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Source: Glossary 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 entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics.
The most basic symbols are the decimal digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), and the letters of the Latin alphabet. The decimal digits are used for representing numbers through the Hindu–Arabic numeral system. Historically, upper-case letters were used for representing points in geometry, and lower-case letters were used for variables and constants. Letters are used for representing many other sorts of mathematical objects. As the number of these sorts has remarkably 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

a
,
A
,
b
,
B

,
…

{\displaystyle \mathbf {a,A,b,B} ,\ldots }

, script typeface

A
,
B

,
…

{\displaystyle {\mathcal {A,B}},\ldots }

(the lower-case script face is rarely used because of the possible confusion with the standard face), German fraktur

a
,
A
,
b
,
B

,
…

{\displaystyle {\mathfrak {a,A,b,B}},\ldots }

, and blackboard bold

N
,
Z
,
Q
,
R
,
C
,
H
,
F

q

{\displaystyle \mathbb {N,Z,Q,R,C,H,F} _{q}}

(the other letters are rarely used in this face, or their use is unconventional).
The use of Latin and Greek letters as symbols for denoting mathematical objects is not described in this article. For such uses, see Variable (mathematics) and List of mathematical constants. However, some symbols that are described here have the same shape as the letter from which they are derived, such as

∏

{\displaystyle \textstyle \prod {}}

and

∑

{\displaystyle \textstyle \sum {}}

.
These letters alone are not sufficient for the needs of mathematicians, and many other symbols are used. Some take their origin in punctuation marks and diacritics traditionally used in typography; others by deforming letter forms, as in the cases of

∈

{\displaystyle \in }

and

∀

{\displaystyle \forall }

. Others, such as + and =, were specially designed for mathematics.

Layout of this article

glossary

Arithmetic operators

Equality, equivalence and similarity

mathematical

Comparison

Set theory

mathematical

Basic logic

Blackboard bold

symbols

Calculus

mathematical

Linear and multilinear algebra

.
2. Denotes a series and, if the series is convergent, the sum of the series. For example,

∑

i
=
0

∞

x

i

i
!

=

e

x

{\displaystyle \textstyle \sum _{i=0}^{\infty }{\frac {x^{i}}{i!}}=e^{x}}

.

∏ (capital-pi notation)
1. Denotes the product of a finite number of terms, which are determined by subscripts and superscripts (which can also be placed below and above), such as in

∏

i
=
1

n

i

2

{\displaystyle \textstyle \prod _{i=1}^{n}i^{2}}

or

∏

0
<
i
<
j
<
n

j
−
i

{\displaystyle \textstyle \prod _{0
.
2. Denotes an infinite product. For example, the Euler product formula for the Riemann zeta function is

ζ
(
z
)
=

∏

n
=
1

∞

1

1
−

p

n

−
z

{\displaystyle \textstyle \zeta (z)=\prod _{n=1}^{\infty }{\frac {1}{1-p_{n}^{-z}}}}

.
3. Also used for the Cartesian product of any number of sets and the direct product of any number of mathematical structures.

⊕

{\displaystyle \oplus }

1. Internal direct sum: if E and F are abelian subgroups of an abelian group V, notation

V
=
E
⊕
F

{\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 F. 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

E
⊕
F

{\displaystyle E\oplus F}

is an abelian group, vector space, or module (respectively) equipped with two monomorphisms

f
:
E
→
E
⊕
F

{\displaystyle f:E\to E\oplus F}

and

g
:
F
→
E
⊕
F

{\displaystyle g:F\to E\oplus F}

such that

E
⊕
F

{\displaystyle E\oplus F}

is the internal direct sum of

f
(
E
)

{\displaystyle f(E)}

and

g
(
F
)

{\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,

E
⊕
F

{\displaystyle E\oplus F}

may denote the exclusive or. Notations E XOR F and

E
⊻
F

{\displaystyle E\veebar F}

are also commonly used; see ⊻.

