Table of Mathematical Symbols
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Table of Mathematical Symbols

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. As formulas are entierely constitued 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 sort of mathematical objects. As the number of these sorts has dramatically increased in modern mathematics, the Greek alphabet and some Hebrew letters are also used. In mathematical formulas, the standard typeface is italic type for Latin letters and lower-case Greek letters, and upright type for upper case Greek letters. For having more symbols, other typefaces are also used, mainly boldface ${\displaystyle \mathbf {a,A,b,B} ,\ldots ,}$ script typeface ${\displaystyle {\mathcal {A,B}},\ldots }$ (the lower-case script face is rarely used because of the possible confusion with the standard face), German fraktur ${\displaystyle {\mathfrak {a,A,b,B}},\ldots ,}$ and blackboard bold ${\displaystyle \mathbb {N,Z,R,C} }$ (the other letters are rarely used, or their use is controversial).

In this article we presentthe main symbols that are used in mathematics and their common use.

For the use of letters as symbols for variables, see variable (mathematics). For their use as symbols for constants, see List of mathematical constants.

## Arithmetic operators

+
1.  Denotes addition and is read as plus; for example, 3 + 2.
2.  Sometimes used instead of ${\displaystyle \sqcup }$ for a disjoint union of sets.
-
1.  Denotes subtraction and is read as minus; for example, 3 - 2.
2.  Denotes the additive inverse and is read as the opposite of; for example, -2.
3.  Also used in place of \ for denoting the set-theoretic complement; see \ in § Set theory.
×
1.  In elementary arithmetic, denotes multiplication, and is read as times; for example 3 × 2.
2.  In geometry and linear algebra, denotes the cross product.
3.  In set theory and category theory, denotes the Cartesian product and the direct product. See also × in § Set theory.
·
1.  Denotes multiplication and is read as times; for example 3 ? 2.
2.  In geometry and linear algebra, denotes the dot product.
3.  Placeholder used for replacing an indeterminate element. For example, "the absolute value is denoted | · |" is clearer than saying that it is denoted as | |.
±
1.  Denotes denotes either a plus sign or a minus sign
2.  Denotes the range of values that a measured quatity may have; for example, 10 ± 2 denotes a unknown value that lies between 8 and 12.
?
Used paired with ±, denotes the opposite sign, that is + if ± is -, and - if ± is +.
÷
Widely used for denoting division in anglophone countries, it is no longer in common use in mathematics and its use is "not recommended".[1] In some countries, it can indicate subtraction.
/
1.  Denotes division and is read as divided by or over. Often replaced by a horizontal bar. For example 3 / 2 or ${\displaystyle {\frac {3}{2}}.}$
2.  Denotes a quotient structure. For example quotient set, quotient group, quotient category, etc.
3.  In number theory and field theory, ${\displaystyle F/E}$ denotes a field extension, where F is an extension field of the field E.
4.  In probability theory, denotes a conditional probability. For example, ${\displaystyle P(A/B)}$ denotes the probability of A, given that B occurs.
?
Denotes square root and is read as square root of. Rarely used in modern mathematics without an horizontal bar delimiting the width of its argument (see the next item). For example ?2.
1.  Denotes square root and is read as square root of. For example .
2.  With an integer greater than 2 as a left superscript, denotes a nth root. For example .

## Equality, equivalence and similarity

=
1.  Denotes equality.
2.  Used for naming a mathematical object in a sentence like "let ${\displaystyle x=E}$", where E is an expression. On a blackbord and in some mathematical texts, this may be abbreviated as ${\displaystyle x\,{\stackrel {\mathrm {def} }{=}}\,E.}$ This is related with the concept of assignment in computer science, which is variously denoted (depending on the used programming language) ${\displaystyle =,:=,==,\leftarrow ,\ldots }$
?
Denotes inequality and means "not equal".
?
Means "is approximatively equal to". For example, ? ? 3.1415.
~
1.  Between two numbers, either it is used in place of ? for meaning "approximatively equal", or it means "has the same order of magnitude as".
2.  Denotes the asymptotic equivalence of two functions or sequences.
3.  Often used for denoting other types of similarity, for example, matrix similarity or similarity of geometric shapes.
4.  Standard notation for an equivalence relation.
?
1.  Denotes an identity, that is an equality that is true whichever values are given to the variables occurring in it.
2.  In number theory, and more specifically in modular arithmetic, denotes the congruence modulo an integer.
${\displaystyle \cong }$
1.  May denote an isomorphism between two mathematical structures, and is read as "is isomorphic to".
2.  In geometry, may denote the congruence of two geometric shapes (that is the equality up to a displacement), and is read "is congruent to".

## Comparison

<
1.  Strict inequality between two numbers; means and is read as "less than".
2.  Commonly used for denoting any strict order.
3.  Between two groups, may mean that the first one is a proper subgroup of the second one.
>
1.  Strict inequality between two numbers; means and is read as "greater than".
2.  Commonly used for denoting any strict order.
3.  Between two groups, may mean that the second one is a proper subgroup of the first one.
1.  Means "less than or equal to". That is, whichever A and B are, A B is equivalent with A < B or A = B.
2.  Between two groups, may mean that the first one is a subgroup of the second one.
>=
1.  Means "greater than or equal to". That is, whichever A and B are, A >= B is equivalent with A > B or A = B.
2.  Between two groups, may mean that the second one is a subgroup of the first one.
>
1.  Mean "much less than" and "much greater than". Generally, much is not formally defined, but means that the lesser quantity can be neglected with respect to the other. This is generally the case when the lesser quantity is smaller than the other by one or several orders of magnitude.
2.  In measure theory, ${\displaystyle \mu \ll \nu }$ means that the measure ${\displaystyle \mu }$ is absolutely continuous with respect to the measure ${\displaystyle \nu .}$
?
1.  A rarely used synonym of . Despite the easy confusion with , some authors use it with a different meaning.
? , ?
Often used for denoting an order or, more generally, a preorder, when it would be confusing or not convenient to use < and >.

## Set theory

?
Denotes the empty set, and is more often written ${\displaystyle \emptyset .}$ Using set builder notation, it may also be denoted ${\displaystyle \{\,\}.}$
?
Denotes set membership, and is read "in" or "belongs to". That is, ${\displaystyle x\in S}$ means that x is an element of the set S.
Means "not in". That is, ${\displaystyle x\notin S}$ means ${\displaystyle \neg (x\in S).}$
?
Denotes set inclusion. However two slightly different definitions are common. It seems that the first one is more commonly used in recent texts, since it allows often avoiding case distinction.
1.  ${\displaystyle A\subset B}$ may mean that A is a subset of B, and is possibly equal to B; that is, every element of A belongs to B; in formula, ${\displaystyle \forall x,\,x\in A\Rightarrow x\in B.}$
2.  ${\displaystyle A\subset B}$ may mean that A is a proper subset of B, that is the two sets are different, and every element of A belongs to B; in formula, ${\displaystyle A\neq B\land \forall x,\,x\in A\Rightarrow x\in B.}$
?
${\displaystyle A\subseteq B}$ means that A is a subset of B. Used for emphasizing that equality is possible, or when the second definition is used for ${\displaystyle A\subset B.}$
?
${\displaystyle A\subsetneq B}$ means that A is a proper subset of B. Used for emphasizing that ${\displaystyle A\neq B,}$ or when the first definition is used for ${\displaystyle A\subset B.}$
? , ? , ?
The same as the preceding ones with the operands reverted. For example, ${\displaystyle B\supset A}$ is equivalent with ${\displaystyle A\subset B.}$
?
Denotes set-theoretic union, that is, ${\displaystyle A\cup B}$ is the set formed by the elements of A and B together. That is, ${\displaystyle A\cup B=\{x\mid (x\in A)\lor (x\in B)\}.}$
?
Denotes set-theoretic intersection, that is, ${\displaystyle A\cap B}$ is the set formed by the elements of both A and B. That is, ${\displaystyle A\cap B=\{x\mid (x\in A)\land (x\in B)\}.}$
\
Denotes set difference; that is, ${\displaystyle A\setminus B}$ is the set formed by the elements of A that are not in B. Sometimes, ${\displaystyle A-B}$ is used instead; see - in § Arithmetic operators.
${\displaystyle \complement }$
1.  With a subscript, denotes a set complement: that is, if ${\displaystyle B\subseteq A,}$ then ${\displaystyle \complement _{A}B=A\setminus B.}$
2.  Without a subscript, denotes the absolute complement; that is, ${\displaystyle \complement A=\complement _{U}A,}$ where U is a set implicitely defined by the context, which contains all sets under consideration. This set U is sometimes called the universe of discourse.
×
1.  Denotes the Cartesian product of two sets. That is, ${\displaystyle A\times B}$ is the set formed by all pairs of an element of A and an element of B.
2.  Denotes the direct product of two mathematical structures of the same type, which is the Cartesian product of the underlying sets, equipped with a structure of the same type. For example, direct product of rings, direct product of topological spaces.
3.  In category theory, denotes the direct product (often called simply product) of two objects, which is a generalization of the preceding concepts of product.
?
Denotes the disjoint union. That is, if A and B are two sets, ${\displaystyle A\sqcup B=A\cup C,}$ where C is a set formed by the elements of B renamed for not belonging to A.
?
1.  Alternative of ${\displaystyle \sqcup }$ for denoting disjoint union.
2.  Denotes the coproduct of mathematical structures or of objects in a category.

