdots & \vdots & \ddots & \vdots \\
 a_{m1} + b_{m1} & a_{m2} + b_{m2} & \cdots & a_{mn} + b_{mn} \\
\end{bmatrix} \\

\end{align}</math>

For example:
: <math>
\begin{align}
  \begin{bmatrix}
    1 & 3 \\
    1 & 0 \\
    1 & 2
  \end{bmatrix}
+
  \begin{bmatrix}
    0 & 0 \\
    7 & 5 \\
    2 & 1
  \end{bmatrix}
&= \begin{bmatrix}
    1+0 & 3+0 \\
    1+7 & 0+5 \\
    1+2 & 2+1
  \end{bmatrix}\\[8mu]
&=
  \begin{bmatrix}
    1 & 3 \\
    8 & 5 \\
    3 & 3
  \end{bmatrix}
\end{align}
</math>

In [[modular arithmetic]], the set of available numbers is restricted to a finite subset of the integers, and addition "wraps around" when reaching a certain value, called the modulus.{{sfnp|Omondi|2020|p=[http://books.google.com/books?id=_m7NDwAAQBAJ&pg=PA142 142]}} For example, the set of integers modulo 12 has twelve elements; it inherits an addition operation from the integers that is central to [[set theory (music)|musical set theory]].{{sfnp|Princeton University Press|2008|p=[http://books.google.com/books?id=ZOfUsvemJDMC&pg=PA938 938]}} The set of integers modulo&nbsp;2 has just two elements; the addition operation it inherits is known in [[Boolean logic]] as the "[[exclusive or]]" function. A similar "wrap around" operation arises in [[geometry]], where the sum of two [[angle|angle measures]] is often taken to be their sum as real numbers modulo&nbsp;2π. This amounts to an addition operation on the [[circle]], which in turn generalizes to addition operations on many-dimensional [[torus|tori]].{{cn|date=June 2025}}

The general theory of abstract algebra allows an "addition" operation to be any [[associative]] and [[commutative]] operation on a set. Basic [[algebraic structure]]s with such an addition operation include [[commutative monoid]]s and [[abelian group]]s.{{cn|date=June 2025}}

[[Linear combination]]s combine multiplication and summation; they are sums in which each term has a multiplier, usually a [[real numbers|real]] or [[complex numbers|complex]] number. Linear combinations are especially useful in contexts where straightforward addition would violate some normalization rule, such as [[mixed strategy|mixing]] of [[strategy (game theory)|strategies]] in [[game theory]] or [[quantum superposition|superposition]] of [[quantum state|states]] in [[quantum mechanics]].{{sfnp|Rieffel|Polak|2011|p=16}}

=== Set theory and category theory ===
A far-reaching generalization of the addition of natural numbers is the addition of [[ordinal number]]s and [[cardinal number]]s in set theory. These give two different generalizations of the addition of natural numbers to the [[transfinite number|transfinite]]. Unlike most addition operations, the addition of ordinal numbers is not commutative.{{sfnp|Cheng|2017|pp=124–132}} Addition of cardinal numbers, however, is a commutative operation closely related to the [[disjoint union]] operation.{{cn|date=June 2025}}

In [[category theory]], disjoint union is seen as a particular case of the [[coproduct]] operation,{{sfnp|Riehl|2016|p=100}} and general coproducts are perhaps the most abstract of all the generalizations of addition. Some coproducts, such as [[direct sum]] and [[wedge sum]], are named to evoke their connection with addition.{{cn|date=June 2025}}

== Related operations ==
Addition, along with subtraction, multiplication, and division, is considered one of the basic operations and is used in [[elementary arithmetic]].

