Glossary of mathematical symbols: Difference between revisions

Latest revision as of 05:16, 18 November 2025

Template:Short description A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula or a mathematical expression. More formally, a mathematical symbol is any grapheme used in mathematical formulas and expressions. As formulas and expressions are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics.

The most basic symbols are the decimal digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), and the letters of the Latin alphabet. The decimal digits are used for representing numbers through the Hindu–Arabic numeral system. Historically, upper-case letters were used for representing points in geometry, and lower-case letters were used for variables and constants. Letters are used for representing many other types of mathematical object. As the number of these types has increased, the Greek alphabet and some Hebrew letters have also come to be used. For more symbols, other typefaces are also used, mainly boldface Template:Tmath, script typeface $𝒜, ℬ, \dots$ (the lower-case script face is rarely used because of the possible confusion with the standard face), German fraktur Template:Tmath, and blackboard bold Template:Tmath (the other letters are rarely used in this face, or their use is unconventional). It is commonplace to use alphabets, fonts and typefaces to group symbols by type (for example, boldface is often used for vectors and uppercase for matrices).

The use of specific Latin and Greek letters as symbols for denoting mathematical objects is not described in this article. For such uses, see Variable § Conventional variable names and List of mathematical constants. However, some symbols that are described here have the same shape as the letter from which they are derived, such as $\prod$ and $\sum$ .

These letters alone are not sufficient for the needs of mathematicians, and many other symbols are used. Some take their origin in punctuation marks and diacritics traditionally used in typography; others by deforming letter forms, as in the cases of $\in$ and $\forall$ . Others, such as Template:Math and Template:Math, were specially designed for mathematics.

Layout of this article

Normally, entries of a glossary are structured by topics and sorted alphabetically. This is not possible here, as there is no natural order on symbols, and many symbols are used in different parts of mathematics with different meanings, often completely unrelated. Therefore, some arbitrary choices had to be made, which are summarized below.
The article is split into sections that are sorted by an increasing level of technicality. That is, the first sections contain the symbols that are encountered in most mathematical texts, and that are supposed to be known even by beginners. On the other hand, the last sections contain symbols that are specific to some area of mathematics and are ignored outside these areas. However, the long section on brackets has been placed near to the end, although most of its entries are elementary: this makes it easier to search for a symbol entry by scrolling.
Most symbols have multiple meanings that are generally distinguished either by the area of mathematics where they are used or by their syntax, that is, by their position inside a formula and the nature of the other parts of the formula that are close to them.
As readers may not be aware of the area of mathematics to which the symbol that they are looking for is related, the different meanings of a symbol are grouped in the section corresponding to their most common meaning.
When the meaning depends on the syntax, a symbol may have different entries depending on the syntax. For summarizing the syntax in the entry name, the symbol $◻$ is used for representing the neighboring parts of a formula that contains the symbol. See Template:Slink for examples of use.
Most symbols have two printed versions. They can be displayed as Unicode characters, or in LaTeX format. With the Unicode version, using search engines and copy-pasting are easier. On the other hand, the LaTeX rendering is often much better (more aesthetic), and is generally considered a standard in mathematics. Therefore, in this article, the Unicode version of the symbols is used (when possible) for labelling their entry, and the LaTeX version is used in their description. So, for finding how to type a symbol in LaTeX, it suffices to look at the source of the article.
For most symbols, the entry name is the corresponding Unicode symbol. So, for searching the entry of a symbol, it suffices to type or copy the Unicode symbol into the search textbox. Similarly, when possible, the entry name of a symbol is also an anchor, which allows linking easily from another Wikipedia article. When an entry name contains special characters such as [,], and |, there is also an anchor, but one has to look at the article source to know it.
Finally, when there is an article on the symbol itself (not its mathematical meaning), it is linked to in the entry name.

