# List 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 in this face, or their use is controversial).

The use of letters as symbols for variables and numerical constants is not described in this article. For these uses, see Variable (mathematics) and List of mathematical constants.

Letters are not sufficient for the need of mathematicians, and many other symbols are used. Some take their origin in punctuation marks and diacritics traditionally used in typography. Other, such as + and =, have been specially designed for mathematics, often by deforming some letters, such as ${\displaystyle \in }$ or ${\displaystyle \forall .}$

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 is easier. On the other hand, the LaTeX rendering is generally much better (more aesthetic). 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.

## 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 the ratio of two quantities.
2.  In some countries, may denote division.
/
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. Also denoted ${\displaystyle P(A\mid B),}$ see "|".
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 3+2.
2.  With an integer greater than 2 as a left superscript, denotes an nth root. For example 32.

## 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, π ≈ 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, AB 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, AB 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 >.

## Brackets

Many sorts of brackets are used in mathematics. There meanings depend not only on their shapes, but also of the nature and the arrangement of what is delimited by them, and sometimes what appear between or after them. For this reason, in the entry titles, the symbol is used for schematizing the syntax that underlies the meaning.

### Parentheses

(□)
Used in an expression for specifying that the sub-expression between the parentheses has to be considered as a single entity; typically used for specifying the order of operations.
□(□)
□(□, □)
□(□, ..., □)
1.  Functional notation: if the first ${\displaystyle \Box }$ is the name (symbol) of a function, denotes the value of the function applied to the expression between the parentheses; for example ${\displaystyle f(x),}$ ${\displaystyle \sin(x+y).}$ In the case of a multivariate function, the parentheses contain several expressions separated by commas, such as ${\displaystyle f(x,y).}$
2.  May also denote a product, such as in ${\displaystyle a(b+c).}$ When the confusion is possible, the context must distinguish which symbols denote functions, and which ones denote variables.
(□, □)
1.  Denotes an ordered pair of mathematical objects, for example ${\displaystyle (\pi ,3.14).}$
2.  If a and b are real numbers, ${\displaystyle -\infty ,}$ or ${\displaystyle +\infty ,}$ and a < b, then ${\displaystyle (a,b)}$ denotes the open interval delimited by a and b. See ]□, □[ for an alternative notation.
3.  If a and b are integers, ${\displaystyle (a,b)}$ may denote the greatest common divisor of a and b. Notation ${\displaystyle \gcd(a,b)}$ is often used instead.
(□, □, □)
If x, y, z are vectors in ${\displaystyle \mathbb {R} ^{3},}$ then ${\displaystyle (x,y,z)}$ may denote the scalar triple product.[citation needed] See also [□,□,□] in § Square brackets.
(□, ..., □)
Denotes a tuple. If there are n objects separated by commas, it is an ntuple.
(□, □, ...)
(□, ..., □, ...)
Denotes an infinite sequence.
${\displaystyle {\begin{pmatrix}\Box &\cdots &\Box \\\vdots &\ddots &\vdots \\\Box &\cdots &\Box \end{pmatrix}}}$
Denotes a matrix. Often denoted with square brackets.
${\displaystyle {\binom {\Box }{\Box }}}$
Denotes a binomial coefficient: Given two nonnegative integers, ${\displaystyle {\binom {n}{k}}}$ is read as "n choose k", and is defined as the integer ${\displaystyle {\frac {n(n-1)\cdots (n-k+1)}{1\cdot 2\cdots k}}={\frac {n!}{k!\,(n-k)!}}}$ (if k = 0, its value is conventionally 1). Using the left-hand-side expression, it is also defined and used for any value real of n.
(/)
Legendre symbol: If p is an odd prime number and a is an integer, the value of ${\displaystyle \left({\frac {a}{p}}\right)}$ is 1 if a is a quadratic residue modulo p; it is –1 if a is a quadratic non-residue modulo p; it is 0 if p divides a.