⊗

{\displaystyle \otimes }

1. Denotes the tensor product of abelian groups, vector spaces, modules, or other mathematical structures, such as in

E
⊗
F
,

{\displaystyle E\otimes F,}

or

E

⊗

K

F
.

{\displaystyle E\otimes _{K}F.}

2. Denotes the tensor product of elements: if

x
∈
E

{\displaystyle x\in E}

and

y
∈
F
,

{\displaystyle y\in F,}

then

x
⊗
y
∈
E
⊗
F
.

{\displaystyle x\otimes y\in E\otimes F.}

□⊤
1. Transpose: if A is a matrix,

A

⊤

{\displaystyle A^{\top }}

denotes the transpose of A, that is, the matrix obtained by exchanging rows and columns of A. Notation

⊤

A

{\displaystyle ^{\top }\!\!A}

is also used. The symbol

⊤

{\displaystyle \top }

is often replaced by the letter T or t.
2. For inline uses of the symbol, see ⊤.

□⊥
1. Orthogonal complement: If W is a linear subspace of an inner product space V, then

W

⊥

{\displaystyle W^{\bot }}

denotes its orthogonal complement, that is, the linear space of the elements of V whose inner products with the elements of W are all zero.
2. Orthogonal subspace in the dual space: If W is a linear subspace (or a submodule) of a vector space (or of a module) V, then

W

⊥

{\displaystyle W^{\bot }}

may denote the orthogonal subspace of W, that is, the set of all linear forms that map W to zero.
3. For inline uses of the symbol, see ⊥.

Advanced group theory

⋉⋊
1. Inner semidirect product: if N and H are subgroups of a group G, such that N is a normal subgroup of G, then

G
=
N
⋊
H

{\displaystyle G=N\rtimes H}

and

G
=
H
⋉
N

{\displaystyle G=H\ltimes N}

mean that G is the semidirect product of N and H, that is, that every element of G can be uniquely decomposed as the product of an element of N and an element of H. (Unlike for the direct product of groups, the element of H may change if the order of the factors is changed.)
2. Outer semidirect product: if N and H are two groups, and

φ

{\displaystyle \varphi }

is a group homomorphism from N to the automorphism group of H, then

N

⋊

φ

H
=
H

⋉

φ

N

{\displaystyle N\rtimes _{\varphi }H=H\ltimes _{\varphi }N}

denotes a group G, unique up to a group isomorphism, which is a semidirect product of N and H, with the commutation of elements of N and H defined by

φ

{\displaystyle \varphi }

.

≀
In group theory,

G
≀
H

{\displaystyle G\wr H}

denotes the wreath product of the groups G and H. It is also denoted as

G
wr
⁡
H

{\displaystyle G\operatorname {wr} H}

or

G
Wr
⁡
H

{\displaystyle G\operatorname {Wr} H}

; see Wreath product § Notation and conventions for several notation variants.

Infinite numbers

∞

{\displaystyle \infty }

(infinity symbol)
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 to form the extended real line.
3.

∞

{\displaystyle \infty }

is the generalized number that is added to the real line to form the projectively extended real line.

c

{\displaystyle {\mathfrak {c}}}

(fraktur 𝔠)

c

{\displaystyle {\mathfrak {c}}}

denotes the cardinality of the continuum, which is the cardinality of the set of real numbers.

ℵ

{\displaystyle \aleph }

(aleph)
With an ordinal i as a subscript, denotes the ith aleph number, that is the ith infinite cardinal. For example,

ℵ

0

{\displaystyle \aleph _{0}}

is the smallest infinite cardinal, that is, the cardinal of the natural numbers.

ℶ

{\displaystyle \beth }

(bet (letter))
With an ordinal i as a subscript, denotes the ith beth number. For example,

ℶ

0

{\displaystyle \beth _{0}}

is the cardinal of the natural numbers, and

ℶ

1

{\displaystyle \beth _{1}}

is the cardinal of the continuum.