## Basic logic

Several logical symbols are widely used in all mathematics, and are listed here. For symbols that are used only in mathematical logic, or are rarely used, see List of logic symbols.

¬
Denotes logical negation, and is read as "not". If E is a logical predicate, ${\displaystyle \neg E}$ is the predicate that evaluates to true if and only if E evaluates to false. For clarity, it is often replaced by the word "not". In programming languages and some mathematical texts, it is often replaced by "~" or "!", which are easier to type on a keyboard.
?
1.  Denotes the logical or and is read as "or". If E and F are logical predicates, ${\displaystyle E\lor F}$ is true if either E, F, or both are true. It is often replaced by the word "or" or the symbol "&".
2.  In lattice theory, denotes the join or least upper bound operation.
3.  In topology, denotes the wedge sum of two pointed spaces.
?
1.  Denotes the logical and and is read as "and". If E and F are logical predicates, ${\displaystyle E\land F}$ is true if E and F are both true. It is often replaced by the word "and".
2.  In lattice theory, denotes the meet or greatest lower bound operation.
3.  In multilinear algebra, geometry, and multivariable calculus denotes the wedge product or the exterior product.
?
1.  Denotes universal quantification and is read "for all". If E is a logical predicate, ${\displaystyle \forall xE}$ means that E is true for all possible values of the variable x.
2.  Often used improperly in plain text as an abbreviation of "for all" or "for every".
?
1.  Denotes existential quantification and is read "there exists ... such that". If E is a logical predicate, ${\displaystyle \exists xE}$ means that there exists at least one value of x for which E is true.
2.  Often used improperly in plain text as an abbreviation of "there exists".
?!
Denotes uniqueness quantification, that is, ${\displaystyle \exists !xP}$ means "there exists exactly one x such that P (is true)". In other words, ${\displaystyle \exists !xP(x)}$ is an abbreviation of ${\displaystyle \exists x\,(P(x)\,\wedge \neg \exists y\,(P(y)\wedge y\neq x)).}$
=>
1.  Denotes material conditional, and is read as "implies". If P and Q are logical predicates, ${\displaystyle P\Rightarrow Q}$ means that if P is true, then Q is also true. Thus, ${\displaystyle P\Rightarrow Q}$ is logically equivalent with ${\displaystyle Q\lor \neg P.}$
2.  Often used improperly in plain text as an abbreviation of "implies".
1.  Denotes logical equivalence, and is read "is equivalent to" or "if and only if. If P and Q are logical predicates, ${\displaystyle P\Leftrightarrow Q}$ is thus an abbreviation of ${\displaystyle (P\Rightarrow Q)\land (Q\Rightarrow P),}$ or of ${\displaystyle (P\land Q)\lor (\neg P\land \neg Q).}$
2.  Often used improperly in plain text as an abbreviation of "if and only if".

## Operators acting on functions or sequences

?
1.  Denotes the sum of a finite number of terms, which are determined by underscripts and superscripts such as in ${\displaystyle \textstyle \sum _{i=1}^{n}i^{2}}$ or ${\displaystyle \textstyle \sum _{0
2.  Denotes a series and, if the series is convergent, the sum of the series. For example ${\displaystyle \textstyle \sum _{i=0}^{\infty }{\frac {x^{i}}{i!}}=e^{i}.}$
?
1.  Without a subscript, denotes an antiderivative. For example, ${\displaystyle \textstyle \int x^{2}dx={\frac {x^{3}}{3}}+C.}$
2.  With a subscript and a superscript, denotes a definite integral, For example, ${\displaystyle \textstyle \int _{a}^{b}x^{2}dx={\frac {b^{3}-a^{3}}{3}}.}$
3.  With a subscript that denotes a curve, denotes a line integral. For example, ${\displaystyle \textstyle \int _{C}f=\int _{a}^{b}f(r(t))r'(t)dt,}$ if r is a parametrization of the curve C, from a to b.
?
Often used, typically in physics, instead of ${\displaystyle \textstyle \int }$ for line integrals over a closed curve.
?, ?
Similar to ${\displaystyle \textstyle \int }$ and ${\displaystyle \textstyle \oint }$ for surface integrals.

## Infinite numbers

?
1.  The symbol is read as infinity. As an upper bound of a summation, an infinite product, an integral, etc., means that the computation is unlimited. Similarly, ${\displaystyle -\infty }$ in a lower bound means that the computation is not limited toward negative values.
2.  ${\displaystyle -\infty }$ and ${\displaystyle +\infty }$ are the generalized numbers that are added to the real line for forming the extended real line
3.  ${\displaystyle \infty }$ is the generalized number that is added to the real line for forming the projectively extended real line.
c
${\displaystyle {\mathfrak {c}}}$ denotes the cardinality of the continuum, which is the cardinality of the set of real numbers.
?
With an ordinal i as a subscript, denotes the ith aleph number, that is the ith infinite cardinal. For example, ${\displaystyle \aleph _{0}}$ is the smallest infinite cardinal, that is, the cardinal of the natural numbers.
?
With an ordinal i as a subscript, denotes the ith beth number. For example, ${\displaystyle \beth _{0}}$ is the cardinal of the natural numbers, and ${\displaystyle \beth _{1}}$ is the cardinal of the continuum.
?
1.  Denotes the first limit ordinal. It is also denoted ${\displaystyle \omega _{0}}$ and can be identified with the ordered set of the natural numbers.
2.  With an ordinal i as a subscript, denotes the ith limit ordinal that has a cardinality greater than that of all preceding ordinals.
3.  In computer science, denotes the (unknown) greatest lower bound for the exponent of the computational complexity of matrix multiplication.
4.  Written as a function of another function, it is used for comparing the asymptotic growth of two functions. See Big O notation § Related asymptotic notations.
5.  In number theory, may denote the prime omega function. That is, ${\displaystyle \omega (n)}$ is the number of distinct prime factors of the integer n.

## Abbreviation of English phrases and logical punctuation

In this section, the symbols that are listed are used as some sort of punctuation marks in mathematics reasoning, or as abbreviations of English phrases. They are generally not used inside a formula. Some were used in classical logic for indicating the logical dependence between sentences written in plain English. Except for the first one, they are normally not used in printed mathematical texts since, for readability, it is generally recommended to have at least one word between two formulas. However, they are still used on a black board for indicating relationships between formulas.