=== Arithmetic ===
[[Subtraction]] can be thought of as a kind of addition—that is, the addition of an [[additive inverse]]. Subtraction is itself a sort of inverse to addition, in that adding <math> x </math> and subtracting <math> x </math> are [[inverse function]]s.{{sfnp|Kay|2021|p=[https://books.google.com/books?id=aw81EAAAQBAJ&pg=PA44 44]}} Given a set with an addition operation, one cannot always define a corresponding subtraction operation on that set; the set of natural numbers is a simple example. On the other hand, a subtraction operation uniquely determines an addition operation, an additive inverse operation, and an additive identity; for this reason, an additive group can be described as a set that is closed under subtraction.<ref>The set still must be nonempty. {{harvtxt|Dummit|Foote|1999}}, p. 48 discuss this criterion written multiplicatively.</ref>

[[Multiplication]] can be thought of as [[Multiplication and repeated addition|repeated addition]]. If a single term {{mvar|x}} appears in a sum <math> n </math> times, then the sum is the product of <math> n </math> and {{mvar|x}}. Nonetheless, this works only for [[natural number]]s.{{sfnp|Musser|Peterson|Burger|2013|p=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA101 101]}} By the definition in general, multiplication is the operation between two numbers, called the multiplier and the multiplicand, that are combined into a single number called the product.

[[File:Csl.JPG|thumb|A circular slide rule]]
In the real and complex numbers, addition and multiplication can be interchanged by the [[exponential function]]:{{sfnp|Rudin|1976|p=178}}
<math display="block"> e^{a+b} = e^a e^b. </math>
This identity allows multiplication to be carried out by consulting a [[mathematical table|table]] of [[logarithm]]s and computing addition by hand; it also enables multiplication on a [[slide rule]]. The formula is still a good first-order approximation in the broad context of [[Lie group]]s, where it relates multiplication of infinitesimal group elements with addition of vectors in the associated [[Lie algebra]].{{sfnp|Lee|2003|p=526|loc=Proposition 20.9}}

There are even more generalizations of multiplication than addition.<ref>{{harvtxt|Linderholm|1971}}, p. 49 observes, "By ''multiplication'', properly speaking, a mathematician may mean practically anything. By ''addition'' he may mean a great variety of things, but not so great a variety as he will mean by 'multiplication'."</ref> In general, multiplication operations always [[distributivity|distribute]] over addition; this requirement is formalized in the definition of a [[ring (mathematics)|ring]]. In some contexts, integers, distributivity over addition, and the existence of a multiplicative identity are enough to determine the multiplication operation uniquely. The distributive property also provides information about the addition operation; by expanding the product <math> (1+1)(a+b) </math> in both ways, one concludes that addition is forced to be commutative. For this reason, ring addition is commutative in general.<ref>{{harvtxt|Dummit|Foote|1999}}, p. 224. For this argument to work, one must assume that addition is a group operation and that multiplication has an identity.</ref>

[[Division (mathematics)|Division]] is an arithmetic operation remotely related to addition. Since <math> a/b = ab^{-1} </math>, division is right distributive over addition: <math> (a+b)/c = a/c + b/c </math>.<ref>For an example of left and right distributivity, see {{harvtxt|Loday|2002}}, p. 15.</ref> However, division is not left distributive over addition, such as <math> 1/(2+2) </math> is not the same as <math> 1/2 + 1/2 </math>.

=== Ordering ===
[[File:XPlusOne.svg|right|thumb|[[Log-log plot]] of {{nowrap|1={{mvar|x}} + 1}} and {{nowrap|1=max ({{mvar|x}}, 1)}} from {{mvar|x}} = 0.001 to 1000<ref>Compare {{harvtxt|Viro|2001}}, p. 2, Figure 1.</ref>]]
The maximum operation <math> \max(a,b) </math> is a binary operation similar to addition. In fact, if two nonnegative numbers <math> a </math> and <math> b </math> are of different [[orders of magnitude]], their sum is approximately equal to their maximum. This approximation is extremely useful in the applications of mathematics, for example, in truncating [[Taylor series]]. However, it presents a perpetual difficulty in [[numerical analysis]], essentially since "max" is not invertible. If <math> b </math> is much greater than <math> a </math>, then a straightforward calculation of <math> (a + b) - b </math> can accumulate an unacceptable [[round-off error]], perhaps even returning zero. See also ''[[Loss of significance]]''.{{cn|date=June 2025}}