Arithmetic operators

Template:Glossary Template:Term Template:Defn Template:Defn Template:Defn

@@ Line 29: / Line 29: @@
 {{defn |no=1 |1=Denotes [[subtraction]] and is read as ''minus''; for example, {{math|3 − 2}}.}}
 {{defn |no=2 |1=Denotes the [[additive inverse]] and is read as ''minus'', '' the negative of'', or ''the opposite of''; for example, {{math|−2}}.}}
-{{defn|no=3|1= Also used in place of {{math|\}} for denoting the [[set-theoretic complement]]; see [[#∖|\]] in {{slink||Set theory}}.}}
+{{defn|no=3|1= Also used in place of {{math|\}} for denoting the [[set-theoretic complement]]; see {{not a typo|[[#∖|\]]}} in {{slink||Set theory}}.}}
 {{term|1=×|content= {{math|1=×}} {{spaces|3|em}}([[multiplication sign]])}}
@@ Line 45: / Line 45: @@
 {{defn|no=2|1=Denotes the range of values that a measured quantity may have; for example, {{math|10 ± 2}} denotes an unknown value that lies between 8 and 12.}}
-{{term|1=∓|content= {{math|1=∓}} {{spaces|3|em}}([[minus-plus sign]])}}
+{{term|1=∓|content= {{math|1=∓}} {{spaces|3|em}}([[minus–plus sign]])}}
 {{defn|1=Used paired with {{math|±}}, denotes the opposite sign; that is, {{math|+}} if {{math|±}} is {{math|−}}, and {{math|−}} if {{math|±}} is {{math|+}}.}}
 {{term|1=÷|content= {{math|1=÷}} {{spaces|3|em}}([[division sign]])}}
-{{defn|1=Widely used for denoting [[division (mathematics)|division]] in [[English-speaking world| Anglophone]] countries, it is no longer in common use in mathematics and its use is "not recommended".<ref name=ISO>[[ISO 80000-2]], Section 9 "Operations", 2-9.6</ref> In some countries, it can indicate subtraction.}}
+{{defn|1=Although widely used for denoting [[division (mathematics)|division]] in [[English-speaking world| Anglophone]] countries, it is no longer in common use in mathematics and its use is "not recommended".<ref name=ISO>[[ISO 80000-2]], Section 9 "Operations", 2-9.6</ref> In some countries, it can indicate subtraction.}}
 {{term|ratio|content= {{math|1=:}} {{spaces|3|em}}([[colon (punctuation)|colon]])}}
@@ Line 71: / Line 71: @@
 {{term|caret|content= {{math|1=^}} {{spaces|3|em}}([[caret]])}}
 {{defn|no=1|[[Exponentiation]] is normally denoted with a [[superscript]]. However, <math>x^y</math> is often denoted {{math|''x''^''y''}} when superscripts are not easily available, such as in [[programming language]]s (including [[LaTeX]]) or plain text [[email]]s.}}
-{{defn|no=2|Not to be confused with [[#∧|∧]]}}
+{{defn|no=2|Not to be confused with [[#∧|∧]].}}
 {{glossary end}}
@@ Line 96: / Line 96: @@
 {{defn|no=3|1=Often used for denoting other types of similarity, for example, [[matrix similarity]] or [[similarity (geometry)|similarity of geometric shapes]].}}
 {{defn|no=4|Standard notation for an [[equivalence relation]].}}
-{{defn|no=5|In [[probability]] and [[statistics]], may specify the [[probability distribution]] of a [[random variable]]. For example, <math>X\sim N(0,1)</math> means that the distribution of the random variable {{mvar|X}} is  [[standard normal distribution|standard normal]].<ref>{{Cite book|url=https://archive.org/details/statisticsdataan0000tamh|title=Statistics and Data Analysis: From Elementary to Intermediate|date=2000 |isbn=978-0-13-744426-7 }}</ref>}}
+{{defn|no=5|In [[probability]] and [[statistics]], may specify the [[probability distribution]] of a [[random variable]]. For example, <math>X\sim N(0,1)</math> means that the distribution of the random variable {{mvar|X}} is  [[standard normal distribution|standard normal]].<ref>{{Cite book|url=https://archive.org/details/statisticsdataan0000tamh|title=Statistics and Data Analysis: From Elementary to Intermediate|date=2000 |isbn=978-0-13-744426-7 |last1=Tamhane |first1=Ajit C. |last2=Dunlop |first2=Dorothy D. |publisher=Prentice Hall }}</ref>}}
 {{defn|no=6|Notation for [[Proportionality_(mathematics)|proportionality]]. See also [[#∝|{{math|∝}}]] for a less ambiguous symbol.}}
@@ Line 133: / Line 133: @@
 {{defn|no=2|In [[measure theory]], <math>\mu\ll\nu</math> means that the measure <math>\mu</math> is absolutely continuous with respect to the measure <math>\nu</math>.}}