### Square brackets

[□]
1.  Sometimes used as a synonym (□) for avoiding nested parentheses.
2.  Equivalence class: given an equivalence relation, ${\displaystyle [x]}$ denotes often the equivalence class of the element x.
3.  Integral part: if x is a real number, [x] denotes often the integral part or truncation of x, that is the integer obtained by removing all digits after the decimal mark. This notation has also been used for other variants of floor and ceiling functions.
4.  Iverson bracket: if P is a predicate, ${\displaystyle [P]}$ may denote the Iverson bracket, that is the function that takes the value 1 for the values of the free variables in P for which P is true, and takes the value 0 otherwise. For example, ${\displaystyle [x=y]}$ is the Kronecker delta function, which equals one if ${\displaystyle x=y,}$ and zero otherwise.
□[□]
Image of a subset: if S is a subset of the domain of the function f, then ${\displaystyle f[S]}$ is sometimes used for denoting the image] of S. When no confusion is possible, notation f(S) is commonly used.
[□, □]
1.  Closed interval: if a and b are real numbers such that ${\displaystyle a\leq b}$, then ${\displaystyle [a,b]}$ denotes the closed interval defined by them.
2.  Commutator (group theory): if a and b belong to a group, then ${\displaystyle [a,b]=a^{-1}b^{-1}ab.}$
3.  Commutator (ring theory): if a and b belong to a ring, then ${\displaystyle [a,b]=ab-ba.}$
4.  Denotes the Lie bracket, the operation of a Lie algebra.
[□ : □]
1.  Degree of a field extension: if F is an extension of a field E, then ${\displaystyle [F:E]}$ denotes the degree of the field extension ${\displaystyle F/E.}$ For example, ${\displaystyle [\mathbb {C} :\mathbb {R} ]=2.}$
2.  Index of a subgroup: if H is a subgroup of a group E, then ${\displaystyle [G:H]}$ denotes the index of H in G. The notation |G:H| is also used
[□, □, □]
If x, y, z are vectors in ${\displaystyle \mathbb {R} ^{3},}$ then ${\displaystyle [x,y,z]}$ may denote the scalar triple product.[2] See also (□,□,□) in § Parentheses.
${\displaystyle {\begin{bmatrix}\Box &\cdots &\Box \\\vdots &\ddots &\vdots \\\Box &\cdots &\Box \end{bmatrix}}}$
Denotes a matrix. Often denoted with parentheses.

### Braces

{ }
Set-builder notation for the empty set, also denoted ${\displaystyle \emptyset }$ or .
{□}
1.  Sometimes used as a synonym (□) and [□] for avoiding nested parentheses.
2.  Set-builder notation for a singleton set: ${\displaystyle \{x\}}$ denotes the set that has x as a single element.
{□, ..., □}
Set-builder notation: denotes the set whose elements are listen between the braces, separated by commas.
{□ : □}
{□ | □}
Set-builder notation: if ${\displaystyle P(x)}$ is a predicate depending on a variable x, then both ${\displaystyle \{x:P(x)\}}$ and ${\displaystyle \{x\mid P(x)\}}$ denote the set formed by the values of x for which ${\displaystyle P(x)}$ is true.
Single brace
1.  Used for emphasizing that several equations have to be considered as simultaneous equations; for example ${\displaystyle \textstyle {\begin{cases}2x+y=1\\3x-y=1\end{cases}}}$
2.  Piecewise definition; for example ${\displaystyle \textstyle |x|={\begin{cases}x&{\text{if }}x\geq 0\\-x&{\text{if }}x<0\end{cases}}}$
3.  Used for grouped annotation of elements in a formula; for example ${\displaystyle \textstyle \underbrace {(a,b,\ldots ,z)} _{26},}$ ${\displaystyle \textstyle \overbrace {1+2+\cdots +100} ^{=5050},}$ ${\displaystyle \textstyle \left.{\begin{bmatrix}A\\B\end{bmatrix}}\right\}m+n{\text{ rows}}}$