The approximation becomes exact in a kind of infinite limit; if either <math> a </math> or <math> b </math> is an infinite [[cardinal number]], their cardinal sum is exactly equal to the greater of the two.<ref>Enderton calls this statement the "Absorption Law of Cardinal Arithmetic"; it depends on the comparability of cardinals and therefore on the [[Axiom of Choice]].</ref> Accordingly, there is no subtraction operation for infinite cardinals.{{sfnp|Enderton|1977|p=[http://books.google.com/books?id=JlR-Ehk35XkC&pg=PA164 164]}}

Maximization is commutative and associative, like addition. Furthermore, since addition preserves the ordering of real numbers, addition distributes over "max" in the same way that multiplication distributes over addition:
<math display="block"> a + \max(b,c) = \max(a+b,a+c).</math>
For these reasons, in [[tropical geometry]] one replaces multiplication with addition and addition with maximization. In this context, addition is called "tropical multiplication", maximization is called "tropical addition", and the tropical "additive identity" is [[extended real number line|negative infinity]].{{sfnp|Mikhalkin|2006|p=1}} Some authors prefer to replace addition with minimization; then the additive identity is positive infinity.{{sfnp|Akian|Bapat|Gaubert|2005|p=4}}

Tying these observations together, tropical addition is approximately related to regular addition through the [[logarithm]]:
<math display="block">\log(a+b) \approx \max(\log a, \log b),</math>
which becomes more accurate as the base of the logarithm increases.{{sfnp|Mikhalkin|2006|p=2}} The approximation can be made exact by extracting a constant <math> h </math>, named by analogy with the [[Planck constant]] from [[quantum mechanics]],{{sfnp|Litvinov|Maslov|Sobolevskii|1999|p=3}} and taking the "[[classical limit]]" as <math> h </math> tends to zero:
<math display="block">\max(a,b) = \lim_{h\to 0}h\log(e^{a/h}+e^{b/h}).</math>
In this sense, the maximum operation is a ''dequantized'' version of addition.{{sfnp|Viro|2001|p=4}}

=== In probability theory ===
[[Convolution]] is used to add two independent [[random variable]]s defined by [[probability distribution|distribution functions]]. Its usual definition combines integration, subtraction, and multiplication.{{sfnp|Gbur|2011|p=300}}

== See also ==
* [[Lunar arithmetic]]
* [[Mental calculation|Mental arithmetic]]
* [[Parallel addition (mathematics)]]
* [[Verbal arithmetic]] (also known as cryptarithms), puzzles involving addition