? , ?
Used for marking the end of a proof and separating it from the current text. The initialism Q.E.D. or QED (quod erat demonstrandum) is often used for the same purprose, either in its upper-case form or in lower case.
?
Abbreviation of "therefore". Placed between two assertions, it means that the first one implies the second one. For example: "All humans are mortal. Socrates is a human. ? Socrates is mortal."
?
Abbreviation of "because" or "since". Placed between two assertions, it means that the first one is implied by the second one. For example: "11 is prime ? it has no positive integer factors other than itself and one."
?
1.  Abbreviation of "such that". For example ${\displaystyle x\ni x>3}$ is normally printed "x such that ${\displaystyle x>3.}$
2.  Sometimes used for reverting the operands of ${\displaystyle \in ;}$ that is, ${\displaystyle S\ni x}$ has the same meaning as ${\displaystyle x\in S.}$ See ? in § Set theory.
?
Abbreviation of "is proportional to"

## Miscellaneous

Symbol
in HTML
Symbol
in TeX
Name Explanation Examples
Category
${\displaystyle !}$
factorial
${\displaystyle n!}$ means the product ${\displaystyle 1\times 2\times \cdots \times n}$. ${\displaystyle 4!=1\times 2\times 3\times 4=24}$

${\displaystyle \to }$
\to
function arrow
from ... to
f: X -> Y means the function f maps the set X into the set Y. Let f: Z -> N ? {0} be defined by f(x) := x2.
?
${\displaystyle \mapsto }$
\mapsto
function arrow
maps to
f: a ? b means the function f maps the element a to the element b. Let f: x ? x + 1 (the successor function).
${\displaystyle \leftarrow }$
\leftarrow
.. if ..
a b means that for the propositions a and b, if b implies a, then a is the converse implication of b.a to the element b. This reads as "a if b", or "not b without a". It is not to be confused with the assignment operator in computer science.
?|
${\displaystyle \langle \ |}$
\langle
the bra ...;
the dual of ...
??| means the dual of the vector |??, a linear functional which maps a ket |?? onto the inner product ??|??.
|?
${\displaystyle |\ \rangle }$
\rangle
the ket ...;
the vector ...
|?? means the vector with label ?, which is in a Hilbert space. A qubit's state can be represented as ?|0?+ ?|1?, where ? and ? are complex numbers s.t. |?|2 + |?|2 = 1.

## Brackets

Symbol
in HTML
Symbol
in TeX
Name Explanation Examples
Category

${\displaystyle {\ \choose \ }}$
{\ \choose\ }
n choose k
${\displaystyle {\begin{pmatrix}n\\k\end{pmatrix}}={\frac {n!/(n-k)!}{k!}}={\frac {(n-k+1)\cdots (n-2)\cdot (n-1)\cdot n}{k!}}}$
means (in the case of n = positive integer) the number of combinations of k elements drawn from a set of n elements.

(This may also be written as C(n, k), C(n; k), nCk, nCk, or ${\displaystyle \left\langle {\begin{matrix}n\\k\end{matrix}}\right\rangle }$.)
${\displaystyle {\begin{pmatrix}36\\5\end{pmatrix}}={\frac {36!/(36-5)!}{5!}}={\frac {32\cdot 33\cdot 34\cdot 35\cdot 36}{1\cdot 2\cdot 3\cdot 4\cdot 5}}=376992}$

${\displaystyle {\begin{pmatrix}.5\\7\end{pmatrix}}={\frac {-5.5\cdot -4.5\cdot -3.5\cdot -2.5\cdot -1.5\cdot -.5\cdot .5}{1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7}}={\frac {33}{2048}}\,\!}$

${\displaystyle \left(\!\!{\ \choose \ }\!\!\right)}$
\left(\!\!{\ \choose\ }\!\!\right)
u multichoose k
${\displaystyle \left(\!\!{u \choose k}\!\!\right)={u+k-1 \choose k}={\frac {(u+k-1)!/(u-1)!}{k!}}}$

(when u is positive integer)
means reverse or rising binomial coefficient.

${\displaystyle \left(\!\!{-5.5 \choose 7}\!\!\right)={\frac {-5.5\cdot -4.5\cdot -3.5\cdot -2.5\cdot -1.5\cdot -.5\cdot .5}{1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7}}={.5 \choose 7}={\frac {33}{2048}}\,\!}$

${\displaystyle \left\{{\begin{array}{lr}\ldots \\\ldots \end{array}}\right.}$
\left\{ \begin{array}{lr} \ldots \\ \ldots \end{array}\right.
is defined as ... if ..., or as ... if ...;
match ... with
everywhere
${\displaystyle f(x)=\left\{{\begin{array}{rl}a,&{\text{if }}p(x)\\b,&{\text{if }}q(x)\end{array}}\right.}$ means the function f(x) is defined as a if the condition p(x) holds, or as b if the condition q(x) holds.

(The body of a piecewise-defined function can have any finite number (not only just two) expression-condition pairs.)

This symbol is also used in type theory for pattern matching the constructor of the value of an algebraic type. For example ${\displaystyle g(n)={\text{match }}n{\text{ with }}\left\{{\begin{array}{rl}x&\rightarrow a\\y&\rightarrow b\end{array}}\right.}$ does pattern matching on the function's arguments and means that g(x) is defined as a, and g(y) is defined as b.

(A pattern matching can have any finite number (not only just two) pattern-expression pairs.)
${\displaystyle |x|=\left\{{\begin{array}{rl}x,&{\text{if }}x\geq 0\\-x,&{\text{if }}x<0\end{array}}\right.}$
${\displaystyle a+b={\text{match }}b{\text{ with }}\left\{{\begin{array}{rl}0&\rightarrow a\\Sn&\rightarrow S(a+n)\end{array}}\right.}$
|...|
${\displaystyle |\ldots |\!\,}$
| \ldots | \!\,
absolute value of; modulus of
|x| means the distance along the real line (or across the complex plane) between x and zero. |3| = 3

|-5| = |5| = 5

| i | = 1

| 3 + 4i | = 5
Euclidean norm or Euclidean length or magnitude
Euclidean norm of
|x| means the (Euclidean) length of vector x. For x = (3,-4)
${\displaystyle |{\textbf {x}}|={\sqrt {3^{2}+(-4)^{2}}}=5}$
determinant of
|A| means the determinant of the matrix A ${\displaystyle {\begin{vmatrix}1&2\\2&9\\\end{vmatrix}}=5}$
cardinality of;
size of;
order of
|X| means the cardinality of the set X.

(# may be used instead as described below.)
|{3, 5, 7, 9}| = 4.
?...?
${\displaystyle \|\ldots \|\!\,}$
\| \ldots \| \!\,
norm of;
length of
? x ? means the norm of the element x of a normed vector space.[2] ? x + y ? x ? + ? y ?
nearest integer to
?x? means the nearest integer to x.

(This may also be written [x], ?x?, nint(x) or Round(x).)
?1? = 1, ?1.6? = 2, ?-2.4? = -2, ?3.49? = 3
{ , }
${\displaystyle {\{\ ,\!\ \}}\!\,}$
{\{\ ,\!\ \}} \!\,
set brackets
the set of ...
{a,b,c} means the set consisting of a, b, and c.[3] N = { 1, 2, 3, ... }
{ : }

{ | }

{ ; }
${\displaystyle \{\ :\ \}\!\,}$
\{\ :\ \} \!\,

${\displaystyle \{\ |\ \}\!\,}$
\{\ |\ \} \!\,

${\displaystyle \{\ ;\ \}\!\,}$
\{\ ;\ \} \!\,
the set of ... such that
{x : P(x)} means the set of all x for which P(x) is true.[3] {x | P(x)} is the same as {x : P(x)}. {n ? N : n2 < 20} = { 1, 2, 3, 4 }
?...?
${\displaystyle \lfloor \ldots \rfloor \!\,}$
\lfloor \ldots \rfloor \!\,
floor;
greatest integer;
entier
?x? means the floor of x, i.e. the largest integer less than or equal to x.

(This may also be written [x], floor(x) or int(x).)
?4? = 4, ?2.1? = 2, ?2.9? = 2, ?-2.6? = -3
?...?
${\displaystyle \lceil \ldots \rceil \!\,}$
\lceil \ldots \rceil \!\,
ceiling
?x? means the ceiling of x, i.e. the smallest integer greater than or equal to x.

(This may also be written ceil(x) or ceiling(x).)
?4? = 4, ?2.1? = 3, ?2.9? = 3, ?-2.6? = -2
?...?
${\displaystyle \lfloor \ldots \rceil \!\,}$
\lfloor \ldots \rceil \!\,
nearest integer to
?x? means the nearest integer to x.

