TABLE OF MATHEMATICAL SYMBOLS

The following table lists many specialized symbols commonly used in mathematics.

Contents

Basic mathematical symbols

See also

External links

Basic mathematical symbols

{| class="wikitable" style="margin:auto; width:100%;"
! rowspan="3" style="font-size:130%;" |Symbol
! style="text-align:left;" |Name
! rowspan="3" style="font-size:130%;" |Explanation
! rowspan="3" style="font-size:130%;" |Examples
|-
! Read as
|-
! style="text-align:right;" |Category
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|=
||equality
| rowspan=3|''x'' = ''y'' means ''x'' and ''y'' represent the same thing or value.
| rowspan=3|1 + 1 = 2
|-
|align=center|is equal to; equals
|-
|align=right|everywhere
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|≠

<>

!=
||inequation
| rowspan=3| ''x'' ≠ ''y'' means that ''x'' and ''y'' do not represent the same thing or value.

(''The symbols'' != ''and'' <> ''are primarily from computer science. They are avoided in mathematical texts.'')
| rowspan=3|1 ≠ 2
|-
|align=center|is not equal to; does not equal
|-
|align=right valign=center|means "not"
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|<

>

≪

≫
||strict inequality
| rowspan=3|''x'' < ''y'' means ''x'' is less than ''y''.

''x'' > ''y'' means ''x'' is greater than ''y''.

''x'' ≪ ''y'' means ''x'' is much less than ''y''.

''x'' ≫ ''y'' means ''x'' is much greater than ''y''.
| rowspan=3|3 < 4
5 > 4

0.003 ≪ 1000000
|-
|align=center|is less than, is greater than, is much less than, is much greater than
|-
|align=right|order theory
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|≤
<=

≥
>=
||inequality
| rowspan=3|''x'' ≤ ''y'' means ''x'' is less than or equal to ''y''.

''x'' ≥ ''y'' means ''x'' is greater than or equal to ''y''.

(''The symbols'' <= ''and'' >= ''are primarily from computer science. They are avoided in mathematical texts.'')
| rowspan=3|3 ≤ 4 and 5 ≤ 5
5 ≥ 4 and 5 ≥ 5
|-
|align=center|is less than or equal to, is greater than or equal to
|-
|align=right|order theory
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|∝
||proportionality
| rowspan=3| ''y'' ∝ ''x'' means that ''y'' = ''kx'' for some constant ''k''.
| rowspan=3|if ''y'' = 2''x'', then ''y'' ∝ ''x''
|-
|align=center|is proportional to; varies as
|-
|align=right valign=center|everywhere
|-
| rowspan=6 bgcolor=#d0f0d0 align=center|+
|addition
| rowspan=3|4 + 6 means the sum of 4 and 6.
| rowspan=3|2 + 7 = 9
|-
|align=center|plus
|-
|align=right|arithmetic
|-
| disjoint union
| rowspan=3|''A''₁ + ''A''₂ means the disjoint union of sets ''A''₁ and ''A''₂.
| rowspan=3|''A''₁ = {1, 2, 3, 4} ∧ ''A''₂ = {2, 4, 5, 7} ⇒
''A''₁ + ''A''₂ = {(1,1), (2,1), (3,1), (4,1), (2,2), (4,2), (5,2), (7,2)}
|-
|align=center|the disjoint union of ... and ...
|-
|align=right|set theory
|-
| rowspan=9 bgcolor=#d0f0d0 align=center|−
|subtraction
| rowspan=3|9 − 4 means the subtraction of 4 from 9.
| rowspan=3|8 − 3 = 5
|-
|align=center|minus
|-
|align=right|arithmetic
|-
|negative sign
| rowspan=3|−3 means the negative of the number 3.
| rowspan=3|−(−5) = 5
|-
|align=center|negative; minus
|-
|align=right|arithmetic
|-
|set-theoretic complement
| rowspan=3|''A'' − ''B'' means the set that contains all the elements of ''A'' that are not in ''B''.