== Notes ==
{{notelist}}

== Footnotes ==
{{reflist}}

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 | last = Schwartzman | first = Steven
 | title = The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English
 | url = https://archive.org/details/wordsofmathemati0000schw
 | url-access = registration
 | publisher = [[Mathematical Association of America|MAA]]
 | year = 1994
 | isbn = 978-0-88385-511-9
}}
* {{cite book
 | last1 = Shmerko | first1 = V. P.
 | last2 = Yanushkevich | first2 = Svetlana N. | author2-link = Svetlana Yanushkevich
 | last3 = Lyshevski | first3 = S. E.
 | year = 2009
 | title = Computer arithmetics for nanoelectronics
 | publisher = [[CRC Press]]
}}
* {{cite book
 | last = Smith | first = Frank
 | title = The Glass Wall: Why Mathematics Can Seem Difficult
 | url = https://archive.org/details/glasswallwhymath0000smit |url-access=registration |publisher=Teachers College Press
 | year = 2002
 | isbn = 978-0-8077-4242-6
}}
* {{cite book
 | last = Smith | first = Karl
 | title = The Nature of Modern Mathematics
 | edition = 3rd
 | publisher = Wadsworth
 | year = 1980
 | isbn = 978-0-8185-0352-8}}
* {{cite book
 | last1 = Sparks | first1 = F.
 | last2 = Rees | first2 = C.
 | title = A Survey of Basic Mathematics
 | publisher = McGraw-Hill
 | year = 1979
 | isbn = 978-0-07-059902-4
}}
* {{cite book
 | last = Stewart | first = James
 | title = Calculus: Early Transcendentals
 | edition = 4th
 | publisher = Brooks/Cole
 | year = 1999
 | isbn = 978-0-534-36298-0
 | url = https://archive.org/details/calculusearlytra00stew
}}
* {{cite book
 | last = Taton | first= René
 | year = 1963
 | language = fr
 | title = Le Calcul Mécanique. Que Sais-Je ? n° 367
 | pages = 20–28
 | publisher = Presses universitaires de France
}}
* {{cite book
 | last1 = Truitt | first1 = T.
 | last2 = Rogers | first2 = A.
 | title = Basics of Analog Computers
 | publisher = John F. Rider
 | year = 1960
 | id = {{LCC|QA76.4|T7}}
}}
* {{cite book
 | last = van de Walle |first=John |title=Elementary and Middle School Mathematics: Teaching developmentally |edition=5e |publisher=Pearson |year=2004 |isbn=978-0-205-38689-5 |url=https://archive.org/details/elementarymiddle00vand }}
* {{Cite book |last=Viro |first=Oleg |year=2001 |url=http://www.math.uu.se/~oleg/dequant/dequantH1.html |title=European Congress of Mathematics: Barcelona, July 10–14, 2000, Volume I. Dequantization of Real Algebraic Geometry on Logarithmic Paper |editor1-first=Carles |editor1-last=Cascuberta |editor2-first=Rosa Maria |editor2-last=Miró-Roig |editor3-first=Joan |editor3-last=Verdera |editor4-first=Sebastià |editor4-last=Xambó-Descamps |publisher=Birkhäuser |location=Basel |isbn=978-3-7643-6417-5 |series=Progress in Mathematics |volume=201 |pages=135–146 |arxiv=math/0005163 |zbl=1024.14026 |bibcode=2000math......5163V }}
* {{cite conference |last=Wynn |first=Karen |book-title=The Development of Mathematical Skills. |title=Numerical competence in infants |publisher=Taylor & Francis |year=1998 |isbn=0-86377-816-X}}
{{Refend}}

== Further reading==
* {{cite conference |last1=Baroody |first1=Arthur |last2=Tiilikainen |first2=Sirpa |title=The Development of Arithmetic Concepts and Skills. Two perspectives on addition development |year=2003 |page=[https://archive.org/details/developmentofari0000unse/page/75 75] |isbn=0-8058-3155-X |publisher=Routledge |url=https://archive.org/details/developmentofari0000unse/page/75 }}
* {{cite book |last1=Davison |first1=David M. |last2=Landau |first2=Marsha S. |last3=McCracken |first3=Leah |last4=Thompson |first4=Linda |title=Mathematics: Explorations & Applications |edition=TE |publisher=Prentice Hall |year=1999 |isbn=978-0-13-435817-8}}
* {{cite book |first1=Lucas N.H. |last1=Bunt |first2=Phillip S. |last2=Jones |first3=Jack D. |last3=Bedient |title=The Historical roots of Elementary Mathematics |url=https://archive.org/details/historicalrootso0000bunt |url-access=registration |publisher=Prentice-Hall |year=1976 |isbn=978-0-13-389015-0}}
* {{cite journal |last=Poonen |first=Bjorn |year=2010 |title=Addition |url=http://www.girlsangle.org/page/bulletin.php |journal=Girls' Angle Bulletin  |volume=3 |issue=3–5 |issn=2151-5743}}
* {{cite conference |first=J. Fred |last=Weaver |title=Addition and Subtraction: A Cognitive Perspective |book-title=Addition and Subtraction: A Cognitive Perspective. Interpretations of Number Operations and Symbolic Representations of Addition and Subtraction |year=1982 |page=60 |isbn=0-89859-171-6 |publisher=Taylor & Francis}}

{{Elementary arithmetic}}
{{Hyperoperations}}
{{Authority control}}

[[Category:Addition| ]]
[[Category:Elementary arithmetic]]
[[Category:Mathematical notation]]
[[Category:Articles with example C code]]