-{{term|less-equal sign|content= <math>\leqq</math>}}
+{{term|less-equal sign|content= <math>\leqq \text{ and } \geqq</math>}}
-{{defn|A rarely used symbol, generally a synonym of {{math|≤}}.}}
+{{defn|Rarely used symbols, generally synonyms of {{math|≤}} and {{math|≥}}, respectively.}}
 {{term|pred-succ|content= <math>\prec \text{ and } \succ</math>}}
@@ Line 143: / Line 143: @@
 == Set theory ==
 {{glossary}}
-{{term|∅|content= {{math|∅}}}}
+{{term|∅|content= {{math|∅}} {{spaces|3|em}}([[null sign]])}}
 {{defn|Denotes the [[empty set]], and is more often written <math>\emptyset</math>. Using [[set-builder notation]], it may also be denoted [[#bb|<math>\{\}</math>]].}}
@@ Line 182: / Line 182: @@
 {{defn|Denotes [[set-theoretic intersection]], that is, <math>A\cap B</math> is the set formed by the elements of both {{mvar|A}} and {{mvar|B}}. That is, <math>A\cap B=\{x\mid (x\in A) \land (x\in B)\}</math>.}}
-{{term|∖|content= {{math|∖}} {{spaces|3|em}}([[backslash]])}}
+{{term|{{not a typo|∖}}|content= {{math|∖}} {{spaces|3|em}}([[backslash]])}}
 {{defn|[[Set difference]]; that is, <math>A\setminus B</math> is the set formed by the elements of {{mvar|A}} that are not in {{mvar|B}}. Sometimes, <math>A-B</math> is used instead; see [[#−|−]] in {{slink||Arithmetic operators}}.}}
 {{term|⊖|{{math|⊖}} or <math>\triangle</math>}}
-{{defn|[[Symmetric difference]]: that is, <math>A\ominus B</math> or <math>A\operatorname{\triangle}B</math> is the set formed by the elements that belong to exactly one of the two sets {{mvar|A}} and {{mvar|B}}.}}
+{{defn|[[Symmetric difference]]: that is, <math>A\ominus B</math> or <math>\ A \operatorname{\triangle}B\ </math> is the set formed by the elements that belong to exactly one of the two sets {{mvar|A}} and {{mvar|B}}.}}
-{{term|∁|content=<math>\complement</math> }}
+{{term|∁|content=<math>\complement</math>}}
-{{defn|no=1|With a subscript, denotes a [[set complement]]: that is, if <math>B\subseteq A</math>, then <math>\complement_A B = A\setminus B</math>.}}
+{{defn|no=1|With a subscript, denotes a [[set complement]]: that is, if <math>\ B\subseteq A\ ,</math> then <math>\ \complement_A B = A\setminus B ~.</math>}}
-{{defn|no=2|Without a subscript, denotes the [[absolute complement]]; that is, <math>\complement A  = \complement_ U A</math>, where {{mvar|U}} is a set implicitly defined by the context, which contains all sets under consideration. This set {{mvar|U}} is sometimes called the [[universe of discourse]]. }}
+{{defn|no=2|Without a subscript, denotes the [[absolute complement]]; that is, <math>\ \complement A  = \complement_ U A\ ,</math> where {{mvar|U}} is a set which contains all possible sets currently under consideration, implicitly defined by the context. This set {{mvar|U}} is sometimes called the [[universe of discourse]]. }}
+{{defn|no=3|Used as a superscript on a set symbol, denotes the [[set complement|complement]] of that set; that is, <math>\ A^\complement = U\ \backslash\ A\ ,</math> where {{mvar|U}} is the universal set, as in definition&nbsp;2.}}
 {{term|cartesian|content= {{math|×}} {{spaces|3|em}}([[multiplication sign]])}}
@@ Line 206: / Line 207: @@
 {{defn|no=2|Denotes the [[coproduct]] of [[mathematical structure]]s or of objects in a [[category (mathematics)|category]].}}
-{{term|よ|content={{math|よ}} or <math>h</math>}}
+{{term|よ|content={{math|よ}} {{spaces|3|em}}([[Yo (kana)|Hiragana Yo]]) or <math>h</math>}}
 {{defn|Denotes the [[Yoneda embedding]] in [[category theory]].}}
@@ Line 271: / Line 272: @@
 The [[blackboard bold]] [[typeface]] is widely used for denoting the basic [[number system]]s. These systems are often also denoted by the corresponding uppercase bold letter. A clear advantage of blackboard bold is that these symbols cannot be confused with anything else. This allows using them in any area of mathematics, without having to recall their definition. For example, if one encounters <math>\mathbb R</math> in [[combinatorics]], one should immediately know that this denotes the [[real number]]s, although combinatorics does not study the real numbers (but it uses them for many proofs).