(This may also be written [x], ||x||, nint(x) or Round(x).)
?2? = 2, ?2.6? = 3, ?-3.4? = -3, ?4.49? = 4, ?4.5? = 5
[ : ]
${\displaystyle [\ :\ ]\!\,}$
[\ :\ ] \!\,
the degree of
[K : F] means the degree of the extension K : F. [Q(?2) : Q] = 2

[C : R] = 2

[R : Q] = ?
[ ]

[ , ]

[ , , ]
${\displaystyle [\ ]\!\,}$
[\ ] \!\,

${\displaystyle [\ ,\ ]\!\,}$
[\ ,\ ] \!\,

${\displaystyle [\ ,\ ,\ ]\!\,}$
the equivalence class of
[a] means the equivalence class of a, i.e. {x : x ~ a}, where ~ is an equivalence relation.

[a]R means the same, but with R as the equivalence relation.
Let a ~ b be true iff a ? b (mod 5).

Then [2] = {..., -8, -3, 2, 7, ...}.

floor;
greatest integer;
entier
[x] means the floor of x, i.e. the largest integer less than or equal to x.

(This may also be written ?x?, floor(x) or int(x). Not to be confused with the nearest integer function, as described below.)
[3] = 3, [3.5] = 3, [3.99] = 3, [-3.7] = -4
nearest integer to
[x] means the nearest integer to x.

(This may also be written ?x?, ||x||, nint(x) or Round(x). Not to be confused with the floor function, as described above.)
[2] = 2, [2.6] = 3, [-3.4] = -3, [4.49] = 4
1 if true, 0 otherwise
[S] maps a true statement S to 1 and a false statement S to 0. [0=5]=0, [7>0]=1, [2 ? {2,3,4}]=1, [5 ? {2,3,4}]=0
image of ... under ...
everywhere
f[X] means { f(x) : x ? X }, the image of the function f under the set X ? dom(f).

(This may also be written as f(X) if there is no risk of confusing the image of f under X with the function application f of X. Another notation is Im f, the image of f under its domain.)
${\displaystyle \sin[\mathbb {R} ]=[-1,1]}$
closed interval
${\displaystyle [a,b]=\{x\in \mathbb {R} :a\leq x\leq b\}}$. 0 and 1/2 are in the interval [0,1].
the commutator of
[g, h] = g-1h-1gh (or ghg-1h-1), if g, h ? G (a group).

[a, b] = ab - ba, if a, b ? R (a ring or commutative algebra).
xy = x[x, y] (group theory).

[AB, C] = A[B, C] + [A, C]B (ring theory).
the triple scalar product of
[a, b, c] = a × b · c, the scalar product of a × b with c. [a, b, c] = [b, c, a] = [c, a, b].
( )

( , )
${\displaystyle (\ )\!\,}$
(\ ) \!\,

${\displaystyle (\ ,\ )\!\,}$
(\ ,\ ) \!\,
function application
of
f(x) means the value of the function f at the element x. If f(x) := x2 - 5, then f(6) = 62 - 5 = 36 - 5=31.
image of ... under ...
everywhere
f(X) means { f(x) : x ? X }, the image of the function f under the set X ? dom(f).

(This may also be written as f[X] if there is a risk of confusing the image of f under X with the function application f of X. Another notation is Im f, the image of f under its domain.)
${\displaystyle \sin(\mathbb {R} )=[-1,1]}$
precedence grouping
parentheses
everywhere
Perform the operations inside the parentheses first. (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.
tuple; n-tuple;
ordered pair/triple/etc;
row vector; sequence
everywhere
An ordered list (or sequence, or horizontal vector, or row vector) of values.

(Note that the notation (a,b) is ambiguous: it could be an ordered pair or an open interval. Set theorists and computer scientists often use angle brackets ? ? instead of parentheses.)

(a, b) is an ordered pair (or 2-tuple).

(a, b, c) is an ordered triple (or 3-tuple).

( ) is the empty tuple (or 0-tuple).

highest common factor;
greatest common divisor; hcf; gcd
number theory
(a, b) means the highest common factor of a and b.

(This may also be written hcf(a, b) or gcd(a, b).)
(3, 7) = 1 (they are coprime); (15, 25) = 5.
( , )

] , [
${\displaystyle (\ ,\ )\!\,}$
(\ ,\ ) \!\,(\ ,\ ) \!\,

${\displaystyle ]\ ,\ [\!\,}$
]\ ,\ [ \!\,]
open interval
${\displaystyle (a,b)=\{x\in \mathbb {R} :a.

(Note that the notation (a,b) is ambiguous: it could be an ordered pair or an open interval. The notation ]a,b[ can be used instead.)

4 is not in the interval (4, 18).

(0, +?) equals the set of positive real numbers.

( , ]

] , ]
${\displaystyle (\ ,\ ]\!\,}$
(\ ,\ ] \!\,

${\displaystyle ]\ ,\ ]\!\,}$
\ ,\ ] \!\,]
half-open interval;
left-open interval
${\displaystyle (a,b]=\{x\in \mathbb {R} :a. (-1, 7] and (-?, -1]
[ , )

[ , [
${\displaystyle [\ ,\ )\!\,}$
[\ ,\ ) \!\,

${\displaystyle [\ ,\ [\!\,}$
[\ ,\ [ \!\,
half-open interval;
right-open interval
${\displaystyle [a,b)=\{x\in \mathbb {R} :a\leq x. [4, 18) and [1, +?)

?,?
${\displaystyle \langle \ \rangle \!\,}$
\langle\ \rangle \!\,

${\displaystyle \langle \ ,\ \rangle \!\,}$
\langle\ ,\ \rangle \!\,
inner product of
?u,v? means the inner product of u and v, where u and v are members of an inner product space.

Note that the notation ?u, v? may be ambiguous: it could mean the inner product or the linear span.

There are many variants of the notation, such as ?u | v? and (u | v), which are described below. For spatial vectors, the dot product notation, x · y is common. For matrices, the colon notation A : B may be used. As ? and ? can be hard to type, the more "keyboard friendly" forms < and > are sometimes seen. These are avoided in mathematical texts.
The standard inner product between two vectors x = (2, 3) and y = (-1, 5) is:
?x, y? = 2 × -1 + 3 × 5 = 13
average
average of
let S be a subset of N for example, ${\displaystyle \langle S\rangle }$ represents the average of all the elements in S. for a time series :g(t) (t = 1, 2,...)

we can define the structure functions Sq(${\displaystyle \tau }$):

${\displaystyle S_{q}=\langle |g(t+\tau )-g(t)|^{q}\rangle _{t}}$
the expectation value of
For a single discrete variable ${\displaystyle x}$ of a function ${\displaystyle f(x)}$, the expectation value of ${\displaystyle f(x)}$ is defined as ${\displaystyle \langle f(x)\rangle =\sum _{x}f(x)P(x)}$, and for a single continuous variable the expectation value of ${\displaystyle f(x)}$ is defined as ${\displaystyle \langle f(x)\rangle =\int _{x}f(x)P(x)}$; where ${\displaystyle P(x)}$ is the PDF of the variable ${\displaystyle x}$.[4]
(linear) span of;
linear hull of
?S? means the span of S ? V. That is, it is the intersection of all subspaces of V which contain S.
?u1, u2, ...? is shorthand for ?{u1, u2, ...}?.

Note that the notation ?u, v? may be ambiguous: it could mean the inner product or the linear span.

The span of S may also be written as Sp(S).