### Other brackets

|□|
1.  Absolute value: if x is a real or complex number, ${\displaystyle |x|}$ denotes its absolute value.
2.  Number of elements: If S is a set, ${\displaystyle |x|}$ may denote its cardinality, that is its number of elements. ${\displaystyle \#S}$ is also often used, see #.
3.  Length of a line segment: If P and Q are two points in a Euclidean space, then ${\displaystyle |PQ|}$ denotes often the length of the line segment that they define, which is the distance from P to Q, and is often denoted ${\displaystyle d(P,Q).}$
|□:□|
Index of a subgroup: if H is a subgroup of a group E, then ${\displaystyle |G:H|}$ denotes the index of H in G. The notation [G:H] is also used
${\displaystyle \textstyle {\begin{vmatrix}\Box &\cdots &\Box \\\vdots &\ddots &\vdots \\\Box &\cdots &\Box \end{vmatrix}}}$
${\displaystyle {\begin{vmatrix}x_{1,1}&\cdots &x_{1,n}\\\vdots &\ddots &\vdots \\x_{n,1}&\cdots &x_{n,n}\end{vmatrix}}}$ denotes the determinant of the square matrix ${\displaystyle {\begin{bmatrix}x_{1,1}&\cdots &x_{1,n}\\\vdots &\ddots &\vdots \\x_{n,1}&\cdots &x_{n,n}\end{bmatrix}}.}$
||□||
Denotes the norm of an element of a normed vector space.
⌊□⌋
Floor function: if x is a real number, ${\displaystyle \lfloor x\rfloor }$ is the largest integer that is not greater than x.
⌈□⌉
Ceil function: if x is a real number, ${\displaystyle \lceil x\rceil }$ is the smallest integer that is not smaller than x.
⌊□⌉
Nearest integer function: if x is a real number, ${\displaystyle \lfloor x\rceil }$ is the integer that is the closest to x.
]□, □[
Open Interval: If a and b are real numbers, ${\displaystyle -\infty ,}$ or ${\displaystyle +\infty ,}$ and ${\displaystyle a then ${\displaystyle ]a,b[}$ denotes the open interval delimited by a and b. See (□, □) for an alternative notation.
(□, □]
]□, □]
Both notations are used for a left-open interval.
[□, □)
[□, □[
Both notations are used for a right-open interval.
⟨□⟩
1.  Generated object: if S is a set of elements in a algebraic structure, ${\displaystyle \langle S\rangle }$ denotes often the object generated by S. If ${\displaystyle S=\{s_{1},\ldots ,s_{n}\},}$ one writes ${\displaystyle \langle s_{1},\ldots ,s_{n}\rangle }$ (that is, braces are omitted). In particular, this may denotes
2.  Often used, mainly in physics, for denoting an expected value. In probability theory, ${\displaystyle E(X)}$ is generally used instead of ${\displaystyle \langle S\rangle .}$
⟨□ , □⟩
⟨□ | □⟩
Both ${\displaystyle \langle x,y\rangle }$ and ${\displaystyle \langle x\mid y\rangle }$ are commonly used for denoting the inner product in an inner product space.
⟨□| and |□⟩
Bra–ket notation or Dirac notation: if x and y are elements of an inner product space, ${\displaystyle |x\rangle }$ is the vector defined by x, and ${\displaystyle \langle y|}$ is the covector defined by y; their inner product is ${\displaystyle \langle y\mid x\rangle .}$

## Set theory

Denotes the empty set, and is more often written ${\displaystyle \emptyset .}$ Using set-builder notation, it may also be denoted { }.
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.
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.
${\displaystyle \coprod }$
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".
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" or the symbol "&".
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".

## Blackboard bold

The blackboard bold typeface is widely used for denoting the basic number systems. These systems are often denoted also 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 encounter ${\displaystyle \mathbb {R} }$ in combinatorics, one should immediately know that this denotes the real numbers, although combinatorics does not study the real numbers (but it uses them for many proofs).

${\displaystyle \mathbb {N} }$
Denotes the set of natural numbers ${\displaystyle \{0,1,2,\ldots \},}$ or sometimes ${\displaystyle \{1,2,\ldots \}.}$ It is often denoted also ${\displaystyle \mathbf {N} .}$
${\displaystyle \mathbb {Z} }$
Denotes the set of integers ${\displaystyle \{\ldots ,-2,-1,0,1,2,\ldots \}.}$ It is often denoted also ${\displaystyle \mathbf {Z} .}$
${\displaystyle \mathbb {Z} _{p}}$
1.  Denotes the set of p-adic integers, where p is a prime number.
2.  Sometimes, ${\displaystyle \mathbb {Z} _{n}}$ denotes the integers modulo n, where n is an integer greater than 1. The notation ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ is also used, and is less ambiguous.
${\displaystyle \mathbb {Q} }$
Denotes the set of rational numbers (fractions of two integers). It is often denoted also ${\displaystyle \mathbf {Q} .}$
${\displaystyle \mathbb {Q} _{p}}$
Denotes the set of p-adic numbers, where p is a prime number.
${\displaystyle \mathbb {R} }$
Denotes the set of real numbers. It is often denoted also ${\displaystyle \mathbf {R} .}$
${\displaystyle \mathbb {C} }$
Denotes the set of complex numbers. It is often denoted also ${\displaystyle \mathbf {C} .}$
${\displaystyle \mathbb {H} }$
Denotes the set of quaternions. It is often denoted also ${\displaystyle \mathbf {H} .}$
${\displaystyle \mathbb {F} _{q}}$
Denotes the finite field with q elements, where q is a prime number or a prime power. It is denoted also GF(q).