 {{glossary}}
-{{term|ℕ|content=<math>\mathbb N</math>}}
+{{term|{{not a typo|ℕ}}|content=<math>\mathbb N</math>}}
 {{defn|Denotes the set of [[natural number]]s <math>\{1, 2,\ldots \},</math> or sometimes <math>\{0, 1, 2, \ldots \}.</math> When the distinction is important and readers might assume either definition, <math>\mathbb{N}_1</math> and <math>\mathbb{N}_0</math> are used, respectively, to denote one of them unambiguously. Notation <math>\mathbf N</math> is also commonly used.}}
-{{term|ℤ|content=<math>\mathbb Z</math>}}
+{{term|{{not a typo|ℤ}}|content=<math>\mathbb Z</math>}}
 {{defn|Denotes the set of [[integer]]s <math>\{\ldots, -2, -1, 0, 1, 2,\ldots \}.</math> It is often denoted also by <math>\mathbf Z.</math>}}
-{{term|ℤp|content=<math>\mathbb{Z}_p</math>}}
+{{term|{{not a typo|ℤp}}|content=<math>\mathbb{Z}_p</math>}}
 {{defn|no=1|Denotes the set of [[p-adic integer|{{mvar|p}}-adic integers]], where {{mvar|p}} is a [[prime number]].}}
 {{defn|no=2|Sometimes, <math>\mathbb Z_n</math> denotes the [[integers modulo n|integers modulo {{mvar|n}}]], where {{mvar|n}} is an [[integer]] greater than 0. The notation <math>\mathbb Z/n\mathbb Z</math> is also used, and is less ambiguous.}}
-{{term|ℚ|content=<math>\mathbb Q</math>}}
+{{term|{{not a typo|ℚ}}|content=<math>\mathbb Q</math>}}
 {{defn|Denotes the set of [[rational number]]s (fractions of two integers). It is often denoted also by <math>\mathbf Q.</math>}}
-{{term|ℚp|content=<math>\mathbb{Q}_p</math>}}
+{{term|{{not a typo|ℚp}}|content=<math>\mathbb{Q}_p</math>}}
 {{defn|Denotes the set of [[p-adic number|{{mvar|p}}-adic numbers]], where {{mvar|p}} is a [[prime number]].}}
-{{term|ℝ|content=<math>\mathbb R</math>}}
+{{term|{{not a typo|ℝ}}|content=<math>\mathbb R</math>}}
 {{defn|Denotes the set of [[real number]]s. It is often denoted also by <math>\mathbf R.</math>}}
-{{term|ℂ|content=<math>\mathbb C</math>}}
+{{term|{{not a typo|ℂ}}|content=<math>\mathbb C</math>}}
 {{defn|Denotes the set of [[complex number]]s. It is often denoted also by <math>\mathbf C.</math>}}
-{{term|ℍ|content=<math>\mathbb H</math>}}
+{{term|{{not a typo|ℍ}}|content=<math>\mathbb H</math>}}
 {{defn|Denotes the set of [[quaternion]]s. It is often denoted also by <math>\mathbf H.</math>}}
-{{term|Fq|content=<math>\mathbb{F}_q</math>}}
+{{term|{{not a typo|Fq}}|content=<math>\mathbb{F}_q</math>}}
 {{defn|Denotes the [[finite field]] with {{mvar|q}} elements, where {{mvar|q}} is a [[prime power]] (including [[prime number]]s). It is denoted also by {{math|GF(''q'')}}.}}
-{{term|O|content=<math>\mathbb O</math>}}
+{{term|{{not a typo|O}}|content=<math>\mathbb O</math>}}
-{{defn|Used on rare occasions to denote the set of [[octonion]]s. It is often denoted also by <math>\mathbf O.</math>}}
+{{defn|Denotes the set of [[octonion]]s. It is often denoted also by <math>\mathbf O.</math>}}
+{{term|{{not a typo|S}}|content=<math>\mathbb S</math>}}
+{{defn|Denotes the set of [[sedenion]]s. It is often denoted also by <math>\mathbf S.</math>}}
+{{term|{{not a typo|T}}|content=<math>\mathbb T</math>}}
+{{defn|Denotes the set of [[trigintaduonion]]s. It is often denoted also by <math>\mathbf T.</math>}}
 {{glossary end}}
@@ Line 310: / Line 317: @@
 {{term|\dot|<math>\dot \Box</math>}}
-{{defn|[[Newton's notation]], most commonly used for the [[derivative]] with respect to time. If {{mvar|x}} is a variable depending on time, then <math>\dot x,</math> read as "x dot", is its derivative with respect to time. In particular, if {{mvar|x}} represents a moving point, then <math>\dot x</math> is its [[velocity]].}}
+{{defn|[[Newton's notation]], most commonly used for the [[derivative]] with respect to time. If {{mvar|x}} is a variable depending on time, then <math>\dot x,</math> read as "{{mvar|x}} dot", is its derivative with respect to time. In particular, if {{mvar|x}} represents a moving point, then <math>\dot x</math> is its [[velocity]].}}