${\displaystyle \left\langle \left({\begin{smallmatrix}1\\0\\0\end{smallmatrix}}\right),\left({\begin{smallmatrix}0\\1\\0\end{smallmatrix}}\right),\left({\begin{smallmatrix}0\\0\\1\end{smallmatrix}}\right)\right\rangle =\mathbb {R} ^{3}}$.
subgroup generated by a set
the subgroup generated by
${\displaystyle \langle S\rangle }$ means the smallest subgroup of G (where S ? G, a group) containing every element of S.
${\displaystyle \left\langle g_{1},g_{2},\dots \right\rangle }$ is shorthand for ${\displaystyle \left\langle \left\{g_{1},g_{2},\dots \right\}\right\rangle }$.
In S3, ${\displaystyle \langle (1\;2)\rangle =\{id,\;(1\;2)\}}$ and ${\displaystyle \langle (1\;2\;3)\rangle =\{id,\;(1\;2\;3),(1\;3\;2))\}}$.
tuple; n-tuple;
ordered pair/triple/etc;
row vector; sequence
everywhere
An ordered list (or sequence, or horizontal vector, or row vector) of values.

(The notation (a,b) is often used as well.)

${\displaystyle \langle a,b\rangle }$ is an ordered pair (or 2-tuple).

${\displaystyle \langle a,b,c\rangle }$ is an ordered triple (or 3-tuple).

${\displaystyle \langle \rangle }$ is the empty tuple (or 0-tuple).

?|?

(|)
${\displaystyle \langle \ |\ \rangle \!\,}$
\langle\ |\ \rangle \!\,

${\displaystyle (\ |\ )\!\,}$
(\ |\ ) \!\,
inner product of
?u | v? means the inner product of u and v, where u and v are members of an inner product space.[5] (u | v) means the same.

Another variant of the notation is ?u, v? which is described above. For spatial vectors, the dot product notation, x · y is common. For matrices, the colon notation A : B may be used. As ? and ? can be hard to type, the more "keyboard friendly" forms < and > are sometimes seen. These are avoided in mathematical texts.

## Other non-letter symbols

Symbol
in HTML
Symbol
in TeX
Name Explanation Examples
Category
${\displaystyle *}$
\ast or *
convolution;
convolved with
f*g means the convolution of f and g.   (Different than f*g, which means the product of g with the complex conjugate of f, as described below.)

(Can also be written in text as   f &lowast; g.)

${\displaystyle (f*g)(t)=\int _{0}^{t}f(\tau )g(t-\tau )\,d\tau }$.
Hodge star;
Hodge dual
*v means the Hodge dual of a vector v. If v is a k-vector within an n-dimensional oriented quadratic space, then *v is an (n-k)-vector. If ${\displaystyle \{e_{i}\}}$ are the standard basis vectors of ${\displaystyle \mathbb {R} ^{5}}$, ${\displaystyle *(e_{1}\wedge e_{2}\wedge e_{3})=e_{4}\wedge e_{5}}$
${\displaystyle ^{*}}$
^\ast or ^*
conjugate
z* means the complex conjugate of z.

(${\displaystyle {\bar {z}}}$ can also be used for the conjugate of z, as described below.)
${\displaystyle (3+4i)^{\ast }=3-4i}$.
the group of units of
R* consists of the set of units of the ring R, along with the operation of multiplication.

This may also be written R×as described above, or U(R).
{\displaystyle {\begin{aligned}(\mathbb {Z} /5\mathbb {Z} )^{\ast }&=\{[1],[2],[3],[4]\}\\&\cong \mathrm {C} _{4}\\\end{aligned}}}
the (set of) hyperreals
*R means the set of hyperreal numbers. Other sets can be used in place of R. *N is the hypernatural numbers.
Kleene star
Corresponds to the usage of * in regular expressions. If ? is a set of strings, then ?* is the set of all strings that can be created by concatenating members of ?. The same string can be used multiple times, and the empty string is also a member of ?*. If ? = ('a', 'b', 'c') then ?* includes '', 'a', 'ab', 'aba', 'abac', etc. The full set cannot be enumerated here since it is countably infinite, but each individual string must have finite length.
${\displaystyle |\!\,}$
given
P(A|B) means the probability of the event A occurring given that B occurs. if X is a uniformly random day of the year P(X is 25 | X is in May) = 1/31
restriction of ... to ...;
restricted to
f|A means the function f is restricted to the set A, that is, it is the function with domain A ? dom(f) that agrees with f. The function f : R -> R defined by f(x) = x2 is not injective, but f|R+ is injective.
such that
such that;
so that
everywhere
| means "such that", see ":" (described below). S = {(x,y) | 0 < y < f(x)}
The set of (x,y) such that y is greater than 0 and less than f(x).
|

?
${\displaystyle \mid \!\,}$
\mid

${\displaystyle \nmid \!\,}$
\nmid
divides
a | b means a divides b.
a ? b means a does not divide b.

(The symbol | can be difficult to type, and its negation is rare, so a regular but slightly shorter vertical bar | character is often used instead.)
Since 15 = 3 × 5, it is true that 3 | 15 and 5 | 15.
||
${\displaystyle \mid \mid \!\,}$
\mid\mid
exactly divides
pa || n means pa exactly divides n (i.e. pa divides n but pa+1 does not). 23 || 360.
?

?

?
${\displaystyle \|\!\,}$
\|
Requires the viewer to support Unicode: \unicode{x2225}, \unicode{x2226}, and \unicode{x22D5}.
\mathrel{\rlap{\,\parallel}} requires \setmathfont{MathJax}.[6]
is parallel to
x ? y means x is parallel to y.
x ? y means x is not parallel to y.
x ? y means x is equal and parallel to y.

(The symbol ? can be difficult to type, and its negation is rare, so two regular but slightly longer vertical bar || characters are often used instead.)
If l ? m and m ? n then l ? n.
is incomparable to
x ? y means x is incomparable to y. {1,2} ? {2,3} under set containment.
${\displaystyle \#\!\,}$
\sharp
cardinality of;
size of;
order of
#X means the cardinality of the set X.

(|...| may be used instead as described above.)
#{4, 6, 8} = 3
connected sum of;
knot sum of;
knot composition of
A#B is the connected sum of the manifolds A and B. If A and B are knots, then this denotes the knot sum, which has a slightly stronger condition. A#Sm is homeomorphic to A, for any manifold A, and the sphere Sm.
primorial
n# is product of all prime numbers less than or equal to n. 12# = 2 × 3 × 5 × 7 × 11 = 2310
${\displaystyle :\!\,}$
such that
such that;
so that
everywhere
: means "such that", and is used in proofs and the set-builder notation (described below). ? n ? N: n is even.
extends;
over
K : F means the field K extends the field F.

This may also be written as K >= F.
R : Q
inner product of matrices
inner product of
A : B means the Frobenius inner product of the matrices A and B.

The general inner product is denoted by ?u, v?, ?u | v? or (u | v), as described below. For spatial vectors, the dot product notation, x·y is common. See also bra-ket notation.
${\displaystyle A:B=\sum _{i,j}A_{ij}B_{ij}}$
index of subgroup
The index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G ${\displaystyle |G:H|={\frac {|G|}{|H|}}}$
divided by
over
everywhere
A : B means the division of A with B (dividing A by B) 10 : 2 = 5
?
${\displaystyle \vdots \!\,}$
\vdots \!\,
vertical ellipsis
everywhere
Denotes that certain constants and terms are missing out (e.g. for clarity) and that only the important terms are being listed. ${\displaystyle P(r,t)=\chi \vdots E(r,t_{1})E(r,t_{2})E(r,t_{3})}$
?
${\displaystyle \wr \!\,}$
\wr \!\,
wreath product of ... by ...
A ? H means the wreath product of the group A by the group H.

This may also be written A wr H.
${\displaystyle \mathrm {S} _{n}\wr \mathrm {Z} _{2}}$ is isomorphic to the automorphism group of the complete bipartite graph on (n,n) vertices.
?

?
${\displaystyle \oplus \!\,}$
\oplus \!\,

${\displaystyle \veebar \!\,}$
\veebar \!\,
xor
The statement A ? B is true when either A or B, but not both, are true. A ? B means the same. A) ? A is always true, A ? A is always false.
direct sum of
The direct sum is a special way of combining several objects into one general object.