## 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 ${\displaystyle \textstyle \sum _{i=0}^{\infty }{\frac {x^{i}}{i!}}=e^{x}.}$
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.
ω
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, and 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

!
Factorial: if n is a positive integer, n! is the product of the n first positive integrs, and is read as "factorial n".
1.  ${\displaystyle A\to B}$ denotes a function with domain A and codomain B. For naming such a function, one writes ${\displaystyle f:A\to B,}$ which is read as "f from A to B".
2.  More generally, ${\displaystyle A\to B}$ denotes a homomorphism or a morphism from A to B.
3.  May denote a logical implication. For the 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.
4.  Over a variable name, means that the variable represents a vector, in a context where ordinary variables represent scalars; for example, ${\displaystyle {\overrightarrow {v}}.}$ Boldface (${\displaystyle \mathbf {v} }$) or a circumflex (${\displaystyle {\hat {v}}}$) are ofen used for the same purpose.
5.  In Euclidean geometry and more generally in Affine geometry, ${\displaystyle {\overrightarrow {PQ}}}$ denotes the vector defined by the two points P and Q, which can be identified with the translation that maps P to Q. The same vector can be denoted also ${\displaystyle Q-P;}$ see Affine space.
Used for defining a function without having to name it. For example, ${\displaystyle x\mapsto x^{2}}$ is the square function.
*
Many different uses in mathematics; see Asterisk § Mathematics.
|
1.  Divisibility: if m and n are two integers, ${\displaystyle m\mid n}$ means that m divides n evenly.
2.  In set-builder notation, it is used as a separator meaning "such that"; see {□ | □}.
3.  Restriction of a function: if f is a function, and S is a subset of its domain, then ${\displaystyle f|_{S}}$ is the function with S as a domain that equals f on S.
4.  Conditional probability: ${\displaystyle P(X\mid E)}$ denotes the probability of X given that the event E occurs. Also denoted ${\displaystyle P(X/E)}$, see "/".
5.  For several uses as brackets (in pairs or with and ) see § other brackets.
Non-divisibility: ${\displaystyle n\nmid m}$ means that n is not a divisor of m.
1.  Denotes parallelism in elementary geometry: if PQ and RS are two lines, ${\displaystyle PQ\parallel RS}$ means that they are parallel.
2.  Parallel, an arithmetical operation used in electrical engineering for modeling parallel resistors: ${\displaystyle x\parallel y={\frac {1}{{\frac {1}{x}}+{\frac {1}{y}}}}.}$
3.  Used in pair as brackets, denotes a norm; see ||□||.
Sometimes used for denoting that two lines are not parallel; for example ${\displaystyle PQ\not \parallel RS.}$
#
1.  Number of elements: ${\displaystyle \#S}$ may denote the cardinality of the set S. An alternative notation is ${\displaystyle |S|;}$ see ||.
2.  Primorial: ${\displaystyle n\#}$ denotes the product of the prime numbers that are not greater than n.
3.  In topology, ${\displaystyle M\#N}$ denotes the connected sum of two manifolds or two knots.
In group theory, ${\displaystyle G\wr H}$ denotes the wreath product of the groups G and H. Its is also denoted as ${\displaystyle G\operatorname {wr} H}$ or ${\displaystyle G\operatorname {Wr} H;}$ see Wreath product § Notation and conventions for several notation variants.
Denotes the d'Alembertian or d'Alembert operator, which is a generalization of the Laplacian to non-Euclidean spaces.
Exclusive or: if E and F are two Boolean variables or predicates, ${\displaystyle E\veebar F}$ denotes the exclusive or. Notations E XOR F and ${\displaystyle E\oplus F}$ are also commonly used; see .
1.  Internal direct sum: if E and F are abelian subgroups of an abelian group V, notation ${\displaystyle V=E\oplus F}$ means that V is the direct sum of E and F; that is, every element of V can be written in a unique way as the sum of an element of E and an element of V. This applies also when E and F are linear subspaces or submodules of the vector space or module V.
2.  Direct sum: if E and F are two abelian groups, vector spaces, or modules, then their direct sum, denoted ${\displaystyle E\oplus F}$ is an abelian group, vector space, or module (respectively) equipped with two monomorphisms ${\displaystyle f:E\to E\oplus F}$ and ${\displaystyle g:F\to E\oplus F}$ such that ${\displaystyle E\oplus F}$ is the internal direct sum of ${\displaystyle f(E)}$ and ${\displaystyle g(F).}$ This definition makes sense because this direct sum is unique up to a unique isomorphism.
3.  Exclusive or: if E and F are two Boolean variables or predicates, ${\displaystyle E\oplus F}$ may denote the exclusive or. Notations E XOR F and ${\displaystyle E\veebar F}$ are also commonly used; see .
1.  Partial derivative: typically used as ${\displaystyle \textstyle {\frac {\partial f(x_{1},\ldots ,x_{n})}{\partial x_{i}}}.}$
2.  Boundary of a topological subspace: if S is a subspace of a topological space, then its boundary, denoted ${\displaystyle \partial S,}$ is the set difference betwen the closure and the interior of S.
[3]
1.  Function composition: if f and g are two functions, then ${\displaystyle g\circ f}$ is the function such that ${\displaystyle (g\circ f)(x)=g(f(x)}$ for every value of x.
2.  Hadamard product of matrices: if A and B are two matrices of the same size, then ${\displaystyle A\circ B}$ is the matrix such that ${\displaystyle (A\circ B)_{i,j}=(A)_{i,j}(B)_{i,j}.}$ Possibly, ${\displaystyle \circ }$ is also used instead of for the Hadamard product of power series.[citation needed]
Hadamard product of power series: if ${\displaystyle \textstyle S=\sum _{i=0}^{\infty }s_{i}x^{i}}$ and ${\displaystyle \textstyle T=\sum _{i=0}^{\infty }t_{i}x^{i}}$, then ${\displaystyle \textstyle S\odot T=\sum _{i=0}^{\infty }s_{i}t_{i}x^{i}.}$ Possibly, ${\displaystyle \odot }$ is also used instead of for the Hadamard product of matrices.[citation needed]