∖ can also be used for set-theoretic complement as described below.
| rowspan=3|{1,2,4} − {1,3,4}  =  {2}
|-
|align=center|minus; without
|-
|align=right|set theory
|-
| rowspan=9 bgcolor=#d0f0d0 align=center|×
|multiplication
| rowspan=3|3 × 4 means the multiplication of 3 by 4.
| rowspan=3|7 × 8 = 56
|-
|align=center|times
|-
|align=right|arithmetic
|-
|Cartesian product
| rowspan=3|''X''×''Y'' means the set of all ordered pairs with the first element of each pair selected from X and the second element selected from Y.
| rowspan=3|{1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)}
|-
|align=center|the Cartesian product of ... and ...; the direct product of ... and ...
|-
|align=right|set theory
|-
|cross product
| rowspan=3|'u' × 'v' means the cross product of vectors 'u' and 'v'
| rowspan=3|(1,2,5) × (3,4,−1) =
(−22, 16, − 2)
|-
|align=center|cross
|-
|align=right|vector algebra
|-
| rowspan=6 bgcolor=#d0f0d0 align=center|·
|multiplication
| rowspan=3|3 · 4 means the multiplication of 3 by 4.
| rowspan=3|7 · 8 = 56
|-
|align=center|times
|-
|align=right|arithmetic
|-
|dot product
| rowspan=3|'u' · 'v' means the dot product of vectors 'u' and 'v'
| rowspan=3|(1,2,5) · (3,4,−1) = 6
|-
|align=center|dot
|-
|align=right|vector algebra
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|÷

⁄
||division
| rowspan=3|6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3.
| rowspan=3|2 ÷ 4 = .5

12 ⁄ 4 = 3
|-
|align=center|divided by
|-
|align=right|arithmetic
|-
| rowspan=6 bgcolor=#d0f0d0 align=center|±
||plus-minus
| rowspan=3|6 ± 3 means both 6 + 3 and 6 - 3.
| rowspan=3|The equation ''x'' = 5 ± √4, has two solutions, ''x'' = 7 and ''x'' = 3.
|-
|align=center|plus or minus
|-
|align=right|arithmetic
|-
||plus-minus
| rowspan=3|10 ± 2 or equivalently 10 ± 20% means the range from 10 − 2 to 10 + 2.
| rowspan=3| If ''a'' = 100 ± 1 mm, then ''a'' ≥ 99 mm and ''a'' ≤ 101 mm.

|-
|align=center|plus or minus
|-
|align=right|measurement
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|
||minus-plus
| rowspan=3|6 ± (3 5) means both 6 + (3 - 5) and 6 - (3 + 5).
| rowspan=3|cos(''x'' ± ''y'') = cos(''x'') cos(''y'') sin(''x'') sin(''y'').
|-
|align=center|minus or plus
|-
|align=right|arithmetic
|-
| rowspan=6 bgcolor=#d0f0d0 align=center|√
||square root
| rowspan=3|√''x'' means the positive number whose square is ''x''.
| rowspan=3|√4 = 2
|-
|align=center|the principal square root of; square root
|-
|align=right|real numbers
|-
||complex square root
| rowspan=3| if ''z'' = ''r'' exp('''i'''φ) is represented in polar coordinates with -'''π''' < φ ≤ '''π''', then √''z'' = √''r'' exp('''i''' φ/2).
| rowspan=3|√(-1) = '''i'''
|-
|align=center|the complex square root of …

square root
|-
|align=right|complex numbers
|-
| rowspan=9 bgcolor=#d0f0d0 align=center||…|
||absolute value or modulus
| rowspan=3| |''x''| means the distance along the real line (or across the complex plane) between ''x'' and zero.
| rowspan=3| |3| = 3