 {{term|\ddot|<math>\ddot \Box</math>}}
@@ Line 317: / Line 324: @@
 {{term|Leibnitz|{{math|{{sfrac|d □|d □}}}}}}
 {{defn|[[Leibniz's notation]] for the [[derivative]], which is used in several slightly different ways.}}
-{{defn|no=1|If {{mvar|y}} is a variable that [[dependent variable|depends]] on {{mvar|x}}, then <math>\textstyle \frac{\mathrm{d}y}{\mathrm{d}x}</math>, read as "d y over d x" (commonly shortened to "d y d x"), is the derivative of {{mvar|y}} with respect to {{mvar|x}}.}}
+{{defn|no=1|If {{mvar|y}} is a variable that [[dependent variable|depends]] on {{mvar|x}}, then <math>\textstyle \frac{\operatorname{d}y}{\operatorname{d}x}</math>, read as "d {{mvar|y}} over d {{mvar|x}}" (commonly shortened to "d {{mvar|y}} d {{mvar|x}}"), is the derivative of {{mvar|y}} with respect to {{mvar|x}}.}}
-{{defn|no=2|If {{mvar|f}} is a [[function (mathematics)|function]] of a single variable {{mvar|x}}, then <math>\textstyle \frac{\mathrm{d}f}{\mathrm{d}x}</math> is the derivative of {{mvar|f}}, and
+{{defn|no=2|If {{mvar|f}} is a [[function (mathematics)|function]] of a single variable {{mvar|x}}, then <math>\textstyle \frac{ \operatorname{d} f }{ \operatorname{d}x }</math> is the derivative of {{mvar|f}}, and
-<math>\textstyle \frac{\mathrm{d}f}{\mathrm{d}x}(a)</math> is the value of the derivative at {{mvar|a}}.}}
+<math>\textstyle \frac{ \operatorname{d} f }{ \operatorname{d}x }(a)</math> is the value of the derivative at {{mvar|a}}.}}
-{{defn|no=3|[[Total derivative]]: If <math>f(x_1, \ldots, x_n)</math> is a [[function (mathematics)|function]] of several variables that [[dependent variable|depend]] on {{mvar|x}}, then <math>\textstyle \frac{\mathrm{d}f}{\mathrm{d}x}</math> is the derivative of {{mvar|f}} considered as a function of {{mvar|x}}. That is, <math>\textstyle \frac{\mathrm{d}f}{dx}=\sum_{i=1}^n \frac{\partial f}{\partial x_i}\,\frac{\mathrm{d}x_i}{\mathrm{d}x}</math>.}}
+{{defn|no=3|[[Total derivative]]: If <math>f(x_1, \ldots, x_n)</math> is a [[function (mathematics)|function]] of several variables that [[dependent variable|depend]] on {{mvar|x}}, then <math>\textstyle \frac{\operatorname{d}f}{\operatorname{d}x}</math> is the derivative of {{mvar|f}} considered as a function of {{mvar|x}}. That is, <math>\textstyle\ \frac{\operatorname{d}f}{\operatorname{d}x}\ =\ \sum_{i=1}^n \frac{ \partial f }{ \partial x_i }\ \frac{\operatorname{d}x_i}{\operatorname{d}x} ~.</math>}}
 {{term|partial|{{math|{{sfrac|∂ □|∂ □}}}}}}
@@ Line 329: / Line 336: @@
 {{term|overline|<math>\overline\Box</math>}}
-{{defn|no=1|[[Complex conjugate]]: If {{mvar|z}} is a [[complex number]], then <math>\overline{z}</math> is its complex conjugate. For example, <math>\overline{a+bi} = a-bi</math>.}}
+{{defn|no=1|[[Complex conjugate]]: If {{mvar|z}} is a [[complex number]], then <math>\overline{z}</math> is its complex conjugate. For example, <math>\overline{a+bi} = a - b\ i ~.</math> }}
 {{defn|no=2|[[Topological closure]]: If {{mvar|S}} is a [[subset]] of a [[topological space]] {{mvar|T}}, then <math>\overline{S}</math> is its topological closure, that is, the smallest [[closed subset]] of {{mvar|T}} that contains {{mvar|S}}.}}
 {{defn|no=3|[[Algebraic closure]]: If {{mvar|F}} is a [[field (mathematics)|field]], then <math>\overline{F}</math> is its algebraic closure, that is, the smallest [[algebraically closed field]] that contains {{mvar|F}}. For example, <math>\overline\mathbb Q</math> is the field of all [[algebraic number]]s.}}
@@ Line 336: / Line 343: @@
 {{term|→|content={{math|→}}}}
-{{defn|no=1|<math>A\to B</math> denotes a [[function (mathematics)|function]] with [[domain of a function|domain]] {{mvar|A}} and [[codomain]] {{mvar|B}}. For naming such a function, one writes <math>f:A \to B</math>, which is read as "{{mvar|f}} from {{mvar|A}} to {{mvar|B}}".}}