(The bun symbol ?, or the coproduct symbol ?, is used; ? is only for logic.)
Most commonly, for vector spaces U, V, and W, the following consequence is used:
U = V ? W (U = V + W) ? (V ? W = {0})
?
${\displaystyle \Box \!\,}$
\Box \!\
D'Alembertian;
wave operator
non-Euclidean Laplacian
It is the generalisation of the Laplace operator in the sense that it is the differential operator which is invariant under the isometry group of the underlying space and it reduces to the Laplace operator if restricted to time independent functions. ${\displaystyle \square ={\frac {1}{c^{2}}}{\partial ^{2} \over \partial t^{2}}-{\partial ^{2} \over \partial x^{2}}-{\partial ^{2} \over \partial y^{2}}-{\partial ^{2} \over \partial z^{2}}}$

## Letter-based symbols

Includes upside-down letters.

### Letter modifiers

Also called diacritics.

Symbol
in HTML
Symbol
in TeX
Name Explanation Examples
Category
${\displaystyle {\bar {a}}}$
\bar{a}, \overline{a}
overbar;
... bar
${\displaystyle {\bar {x}}}$ (often read as "x bar") is the mean (average value of ${\displaystyle x_{i}}$). ${\displaystyle x=\{1,2,3,4,5\};{\bar {x}}=3}$.
finite sequence, tuple
${\displaystyle {\overline {a}}}$ means the finite sequence/tuple ${\displaystyle (a_{1},a_{2},...,a_{n}).}$. ${\displaystyle {\overline {a}}:=(a_{1},a_{2},...,a_{n})}$.
algebraic closure of
${\displaystyle {\overline {F}}}$ is the algebraic closure of the field F. The field of algebraic numbers is sometimes denoted as ${\displaystyle {\overline {\mathbb {Q} }}}$ because it is the algebraic closure of the rational numbers ${\displaystyle {\mathbb {Q} }}$.
conjugate
${\displaystyle {\overline {z}}}$ means the complex conjugate of z.

(z*can also be used for the conjugate of z, as described above.)
${\displaystyle {\overline {3+4i}}=3-4i}$.
(topological) closure of
${\displaystyle {\overline {S}}}$ is the topological closure of the set S.

This may also be denoted as cl(S) or Cl(S).
In the space of the real numbers, ${\displaystyle {\overline {\mathbb {Q} }}=\mathbb {R} }$ (the rational numbers are dense in the real numbers).
${\displaystyle {\overset {\rightharpoonup }{a}}}$
${\displaystyle {\overset {\rightharpoonup }{a}}}$
\overset{\rightharpoonup}{a}
harpoon
â
${\displaystyle {\hat {a}}}$
\hat a
hat
${\displaystyle \mathbf {\hat {a}} }$ (pronounced "a hat") is the normalized version of vector ${\displaystyle \mathbf {a} }$, having length 1.
estimator for
${\displaystyle {\hat {\theta }}}$ is the estimator or the estimate for the parameter ${\displaystyle \theta }$. The estimator ${\displaystyle \mathbf {\hat {\mu }} ={\frac {\sum _{i}x_{i}}{n}}}$ produces a sample estimate ${\displaystyle \mathbf {\hat {\mu }} (\mathbf {x} )}$ for the mean ${\displaystyle \mu }$.
${\displaystyle '}$
'
... prime;
derivative of
f ?(x) means the derivative of the function f at the point x, i.e., the slope of the tangent to f at x.

(The single-quote character ' is sometimes used instead, especially in ASCII text.)
If f(x) := x2, then f ?(x) = 2x.
${\displaystyle {\dot {\,}}}$
\dot{\,}
... dot;
time derivative of
${\displaystyle {\dot {x}}}$ means the derivative of x with respect to time. That is ${\displaystyle {\dot {x}}(t)={\frac {\partial }{\partial t}}x(t)}$. If x(t) := t2, then ${\displaystyle {\dot {x}}(t)=2t}$.

### Symbols based on Latin letters

Symbol
in HTML
Symbol
in TeX
Name Explanation Examples
Category
B

B
${\displaystyle \mathbb {B} }$
\mathbb{B}

${\displaystyle \mathbf {B} }$
\mathbf{B}
B;
the (set of) boolean values;
the (set of) truth values;
B means either {0, 1}, {false, true}, {F, T}, or ${\displaystyle \left\{\bot ,\top \right\}}$. False) ? B
C

C
${\displaystyle \mathbb {C} }$
\mathbb{C}

${\displaystyle \mathbf {C} }$
\mathbf{C}
C;
the (set of) complex numbers
C means {a + b i : a,b ? R}. i ? C

${\displaystyle \partial }$
\partial
partial;
d
?f/?xi means the partial derivative of f with respect to xi, where f is a function on (x1, ..., xn). If f(x,y) := x2y, then ?f/?x = 2xy,
boundary of
?M means the boundary of M ?{x : ||x|| x : ||x|| = 2}
degree of
?f means the degree of the polynomial f.

(This may also be written deg f.)
?(x2 - 1) = 2
E

E
${\displaystyle \mathbb {E} }$
\mathbb E

${\displaystyle \mathrm {E} }$
\mathrm{E}
expected value
the value of a random variable one would "expect" to find if one could repeat the random variable process an infinite number of times and take the average of the values obtained ${\displaystyle \mathbb {E} [X]={\frac {x_{1}p_{1}+x_{2}p_{2}+\dotsb +x_{k}p_{k}}{p_{1}+p_{2}+\dotsb +p_{k}}}}$
${\displaystyle \mathbb {F} }$
\mathbb{F}
Galois field, or finite field
${\displaystyle \mathbb {F} _{p^{n}}}$, for any prime p and integer n, is the unique finite field with order ${\displaystyle p^{n}}$, often written ${\displaystyle \mathrm {GF} (p^{n})}$, and sometimes also known as ${\displaystyle \mathbf {Z} /p\mathbf {Z} }$, ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$, or ${\displaystyle \mathbb {Z} _{p}}$, although this last notation is ambiguous. ${\displaystyle \left(\mathbb {F} _{2^{255}-19}\right)^{2}}$ is ${\displaystyle \mathrm {GF} (2^{255}-19)^{2}}$, the finite field in whose quadratic extension the popular elliptic curve Curve25519 is computed.
H

H
${\displaystyle \mathbb {H} }$
\mathbb{H}

${\displaystyle \mathbf {H} }$
\mathbf{H}
quaternions or Hamiltonian quaternions
H;
the (set of) quaternions
H means {a + b i + c j + d k : a,b,c,d ? R}.
I

I
${\displaystyle \mathbb {I} }$
\mathbb{I}

${\displaystyle \mathbf {I} }$
\mathbf{I}
the indicator of
The indicator function of a subset A of a set X is a function ${\displaystyle \mathbf {1} _{A}\colon X\to \{0,1\}}$ defined as : ${\displaystyle \mathbf {1} _{A}(x):={\begin{cases}1&{\text{if }}x\in A,\\0&{\text{if }}x\notin A.\end{cases}}}$

Note that the indicator function is also sometimes denoted 1.

N

N
${\displaystyle \mathbb {N} }$
\mathbb{N}

${\displaystyle \mathbf {N} }$
\mathbf{N}
the (set of) natural numbers
N means either { 0, 1, 2, 3, ...} or { 1, 2, 3, ...}.

The choice depends on the area of mathematics being studied; e.g. number theorists prefer the latter; analysts, set theorists and computer scientists prefer the former. To avoid confusion, always check an author's definition of N.

Set theorists often use the notation ? (for least infinite ordinal) to denote the set of natural numbers (including zero), along with the standard ordering relation
N = {|a| : a ? Z} or N = {|a| > 0: a ? Z}
?