ω

{\displaystyle \omega }

(omega)
1. Denotes the first limit ordinal. It is also denoted

ω

0

{\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,

ω
(
n
)

{\displaystyle \omega (n)}

is the number of distinct prime factors of the integer n.

Brackets

Many sorts of brackets are used in mathematics. Their meanings depend not only on their shapes, but also on the nature and the arrangement of what is delimited by them, and sometimes what appears between or before them. For this reason, in the entry titles, the symbol □ is used as a placeholder 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,

f
(
x
)

{\displaystyle f(x)}

,

sin
⁡
(
x
+
y
)

{\displaystyle \sin(x+y)}

. In the case of a multivariate function, the parentheses contain several expressions separated by commas, such as

f
(
x
,
y
)

{\displaystyle f(x,y)}

.
2. May also denote a product, such as in

a
(
b
+
c
)

{\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,

(
π
,
0
)

{\displaystyle (\pi ,0)}

.
2. If a and b are real numbers,

−
∞

{\displaystyle -\infty }

, or

+
∞

{\displaystyle +\infty }

, and a < b, then

(
a
,
b
)

{\displaystyle (a,b)}

denotes the open interval delimited by a and b. See ]□, □[ for an alternative notation.
3. If a and b are integers,

(
a
,
b
)

{\displaystyle (a,b)}

may denote the greatest common divisor of a and b. Notation

gcd
(
a
,
b
)

{\displaystyle \gcd(a,b)}

is often used instead.

(□, □, □)
If x, y, z are vectors in

R

3

{\displaystyle \mathbb {R} ^{3}}

, then

(
x
,
y
,
z
)

{\displaystyle (x,y,z)}

may denote the scalar triple product. See also [□,□,□] in § Square brackets.

(□, ..., □)
Denotes a tuple. If there are n objects separated by commas, it is an n-tuple.

(□, □, ...)(□, ..., □, ...)
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,

(

n
k

)

{\displaystyle {\binom {n}{k}}}

is read as "n choose k", and is defined as the integer

n
(
n
−
1
)
⋯
(
n
−
k
+
1
)

1
⋅
2
⋯
k

=

n
!

k
!

(
n
−
k
)
!

{\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 denotes a polynomial in n, and is thus defined and used for any real or complex value of n.

(

◻
◻

)

{\displaystyle \left({\frac {\Box }{\Box }}\right)}

Legendre symbol: If p is an odd prime number and a is an integer, the value of

(

a
p

)

{\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. The same notation is used for the Jacobi symbol and Kronecker symbol, which are generalizations where p is respectively any odd positive integer, or any integer.

= Square brackets

=

[□]
1. Sometimes used as a synonym of (□) for avoiding nested parentheses.
2. Equivalence class: given an equivalence relation,

[
x
]

{\displaystyle [x]}

often denotes the equivalence class of the element x.
3. Integral part: if x is a real number,

[
x
]

{\displaystyle [x]}

often denotes 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,

[
P
]

{\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,

[
x
=
y
]

{\displaystyle [x=y]}

is the Kronecker delta function, which equals one if

x
=
y

{\displaystyle x=y}

, and zero otherwise.
5. In combinatorics or computer science, sometimes

[
n
]

{\displaystyle [n]}

with

n
∈

N

{\displaystyle n\in \mathbb {N} }

denotes the set

{
1
,
2
,
3
,
…
,
n
}

{\displaystyle \{1,2,3,\ldots ,n\}}

of positive integers up to n, with

[
0
]
=
∅

{\displaystyle [0]=\emptyset }

.