Symbol
in HTML
Symbol
in TeX
Name Explanation Examples
Category
${\displaystyle \top }$
\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
xy means x is perpendicular to y; or more generally x is orthogonal to y. If lm and mn 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
xy means x has no factor greater than 1 in common with y. 34 ⊥ 55
is independent of
AB 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 AB, 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
xy means that x is comparable to y. {e, π} ⊥ {1, 2, e, 3, π} under set containment.
${\displaystyle \otimes }$
\otimes
tensor product of
${\displaystyle V\otimes U}$ means the tensor product of V and U.[4] ${\displaystyle V\otimes _{R}U}$ 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
Nφ H is the semidirect product of N (a normal subgroup) and H (a subgroup), with respect to φ. Also, if G = Nφ H, then G is said to split over N.

(⋊ may also be written the other way round, as ⋉, or as ×.)
${\displaystyle D_{2n}\cong \mathrm {C} _{n}\rtimes \mathrm {C} _{2}}$

## Modifiers (diacritics and superscripts)

These are diacritics or symbols occur

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).
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 ${\displaystyle {\dot {x}}(t)=2t}$.
${\displaystyle {}^{\dagger }}$
{}^\dagger
conjugate transpose;
A means the transpose of the complex conjugate of A.[5]

This may also be written A∗T, AT∗, A, AT or 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′, At or Atr.
If A = (aij) then AT = (aji).

## Letters as symbols

Symbol
in HTML
Symbol
in TeX
Name Explanation Examples
Category

${\displaystyle O}$
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 ${\displaystyle f(x)=O(g(x)){\mbox{ as }}x\to \infty \,}$

${\displaystyle \Gamma }$
\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
${\displaystyle \delta (x)={\begin{cases}\infty ,&x=0\\0,&x\neq 0\end{cases}}}$ δ(x)
Kronecker delta of
${\displaystyle \delta _{ij}={\begin{cases}1,&i=j\\0,&i\neq j\end{cases}}}$ δij
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
AB (or AB) means the set of elements in exactly one of A or B.

(Not to be confused with delta, Δ, 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
Δx means a (non-infinitesimal) change in x.

(If the change becomes infinitesimal, δ 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 ƒ is a twice-differentiable real-valued function, then the Laplacian of ƒ 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
prime-counting function of
${\displaystyle \pi (x)}$ counts the number of prime numbers less than or equal to ${\displaystyle x}$. ${\displaystyle \pi (10)=4}$
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. ${\displaystyle \sigma ={\sqrt {\dfrac {\Sigma (x_{i}-\mu )^{2}}{N}}}}$