+{{defn|no=1|<math>A \to B</math> denotes a [[function (mathematics)|function]] with [[domain of a function|domain]] {{mvar|A}} and [[codomain]] {{mvar|B}}. For naming such a function, one writes <math>\ f: A \to B\ ,</math> which is read as "function &thinsp;{{mvar|f}}&thinsp; maps any from set &thinsp;{{mvar|A}}&thinsp; into set {{nobr|&thinsp;{{mvar|B}}&thinsp;",}} or {{nobr|"&thinsp;{{mvar|f}}&thinsp; maps}} &thinsp;{{mvar|A}}&thinsp; to {{nobr|&thinsp;{{mvar|B}}&thinsp;" .}} }}
-{{defn|no=2|More generally, <math>A\to B</math> denotes a [[homomorphism]] or a [[morphism]] from {{mvar|A}} to {{mvar|B}}.}}
+{{defn|no=2|More generally, <math>A \to B</math> denotes a [[homomorphism]] or a [[morphism]] from {{mvar|A}} to {{mvar|B}}.}}
 {{defn|no=3|May denote a [[logical implication]]. For the [[material conditional|material implication]] that is widely used in mathematics reasoning, it is nowadays generally replaced by [[#⇒|⇒]]. In [[mathematical logic]], it remains used for denoting implication, but its exact meaning depends on the specific theory that is studied.}}
-{{defn|no=4|Over a [[variable (mathematics)|variable name]], means that the variable represents a [[vector (mathematics and physics)|vector]], in a context where ordinary variables represent [[scalar (mathematics)|scalar]]s; for example, <math>\overrightarrow v</math>. Boldface (<math>\mathbf v</math>) or a [[circumflex]] (<math>\hat v</math>) are often used for the same purpose.}}
+{{defn|no=4|Over a [[variable (mathematics)|variable name]], means that the variable represents a [[vector (mathematics and physics)|vector]], in a context where ordinary variables represent [[scalar (mathematics)|scalar]]s; for example, <math>\overrightarrow v</math>. Boldface (<math>\mathbf v</math>), blackboard bold ({{math|𝕧}}), or a [[circumflex]] (<math>\hat v</math>) are often used for the same purpose.}}
-{{defn|no=5|In [[Euclidean geometry]] and more generally in [[affine geometry]], <math>\overrightarrow{PQ}</math> denotes the [[vector (mathematics and physics)|vector]] defined by the two points {{mvar|P}} and {{mvar|Q}}, which can be identified with the [[Translation (geometry)|translation]] that maps {{mvar|P}} to {{mvar|Q}}. The same vector can be denoted also {{tmath|1= Q-P }}; see ''[[Affine space]]''.}}
+{{defn|no=5|In [[Euclidean geometry]] and more generally in [[affine geometry]], <math>\ \overrightarrow{PQ}\ </math> denotes the [[vector (mathematics and physics)|vector]] defined by the two points {{mvar|P}} and {{mvar|Q}}, which can be identified with the [[Translation (geometry)|translation]] that maps {{mvar|P}} to {{mvar|Q}}. The same vector can be denoted also {{tmath|1= Q-P }}; see ''[[Affine space]]''.}}
 {{term|↦|content={{math|↦}}}}
@@ Line 346: / Line 353: @@
 {{term|∘|{{math|○}}<ref>The [[LaTeX]] equivalent to both [[Unicode]] symbols ∘ and ○ is \circ. The Unicode symbol that has the same size as \circ depends on the browser and its implementation. In some cases ∘ is so small that it can be confused with an [[interpoint]], and ○ looks similar as \circ. In other cases, ○ is too large for denoting a binary operation, and it is ∘ that looks like \circ. As LaTeX is commonly considered as the standard for mathematical typography, and it does not distinguish these two Unicode symbols, they are considered here as having the same mathematical meaning.</ref>}}
-{{defn|no=1|[[Function composition]]: If {{mvar|f}} and {{mvar|g}} are two functions, then <math>g\circ f</math> is the function such that <math>(g\circ f)(x)=g(f(x))</math> for every value of {{mvar|x}}.}}
+{{defn|no=1|[[Function composition]]: If {{mvar|f}} and {{mvar|g}} are two functions, then <math>g\circ f</math> is the function such that <math>(g\circ f)(x) = g(f(x))</math> for every value of {{mvar|x}}.}}
-{{defn|no=2|[[Hadamard product (matrices)|Hadamard product of matrices]]: If {{mvar|A}} and {{mvar|B}} are two matrices of the same size, then <math>A\circ B</math> is the matrix such that <math>(A\circ B)_{i,j} = (A)_{i,j}(B)_{i,j}</math>. Possibly, <math>\circ</math> is also used instead of [[#⊙|{{math|⊙}}]] for the [[Hadamard product (series)|Hadamard product of power series]].{{citation needed|date=November 2020}}}}