□[□]
Image of a subset: if S is a subset of the domain of the function f, then

f
[
S
]

{\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

a
≤
b

{\displaystyle a\leq b}

, then

[
a
,
b
]

{\displaystyle [a,b]}

denotes the closed interval defined by them.
2. Commutator (group theory): if a and b belong to a group, then

[
a
,
b
]
=

a

−
1

b

−
1

a
b

{\displaystyle [a,b]=a^{-1}b^{-1}ab}

.
3. Commutator (ring theory): if a and b belong to a ring, then

[
a
,
b
]
=
a
b
−
b
a

{\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

[
F
:
E
]

{\displaystyle [F:E]}

denotes the degree of the field extension

F

/

E

{\displaystyle F/E}

. For example,

[

C

:

R

]
=
2

{\displaystyle [\mathbb {C} :\mathbb {R} ]=2}

.
2. Index of a subgroup: if H is a subgroup of a group E, then

[
G
:
H
]

{\displaystyle [G:H]}

denotes the index of H in G. The notation |G:H| is also used

[□, □, □]
If x, y, z are vectors in

R

3

{\displaystyle \mathbb {R} ^{3}}

, then

[
x
,
y
,
z
]

{\displaystyle [x,y,z]}

may denote the scalar triple product. 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 of (□) and [□] for avoiding nested parentheses.
2. Set-builder notation for a singleton set:

{
x
}

{\displaystyle \{x\}}

denotes the set that has x as a single element.

{□, ..., □}
Set-builder notation: denotes the set whose elements are listed between the braces, separated by commas.

{□ : □} {□ | □}
Set-builder notation: if

P
(
x
)

{\displaystyle P(x)}

is a predicate depending on a variable x, then both

{
x
:
P
(
x
)
}

{\displaystyle \{x:P(x)\}}

and

{
x
∣
P
(
x
)
}

{\displaystyle \{x\mid P(x)\}}

denote the set formed by the values of x for which

P
(
x
)

{\displaystyle P(x)}

is true.

Single brace
1. Used for emphasizing that several equations have to be considered as simultaneous equations; for example,

{

2
x
+
y
=
1

3
x
−
y
=
1

{\displaystyle \textstyle {\begin{cases}2x+y=1\\3x-y=1\end{cases}}}

.
2. Piecewise definition; for example,

|

x

|

=

{

x

if

x
≥
0

−
x

if

x
<
0

{\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,

(
a
,
b
,
…
,
z
)

⏟

26

{\displaystyle \textstyle \underbrace {(a,b,\ldots ,z)} _{26}}

,

1
+
2
+
⋯
+
100

⏞

=
5050

{\displaystyle \textstyle \overbrace {1+2+\cdots +100} ^{=5050}}

,

[

A

B

]

}

m
+
n

rows

{\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,

|

x

|

{\displaystyle |x|}

denotes its absolute value.
2. Number of elements: If S is a set,

|

S

|

{\displaystyle |S|}

may denote its cardinality, that is, its number of elements.

#
S

{\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

|

P
Q

|

{\displaystyle |PQ|}

often denotes the length of the line segment that they define, which is the distance from P to Q, and is often denoted

d
(
P
,
Q
)

{\displaystyle d(P,Q)}

.
4. For a similar-looking operator, see |.

|□:□|
Index of a subgroup: if H is a subgroup of a group G, then

|

G
:
H

|

{\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}}}

|

x

1
,
1

⋯

x

1
,
n

⋮

⋱

⋮

x

n
,
1

⋯

x

n
,
n

|

{\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

[

x

1
,
1

⋯

x

1
,
n

⋮

⋱

⋮

x

n
,
1

⋯

x

n
,
n

]

{\displaystyle {\begin{bmatrix}x_{1,1}&\cdots &x_{1,n}\\\vdots &\ddots &\vdots \\x_{n,1}&\cdots &x_{n,n}\end{bmatrix}}}

.

||□||
1. Denotes the norm of an element of a normed vector space.
2. For the similar-looking operator named parallel, see ∥.

⌊□⌋
Floor function: if x is a real number,

⌊
x
⌋

{\displaystyle \lfloor x\rfloor }

is the greatest integer that is not greater than x.

⌈□⌉
Ceiling function: if x is a real number,

⌈
x
⌉

{\displaystyle \lceil x\rceil }

is the lowest integer that is not lesser than x.