?
${\displaystyle \circ }$
\circ

${\displaystyle \odot }$[7][8]
\odot
entrywise product, elementwise product, circled dot
For two matrices (or vectors) of the same dimensions ${\displaystyle A,B\in {\mathbb {R} }^{m\times n}}$ the Hadamard product is a matrix of the same dimensions ${\displaystyle A\circ B\in {\mathbb {R} }^{m\times n}}$ with elements given by ${\displaystyle (A\circ B)_{i,j}=(A\odot B)_{i,j}=(A)_{i,j}\cdot (B)_{i,j}}$. ${\displaystyle {\begin{bmatrix}1&2\\2&4\\\end{bmatrix}}\circ {\begin{bmatrix}1&2\\0&0\\\end{bmatrix}}={\begin{bmatrix}1&2\\2&4\\\end{bmatrix}}\odot {\begin{bmatrix}1&2\\0&0\\\end{bmatrix}}={\begin{bmatrix}1&4\\0&0\\\end{bmatrix}}}$
?
${\displaystyle \circ }$
\circ
composed with
f ? g is the function such that (f ? g)(x) = f(g(x)).[9] if f(x) := 2x, and g(x) := x + 3, then (f ? g)(x) = 2(x + 3).
${\displaystyle O}$
O
big-oh of
The Big O notation describes the limiting behavior of a function, when the argument tends towards a particular value or infinity. If f(x) = 6x4 - 2x3 + 5 and g(x) = x4, then ${\displaystyle f(x)=O(g(x)){\mbox{ as }}x\to \infty \,}$

P

P
${\displaystyle \mathbb {P} }$
\mathbb{P}

${\displaystyle \mathbf {P} }$
\mathbf{P}
P;
the set of prime numbers
P is often used to denote the set of prime numbers. ${\displaystyle 2\in \mathbb {P} ,3\in \mathbb {P} ,8\notin \mathbb {P} }$
P;
the projective space;
the projective line;
the projective plane
P means a space with a point at infinity. ${\displaystyle \mathbb {P} ^{1}}$,${\displaystyle \mathbb {P} ^{2}}$
the space of all possible polynomials
P means anxn + an-1xn-1...a1x+a0
Pn means the space of all polynomials of degree less than or equal to n
${\displaystyle 2x^{3}-3x^{2}+2\in \mathbb {P} _{3}}$
the probability of
P(X) means the probability of the event X occurring.

This may also be written as P(X), Pr(X), P[X] or Pr[X].
If a fair coin is flipped, P(Heads) = P(Tails) = 0.5.
the Power set of
Given a set S, the power set of S is the set of all subsets of the set S. The power set of S0 is

denoted by P(S).

The power set P({0, 1, 2}) is the set of all subsets of {0, 1, 2}. Hence,

P({0, 1, 2}) = {?, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2} }.

Q

Q
${\displaystyle \mathbb {Q} }$
\mathbb{Q}

${\displaystyle \mathbf {Q} }$
\mathbf{Q}
Q;
the (set of) rational numbers;
the rationals
Q means {p/q : p ? Z, q ? N}. 3.14000... ? Q

? ? Q
Qp

Qp
${\displaystyle \mathbb {Q} _{p}}$
\mathbb{Q}_p

${\displaystyle \mathbf {Q} _{p}}$
\mathbf{Q}_p
Q means {p/q : p ? Z, q ? N}.

R

R
${\displaystyle \mathbb {R} }$
\mathbb{R}

${\displaystyle \mathbf {R} }$
\mathbf{R}
R;
the (set of) real numbers;
the reals
R means the set of real numbers. ? ? R

?(-1) ? R

${\displaystyle {}^{\dagger }}$
{}^\dagger
conjugate transpose;
A+ means the transpose of the complex conjugate of A.[10]

This may also be written A*T, AT*, A*, ATor AT.
If A = (aij) then A+ = (aji).
${\displaystyle {}^{\mathsf {T}}}$
{}^{\mathsf{T}}
transpose
AT means A, but with its rows swapped for columns.

This may also be written A?, Ator Atr.
If A = (aij) then AT = (aji).
${\displaystyle \top }$
\top
the top element
? means the largest element of a lattice. ?x : x ? ? = x
the top type; top
? means the top or universal type; every type in the type system of interest is a subtype of top. ? types T, T <
top, verum
The statement ? is unconditionally true. A => ? is always true.
${\displaystyle \bot }$
\bot
bottom, falsum, falsity
The statement ? is unconditionally false. ? => A is always true.
is perpendicular to
x ? y means x is perpendicular to y; or more generally x is orthogonal to y. If l ? m and m ? n in the plane, then l || n.
orthogonal/ perpendicular complement of;
perp
W? means the orthogonal complement of W (where W is a subspace of the inner product space V), the set of all vectors in V orthogonal to every vector in W. Within ${\displaystyle \mathbb {R} ^{3}}$, ${\displaystyle (\mathbb {R} ^{2})^{\perp }\cong \mathbb {R} }$.
is coprime to
x ? y means x has no factor greater than 1 in common with y. 34 ? 55
is independent of
A ? B means A is an event whose probability is independent of event B. The double perpendicular symbol (${\displaystyle \perp \!\!\!\perp }$) is also commonly used for the purpose of denoting this, for instance: ${\displaystyle A\perp \!\!\!\perp B}$ (In LaTeX, the command is: "A \perp\!\!\!\perp B".) If A ? B, then P(A|B) = P(A).
the bottom element
? means the smallest element of a lattice. ?x : x ? ? = x
the bottom type;
bot
? means the bottom type (a.k.a. the zero type or empty type); bottom is the subtype of every type in the type system. ? types T, ? <: T
is comparable to
x ? y means that x is comparable to y. {e, ?} ? {1, 2, e, 3, ?} under set containment.

U

U
${\displaystyle \mathbb {U} }$
\mathbb{U}

${\displaystyle \mathbf {U} }$
\mathbf{U}
U;
the universal set;
the set of all numbers;
all numbers considered
U means "the set of all elements being considered."
It may represent all numbers both real and complex, or any subset of these--hence the term "universal".
U = {R,C} includes all numbers.

If instead, U = {Z,C}, then ? ? U.

?
${\displaystyle \otimes }$
\otimes
tensor product of
${\displaystyle V\otimes U}$ means the tensor product of V and U.[11]${\displaystyle V\otimes _{R}U}$ means the tensor product of modules V and U over the ring R. {1, 2, 3, 4} ? {1, 1, 2} =
{{1, 1, 2}, {2, 2, 4}, {3, 3, 6}, {4, 4, 8}}
?

?
${\displaystyle \ltimes }$
\ltimes

${\displaystyle \rtimes }$
\rtimes
the semidirect product of
N ??H is the semidirect product of N (a normal subgroup) and H (a subgroup), with respect to ?. Also, if G = N ??H, then G is said to split over N.

(? may also be written the other way round, as ?, or as ×.)
${\displaystyle D_{2n}\cong \mathrm {C} _{n}\rtimes \mathrm {C} _{2}}$
the semijoin of
R ? S is the semijoin of the relations R and S, the set of all tuples in R for which there is a tuple in S that is equal on their common attribute names. R ${\displaystyle \ltimes }$ S = ${\displaystyle \Pi }$a1,..,an(R ${\displaystyle \bowtie }$ S)
?
${\displaystyle \bowtie }$
\bowtie
the natural join of
R ? S is the natural join of the relations R and S, the set of all combinations of tuples in R and S that are equal on their common attribute names.

Z

Z
${\displaystyle \mathbb {Z} }$
\mathbb{Z}

${\displaystyle \mathbf {Z} }$
\mathbf{Z}
the (set of) integers
Z means {..., -3, -2, -1, 0, 1, 2, 3, ...}.

Z+ or Z> means {1, 2, 3, ...} .
Z>= means {0, 1, 2, 3, ...} .
Z* is used by some authors to mean {0, 1, 2, 3, ...}[12] and others to mean {... -2, -1, 1, 2, 3, ... }.[13]

Z = {p, -p : p ? N ? {0}}
Zn

Zp

Zn

Zp
${\displaystyle \mathbb {Z} _{n}}$
\mathbb{Z}_n

${\displaystyle \mathbb {Z} _{p}}$
\mathbb{Z}_p

${\displaystyle \mathbf {Z} _{n}}$
\mathbf{Z}_n

${\displaystyle \mathbf {Z} _{p}}$
the (set of) integers modulo n
Zn means {[0], [1], [2], ...[n-1]} with addition and multiplication modulo n.

Note that any letter may be used instead of n, such as p. To avoid confusion with p-adic numbers, use Z/pZ or Z/(p) instead.
Z3 = {[0], [1], [2]}

Note that any letter may be used instead of p, such as n or l.