+{{defn|no=2|[[Hadamard product (matrices)|Hadamard product of matrices]]: If {{mvar|A}} and {{mvar|B}} are two matrices of the same size, then <math>\ A\circ B\ </math> is the matrix such that <math>(A\circ B)_{i,j} = (A)_{i,j}(B)_{i,j}</math>. Possibly, <math>\ \circ\ </math> is also used instead of [[#⊙|{{math|⊙}}]] for the [[Hadamard product (series)|Hadamard product of power series]].{{citation needed|date=November 2020}}}}
 {{term|∂|{{math|∂}}}}
-{{defn|no=1|[[Boundary (topology)|Boundary]] of a [[topological subspace]]: If {{mvar|S}} is a subspace of a topological space, then its ''boundary'', denoted <math>\partial S</math>, is the [[set difference]] between the [[closure (topology)|closure]] and the [[interior (topology)|interior]] of {{mvar|S}}.}}
+{{defn|no=1|[[Boundary (topology)|Boundary]] of a [[topological subspace]]: If {{mvar|S}} is a subspace of a topological space, then its ''boundary'', denoted <math>\ \partial S\ ,</math> is the [[set difference]] between the [[closure (topology)|closure]] and the [[interior (topology)|interior]] of {{mvar|S}}.}}
 {{defn|no=2|[[Partial derivative]]: see [[#partial|{{math|{{sfrac|∂□|∂□}}}}]].}}
 {{term|integral|content={{math|∫}}}}
-{{defn|no=1|1=Without a subscript, denotes an [[antiderivative]]. For example, <math>\textstyle\int x^2 dx = \frac{x^3}3 +C</math>.}}
+{{defn|no=1|1=Without a subscript, denotes an [[antiderivative]]. For example, <math>\ \textstyle\int x^2 \operatorname{d} x = \frac{\; x^3}{ 3 } + C ~.</math>}}
-{{defn|no=2|1=With a subscript and a superscript, or expressions placed below and above it, denotes a [[definite integral]]. For example, <math>\textstyle \int_a^b x^2dx = \frac{b^3-a^3}{3}</math>.}}
+{{defn|no=2|1=With a subscript and a superscript, or expressions placed below and above it, denotes a [[definite integral]]. For example, <math>\textstyle \int_a^b x^2 \operatorname{d} x = \frac{\;b^3 - a^3}{ 3 } ~.</math>}}
-{{defn|no=3|1=With a subscript that denotes a curve, denotes a [[line integral]]. For example, <math>\textstyle\int_C f=\int_a^b f(r(t))r'(t)\operatorname{d}t</math>, if {{mvar|r}} is a parametrization of the curve {{mvar|C}}, from {{mvar|a}} to {{mvar|b}}.}}
+{{defn|no=3|1=With a subscript that denotes a curve, denotes a [[line integral]]. For example, <math>\textstyle\ \int_C f\ =\ \int_a^b\ f\!\bigl(\ \!r\!\left(t\right)\ \!\!\bigr)\ r'\!\left(t\right) \operatorname{d} t\ ,</math> if {{mvar|r}} is a parametrization of the curve {{mvar|C}}, from point {{mvar|t {{=}} a}} to {{mvar|t {{=}} b}}.}}
 {{term|oint|content={{math|∮}}}}
-{{defn|Often used, typically in physics, instead of <math>\textstyle\int</math> for [[line integral]]s over a [[closed curve]].}}
+{{defn|Often used, typically in physics, instead of <math>\textstyle \int</math> for [[line integral]]s over a [[closed curve]].}}
 {{term|iint|content={{math|∬, ∯}}}}
 {{defn|Similar to <math>\textstyle\int</math> and <math>\textstyle\oint</math> for [[surface integral]]s.}}
+{{term|iiint|content={{math|∭, ∰}}}}
+{{defn|Similar to <math>\textstyle\int</math> and <math>\textstyle\oint</math> for [[volume integral]]s.}}
 {{term|∇|[[nabla symbol|<math>\boldsymbol{\nabla}</math> or <math>\vec{\nabla}</math>]]}}
-{{defn|[[del|Nabla]], the [[gradient]], vector derivative operator <math>\textstyle \left(\frac \partial {\partial x}, \frac \partial {\partial y}, \frac \partial {\partial z}\right)</math>, also called ''del'' or ''grad'',}} or the [[covariant derivative]].
+{{defn|[[del|Nabla]], the [[gradient]], vector derivative operator <math>\textstyle\ \left(\frac{ \partial }{\ \partial x\ }, \frac{ \partial }{\ \partial y\ }, \frac{ \partial }{\ \partial z\ }\right)\ ,</math> also called ''del'' or ''grad'',}} or the [[covariant derivative]].