⌊□⌉
Nearest integer function: if x is a real number,

⌊
x
⌉

{\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

a
<
b

{\displaystyle a
, then

]
a
,
b
[

{\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 an algebraic structure,

⟨
S
⟩

{\displaystyle \langle S\rangle }

denotes often the object generated by S. If

S
=
{

s

1

,
…
,

s

n

}

{\displaystyle S=\{s_{1},\ldots ,s_{n}\}}

, one writes

⟨

s

1

,
…
,

s

n

⟩

{\displaystyle \langle s_{1},\ldots ,s_{n}\rangle }

(that is, braces are omitted). In particular, this may denote
the linear span in a vector space (also often denoted Span(S)),
the generated subgroup in a group,
the generated ideal in a ring,
the generated submodule in a module.
2. Often used, mainly in physics, for denoting an expected value. In probability theory,

E
(
X
)

{\displaystyle E(X)}

is generally used instead of

⟨
S
⟩

{\displaystyle \langle S\rangle }

.
⟨□, □⟩⟨□ | □⟩
Both

⟨
x
,
y
⟩

{\displaystyle \langle x,y\rangle }

and

⟨
x
∣
y
⟩

{\displaystyle \langle x\mid y\rangle }

are commonly used for denoting the inner product in an inner product space.

⟨
◻

|

and

|

◻
⟩

{\displaystyle \langle \Box |{\text{ and }}|\Box \rangle }

Bra–ket notation or Dirac notation: if x and y are elements of an inner product space,

|

x
⟩

{\displaystyle |x\rangle }

is the vector defined by x, and

⟨
y

|

{\displaystyle \langle y|}

is the covector defined by y; their inner product is

⟨
y
∣
x
⟩

{\displaystyle \langle y\mid x\rangle }

.

Symbols that do not belong to formulas

In this section, the symbols that are listed are used as some sorts of punctuation marks in mathematical reasoning, or as abbreviations of natural language 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 language. Except for the first two, 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 (Latin: quod erat demonstrandum, "as was to be shown") is often used for the same purpose, either in its upper-case form or in lower case.

☡
Bourbaki dangerous bend symbol: Sometimes used in the margin to forewarn readers against serious errors, where they risk falling, or to mark a passage that is tricky on a first reading because of an especially subtle argument.

∴
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,

x
∋
x
>
3

{\displaystyle x\ni x>3}

is normally printed "x such that

x
>
3

{\displaystyle x>3}

".
2. Sometimes used for reversing the operands of

∈

{\displaystyle \in }

; that is,

S
∋
x

{\displaystyle S\ni x}

has the same meaning as

x
∈
S

{\displaystyle x\in S}

. See ∈ in § Set theory.

∝
Abbreviation of "is proportional to".

Miscellaneous

!
1. Factorial: if n is a positive integer, n! is the product of the first n positive integers, and is read as "n factorial".
2. Double factorial: if n is a positive integer, n!! is the product of all positive integers up to n with the same parity as n, and is read as "the double factorial of n".
3. Subfactorial: if n is a positive integer, !n is the number of derangements of a set of n elements, and is read as "the subfactorial of n".

*
Many different uses in mathematics; see Asterisk § Mathematics.

|
1. Divisibility: if m and n are two integers,

m
∣
n

{\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

f

|

S

{\displaystyle f|_{S}}

is the function with S as a domain that equals f on S.
4. Conditional probability:

P
(
X
∣
E
)

{\displaystyle P(X\mid E)}

denotes the probability of X given that the event E occurs. Also denoted

P
(
X

/

E
)

{\displaystyle P(X/E)}

; see "/".
5. For several uses as brackets (in pairs or with ⟨ and ⟩) see § Other brackets.