### Symbols based on Hebrew or Greek letters

Symbol
in HTML
Symbol
in TeX
Name Explanation Examples
Category
${\displaystyle \Gamma }$
\Gamma
Gamma function
${\displaystyle \Gamma (z)=\int _{0}^{\infty }x^{z-1}e^{-x}\,dx,\ \qquad \Re (z)>0\ .}$ {\displaystyle {\begin{aligned}\Gamma (1)&=\int _{0}^{\infty }x^{1-1}e^{-x}\,dx\\[6pt]&={\Big [}-e^{-x}{\Big ]}_{0}^{\infty }\\[6pt]&=\lim _{x\to \infty }(-e^{-x})-(-e^{-0})\\[6pt]&=0-(-1)\\[6pt]&=1.\end{aligned}}}
${\displaystyle \delta }$
\delta
Dirac delta of
${\displaystyle \delta (x)={\begin{cases}\infty ,&x=0\\0,&x\neq 0\end{cases}}}$ ?(x)
Kronecker delta of
${\displaystyle \delta _{ij}={\begin{cases}1,&i=j\\0,&i\neq j\end{cases}}}$ ?ij
Functional derivative of
{\displaystyle {\begin{aligned}\left\langle {\frac {\delta F[\varphi (x)]}{\delta \varphi (x)}},f(x)\right\rangle &=\int {\frac {\delta F[\varphi (x)]}{\delta \varphi (x')}}f(x')dx'\\&=\lim _{\varepsilon \to 0}{\frac {F[\varphi (x)+\varepsilon f(x)]-F[\varphi (x)]}{\varepsilon }}\\&=\left.{\frac {d}{d\epsilon }}F[\varphi +\epsilon f]\right|_{\epsilon =0}.\end{aligned}}} ${\displaystyle {\frac {\delta V(r)}{\delta \rho (r')}}={\frac {1}{4\pi \epsilon _{0}|r-r'|}}}$
?

?

?
${\displaystyle \vartriangle }$
\vartriangle

${\displaystyle \ominus }$
\ominus

${\displaystyle \oplus }$
\oplus
symmetric difference
A ? B (or A ? B) means the set of elements in exactly one of A or B.

(Not to be confused with delta, ?, described below.)
{1,5,6,8} ? {2,5,8} = {1,2,6}

{3,4,5,6} ? {1,2,5,6} = {1,2,3,4}
${\displaystyle \Delta }$
\Delta
delta;
change in
?x means a (non-infinitesimal) change in x.

(If the change becomes infinitesimal, ? and even d are used instead. Not to be confused with the symmetric difference, written ?, above.)
${\displaystyle {\tfrac {\Delta y}{\Delta x}}}$ is the gradient of a straight line.
Laplace operator
The Laplace operator is a second order differential operator in n-dimensional Euclidean space If ? is a twice-differentiable real-valued function, then the Laplacian of ? is defined by ${\displaystyle \Delta f=\nabla ^{2}f=\nabla \cdot \nabla f}$
${\displaystyle \nabla }$
\nabla
?f (x1, ..., xn) is the vector of partial derivatives (?f / ?x1, ..., ?f / ?xn). If f (x,y,z) := 3xy + z², then ?f = (3y, 3x, 2z)
del dot;
divergence of
${\displaystyle \nabla \cdot {\vec {v}}={\partial v_{x} \over \partial x}+{\partial v_{y} \over \partial y}+{\partial v_{z} \over \partial z}}$ If ${\displaystyle {\vec {v}}:=3xy\mathbf {i} +y^{2}z\mathbf {j} +5\mathbf {k} }$, then ${\displaystyle \nabla \cdot {\vec {v}}=3y+2yz}$.
curl of
${\displaystyle \nabla \times {\vec {v}}=\left({\partial v_{z} \over \partial y}-{\partial v_{y} \over \partial z}\right)\mathbf {i} }$
${\displaystyle +\left({\partial v_{x} \over \partial z}-{\partial v_{z} \over \partial x}\right)\mathbf {j} +\left({\partial v_{y} \over \partial x}-{\partial v_{x} \over \partial y}\right)\mathbf {k} }$
If ${\displaystyle {\vec {v}}:=3xy\mathbf {i} +y^{2}z\mathbf {j} +5\mathbf {k} }$, then ${\displaystyle \nabla \times {\vec {v}}=-y^{2}\mathbf {i} -3x\mathbf {k} }$.
${\displaystyle \pi }$
\pi
pi;
3.1415926...;
?355÷113
Used in various formulas involving circles; ? is equivalent to the amount of area a circle would take up in a square of equal width with an area of 4 square units, roughly 3.14159. It is also the ratio of the circumference to the diameter of a circle. A = ?R2 = 314.16 -> R = 10
prime-counting function of
${\displaystyle \pi (x)}$ counts the number of prime numbers less than or equal to ${\displaystyle x}$. ${\displaystyle \pi (10)=4}$
Projection of
${\displaystyle \pi _{a_{1},\ldots ,a_{n}}(R)}$ restricts ${\displaystyle R}$ to the ${\displaystyle \{a_{1},\ldots ,a_{n}\}}$ attribute set. ${\displaystyle \pi _{\text{Age,Weight}}({\text{Person}})}$
the nth Homotopy group of
${\displaystyle \pi _{n}(X)}$ consists of homotopy equivalence classes of base point preserving maps from an n-dimensional sphere (with base point) into the pointed space X. ${\displaystyle \pi _{i}(S^{4})=\pi _{i}(S^{7})\oplus \pi _{i-1}(S^{3})}$
${\displaystyle \prod }$
\prod
product over ... from ... to ... of
${\displaystyle \prod _{k=1}^{n}a_{k}}$ means ${\displaystyle a_{1}a_{2}\dots a_{n}}$. ${\displaystyle \prod _{k=1}^{4}(k+2)=(1+2)(2+2)(3+2)(4+2)=3\times 4\times 5\times 6=360}$
the Cartesian product of;
the direct product of
${\displaystyle \prod _{i=0}^{n}{Y_{i}}}$ means the set of all (n+1)-tuples
(y0, ..., yn).
${\displaystyle \prod _{n=1}^{3}{\mathbb {R} }=\mathbb {R} \times \mathbb {R} \times \mathbb {R} =\mathbb {R} ^{3}}$
${\displaystyle \sigma }$
\sigma
population standard deviation
A measure of spread or variation of a set of values in a sample population set. ${\displaystyle \sigma ={\sqrt {\dfrac {\Sigma (x_{i}-\mu )^{2}}{N}}}}$

## Variations

In mathematics written in Persian or Arabic, some symbols may be reversed to make right-to-left writing and reading easier.[14]

## References

1. ^ ISO 80000-2, Section 9 "Operations", 2-9.6
2. ^ Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, Cambridge University Press, p. 66, ISBN 978-0-521-63503-5, OCLC 43641333
3. ^ a b Goldrei, Derek (1996), Classic Set Theory, Chapman and Hall, p. 3, ISBN 978-0-412-60610-6
4. ^ "Expectation Value". MathWorld. Wolfram Research. Retrieved .
5. ^ Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, Cambridge University Press, p. 62, ISBN 978-0-521-63503-5, OCLC 43641333
6. ^ "Parallel Symbol in TeX". Google Groups. Retrieved 2017.
8. ^ What does a dot in a circle mean?
9. ^ Goldrei, Derek (1996), Classic Set Theory, Chapman and Hall, p. 5, ISBN 978-0-412-60610-6
10. ^ Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, Cambridge University Press, pp. 69-70, ISBN 978-0-521-63503-5, OCLC 43641333
11. ^ Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, Cambridge University Press, pp. 71-72, ISBN 978-0-521-63503-5, OCLC 43641333
12. ^ Z^* from Wolfram MathWorld
13. ^ Turner, L.K.; Budden, F.J.; Knighton, D. (1975). Advanced mathematics : a unified course. 2. Longman. ISBN 9780582352407.
14. ^ M. Benatia, A. Lazrik, and K. Sami, "Arabic mathematical symbols in Unicode Archived 2011-07-22 at the Wayback Machine", 27th Internationalization and Unicode Conference, 2005.