 {{term|Laplacian|{{math|&nabla;<sup>2</sup>}} or {{math|&nabla;&sdot;&nabla;}}}}
-{{defn|[[Laplace operator]] or ''Laplacian'': <math>\textstyle \frac {\partial^2}{\partial x^2} + \frac {\partial^2}{\partial y^2} + \frac {\partial^2}{\partial z^2}</math>. The forms <math>\nabla^2</math> and <math>\boldsymbol\nabla \cdot \boldsymbol\nabla</math> represent the dot product of the [[#∇|gradient]] (<math>\boldsymbol{\nabla}</math> or <math>\vec{\nabla}</math>) with itself. Also notated {{math|&Delta;}} (next item).}}
+{{defn|[[Laplace operator]] or ''Laplacian'': <math>\textstyle\ \frac{ \partial^2 }{\ \partial x^2 } + \frac{ \partial^2 }{\ \partial y^2 } + \frac{ \partial^2 }{\ \partial z^2 } ~.</math> The forms <math>\nabla^2</math> and <math>\boldsymbol\nabla \cdot \boldsymbol\nabla</math> represent the dot product of the [[#∇|gradient]] (<math>\boldsymbol{\nabla}</math> or <math>\vec{\nabla}</math>) with itself. Also notated {{math|&Delta;}} (next item).}}
 {{term|Delta|{{math|&Delta;}}}}
@@ Line 376: / Line 386: @@
 {{term|four-gradient|<math>\boldsymbol{\partial}</math> or <math>\partial_\mu</math>}}
-(Note: the notation <math>\Box</math> is not recommended for the four-gradient since both <math>\Box</math> and <math>{\Box}^2</math> are used to denote the [[d'Alembertian]]; see below.)
+(Note: Although <math>\ \vec\Box\ </math> may be used unambiguously, the notation <math>\boldsymbol\Box</math> is not recommended for the four-gradient since both <math>\Box</math> and <math>{\Box}^2</math> are used to denote the [[d'Alembertian]]; see below.)
-{{defn|Quad, the 4-vector gradient operator or [[four-gradient]], <math>\textstyle \left( \frac \partial {\partial t}, \frac \partial {\partial x}, \frac \partial {\partial y}, \frac \partial {\partial z}\right)</math>.}}
+{{defn|Quad, the 4-vector gradient operator or [[four-gradient]], <math>\textstyle \left( \frac{\ \partial }{\ \partial t\ }, \frac{\ \partial }{\ \partial x\ }, \frac{\ \partial }{\ \partial y\ }, \frac{\ \partial }{\ \partial z\ }\right) ~.</math>}}
-{{term|d'Alembertian|<math>\Box</math> or <math>{\Box}^2</math>}} (here an actual box, not a placeholder)
+{{term|d'Alembertian|<math>\Box</math> or <math>{\Box}^2</math>}} (here the symbol is an actual box, not a placeholder)
-{{defn|Denotes the [[d'Alembertian]] or squared [[four-gradient]], which is a generalization of the [[Laplacian]] to four-dimensional spacetime. In flat spacetime with Euclidean coordinates, this may mean either <math>~ \textstyle - \frac {\partial^2}{\partial t^2} + \frac {\partial^2}{\partial x^2} + \frac {\partial^2}{\partial y^2} + \frac {\partial^2}{\partial z^2} ~\;</math>  or  <math>\;~ \textstyle + \frac {\partial^2}{\partial t^2} - \frac {\partial^2}{\partial x^2} - \frac {\partial^2}{\partial y^2} - \frac {\partial^2}{\partial z^2} ~\;</math>; the sign convention must be specified. In curved spacetime (or flat spacetime with non-Euclidean coordinates), the definition is more complicated. Also called ''box'' or ''quabla''.}}
+{{defn|Denotes the [[d'Alembertian]] or squared [[four-gradient]], which is a generalization of the [[Laplacian]] to four-dimensional spacetime. In flat spacetime with Euclidean coordinates, this may mean either <math>~ \textstyle - \frac {\partial^2}{\partial t^2} + \frac{\ \partial^2 }{\ \partial x^2 } + \frac{\ \partial^2 }{\ \partial y^2 } + \frac{\ \partial^2 }{\ \partial z^2 } ~\;</math>  or  <math>\;~ \textstyle + \frac{\ \partial^2 }{\ \partial t^2 } - \frac{\ \partial^2 }{\ \partial x^2 } - \frac{\ \partial^2 }{\ \partial y^2 } - \frac{\ \partial^2 }{\ \partial z^2 } ~\;;</math> the sign convention must be specified. In curved spacetime (or flat spacetime with non-Euclidean coordinates), the definition is more complicated. Also called ''box'' or ''quabla''.}}
 {{glossary end}}
@@ Line 719: / Line 729: @@
 == External links ==
-* [http://jeff560.tripod.com/mathsym.html Jeff Miller: ''Earliest Uses of Various Mathematical Symbols'']
+* [https://mathshistory.st-andrews.ac.uk/Miller/mathsym/ Jeff Miller: ''Earliest Uses of Various Mathematical Symbols'']
 * [http://www.numericana.com/answer/symbol.htm Numericana: ''Scientific Symbols and Icons'']
 * [http://us.metamath.org/symbols/symbols.html GIF and PNG Images for Math Symbols]

Glossary of mathematical symbols: Difference between revisions

Latest revision as of 05:16, 18 November 2025

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