∤
Non-divisibility:

n
∤
m

{\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,

P
Q
∥
R
S

{\displaystyle PQ\parallel RS}

means that they are parallel.
2. Parallel, an arithmetical operation used in electrical engineering for modeling parallel resistors:

x
∥
y
=

1

1
x

+

1
y

{\displaystyle x\parallel y={\frac {1}{{\frac {1}{x}}+{\frac {1}{y}}}}}

.
3. Used in pairs as brackets, denotes a norm; see ||□||.
4. Concatenation: Typically used in computer science,

x

|
|

y

{\displaystyle x\mathbin {\vert \vert } y}

is said to represent the value resulting from appending the digits of y to the end of x.
5.

D

KL

(
P
∥
Q
)

{\displaystyle {\displaystyle D_{\text{KL}}(P\parallel Q)}}

, denotes a statistical distance or measure of how one probability distribution P is different from a second, reference probability distribution Q.

∦
Sometimes used for denoting that two lines are not parallel; for example,

P
Q
∦
R
S

{\displaystyle PQ\not \parallel RS}

.

⊥

{\displaystyle \perp }

1. Denotes perpendicularity and orthogonality. For example, if A, B, C are three points in a Euclidean space, then

A
B
⊥
A
C

{\displaystyle AB\perp AC}

means that the line segments AB and AC are perpendicular, and form a right angle.
2. For the similar symbol, see

⊥

{\displaystyle \bot }

.

⊙
Hadamard product of power series: if

S
=

∑

i
=
0

∞

s

i

x

i

{\displaystyle \textstyle S=\sum _{i=0}^{\infty }s_{i}x^{i}}

and

T
=

∑

i
=
0

∞

t

i

x

i

{\displaystyle \textstyle T=\sum _{i=0}^{\infty }t_{i}x^{i}}

, then

S
⊙
T
=

∑

i
=
0

∞

s

i

t

i

x

i

{\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.

= Related articles

=
Language of mathematics
Mathematical notation
Notation in probability and statistics
Physical constants

= Related lists

=
List of logic symbols
List of mathematical constants
Table of mathematical symbols by introduction date
Blackboard bold
Greek letters used in mathematics, science, and engineering
Latin letters used in mathematics, science, and engineering
List of common physics notations
List of letters used in mathematics, science, and engineering
List of mathematical abbreviations
List of typographical symbols and punctuation marks
ISO 31-11 (Mathematical signs and symbols for use in physical sciences and technology)
List of APL functions

= Unicode symbols

=
Unicode block
Mathematical Alphanumeric Symbols (Unicode block)
List of Unicode characters
Letterlike Symbols
Mathematical operators and symbols in Unicode
Miscellaneous Mathematical Symbols: A, B, Technical
Arrow (symbol) and Miscellaneous Symbols and Arrows
Number Forms
Geometric Shapes

References

External links

Jeff Miller: Earliest Uses of Various Mathematical Symbols
Numericana: Scientific Symbols and Icons
GIF and PNG Images for Math Symbols
Mathematical Symbols in Unicode
Detexify: LaTeX Handwriting Recognition Tool
Some Unicode charts of mathematical operators and symbols:
Index of Unicode symbols
Range 2100–214F: Unicode Letterlike Symbols
Range 2190–21FF: Unicode Arrows
Range 2200–22FF: Unicode Mathematical Operators
Range 27C0–27EF: Unicode Miscellaneous Mathematical Symbols–A
Range 2980–29FF: Unicode Miscellaneous Mathematical Symbols–B
Range 2A00–2AFF: Unicode Supplementary Mathematical Operators
Some Unicode cross-references:
Short list of commonly used LaTeX symbols and Comprehensive LaTeX Symbol List
MathML Characters - sorts out Unicode, HTML and MathML/TeX names on one page
Unicode values and MathML names
Unicode values and Postscript names from the source code for Ghostscript

Layout of this article

Arithmetic operators

Equality, equivalence and similarity

Comparison

Set theory

Basic logic

Blackboard bold

Calculus

Linear and multilinear algebra

Advanced group theory

Infinite numbers

Brackets

= Parentheses

= Square brackets

= Braces

= Other brackets

Symbols that do not belong to formulas

Miscellaneous

See also

= Related articles

= Related lists

= Unicode symbols

References

External links

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