|–5| = |5|

| '''i''' | = 1

| 3 + 4'''i''' | = 5
|-
|align=center|absolute value (modulus) of
|-
|align=right|numbers
|-
||Euclidean distance
| rowspan=3| |'x' – 'y'| means the Euclidean distance between 'x' and 'y'.
| rowspan=3| For 'x' = (1,1), and 'y' = (4,5),
|'x' – 'y'| = √([1–4]² + [1–5]²) = 5
|-
|align=center|Euclidean distance between; Euclidean norm of
|-
|align=right|Geometry
|-
||Determinant
| rowspan=3| |''A''| means the determinant of the matrix 'A'
| rowspan=3|
1&2 \
2&4 \
end{vmatrix} = 0
|-
|align=center|determinant of
|-
|align=right|Matrix theory
|-
| rowspan=6 bgcolor=#d0f0d0 align=center||
||divides
| rowspan=3| A single vertical bar is used to denote divisibility.
''a''|''b'' means ''a'' divides ''b''.
| rowspan=3| Since 15 = 3×5, it is true that 3|15 and 5|15.
|-
|align=center|divides
|-
|align=right|Number Theory
|-
||Conditional probability
| rowspan=3| A single vertical bar is used to describe the probability of an event given another event happening.
''P''(''A''|''B'')'' means ''a'' given ''b''.
| rowspan=3| If A=0.4 and B=0.5, ''P''(''A''|''B'')''=((0.4)(0.5))/(0.5)=0.4
|-
|align=center|Given
|-
|align=right|Probability
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|!
||factorial
| rowspan=3|''n'' '!' is the product 1 × 2× ... × ''n''.
| rowspan=3|4'!' = 1 × 2 × 3 × 4 = 24
|-
|align=center|factorial
|-
|align=right|combinatorics
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|T
||transpose
| rowspan=3| Swap rows for columns
| rowspan=3| $A\_\{ij\}\; =\; (A^T)\_\{ji\}$
|-
|align=center|transpose
|-
|align=right|matrix operations
|-
| rowspan=6 bgcolor=#d0f0d0 align=center|~
||probability distribution
| rowspan=3| ''X ~ D'', means the random variable ''X'' has the probability distribution ''D''.
| rowspan=3|''X ~ N(0,1), the standard normal distribution
|-
|align=center|has distribution
|-
|align=right|statistics
|-
||Row equivalence
| rowspan=3| ''A''~''B'' means that ''B'' can be generated by using a series of elementary row operations on ''A''
| rowspan=3|
1&2 \
2&4 \
end{bmatrix} sim egin{bmatrix}
1&2 \
0&0 \
end{bmatrix}
|-
|align=center|is row equivalent to
|-
|align=right|Matrix theory
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|⇒

→

⊃
||material implication
| rowspan=3|''A'' ⇒ ''B'' means if ''A'' is true then ''B'' is also true; if ''A'' is false then nothing is said about ''B''.

→ may mean the same as ⇒, or it may have the meaning for functions given below.

⊃ may mean the same as ⇒, or it may have the meaning for superset given below.
| rowspan=3|''x'' = 2  ⇒  ''x''² = 4 is true, but ''x''² = 4   ⇒  ''x'' = 2 is in general false (since ''x'' could be −2).
|-
|align=center|implies; if … then
|-
|align=right|propositional logic, Heyting algebra
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|⇔

↔
||material equivalence
| rowspan=3|''A'' ⇔ ''B'' means ''A'' is true if ''B'' is true and ''A'' is false if ''B'' is false.
| rowspan=3|''x'' + 5 = ''y'' +2  ⇔  ''x'' + 3 = ''y''
|-
|align=center|if and only if; iff
|-
|align=right|propositional logic
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|¬

˜
||logical negation
| rowspan=3|The statement ¬''A'' is true if and only if ''A'' is false.

A slash placed through another operator is the same as "¬" placed in front.

(''The symbol'' ~ ''has many other uses, so'' ¬ '' or the slash notation is preferred.'')
| rowspan=3|¬(¬''A'') ⇔ ''A''
''x'' ≠ ''y''  ⇔  ¬(''x'' =  ''y'')
|-
|align=center|not
|-
|align=right|propositional logic
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|∧
||logical conjunction or 'meet' in a lattice
| rowspan=3|The statement ''A'' ∧ ''B'' is true if ''A'' and ''B'' are both true; else it is false.

For functions ''A''(x) and ''B''(x), ''A''(x) ∧ ''B''(x) is used to mean min(A(x), B(x)).
| rowspan=3|''n'' < 4  ∧  ''n'' >2  ⇔  ''n'' = 3 when ''n'' is a natural number.
|-
|align=center|and; min
|-
|align=right|propositional logic, lattice theory
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|∨
||logical disjunction or 'join' in a lattice
| rowspan=3|The statement ''A'' ∨ ''B'' is true if ''A'' or ''B'' (or both) are true; if both are false, the statement is false.

For functions ''A''(x) and ''B''(x), ''A''(x) ∨ ''B''(x) is used to mean max(A(x), B(x)).
| rowspan=3|''n'' ≥ 4  ∨  ''n'' ≤ 2  ⇔ ''n'' ≠ 3 when ''n'' is a natural number.
|-
|align=center|or; max
|-
|align=right|propositional logic, lattice theory
|-
| rowspan=6 bgcolor=#d0f0d0 align=center|
⊕

||exclusive or
| rowspan=3| The statement ''A'' ⊕ ''B'' is true when either A or B, but not both, are true. ''A'' ''B'' means the same.
| rowspan=3| (¬''A'') ⊕ ''A'' is always true, ''A'' ⊕ ''A'' is always false.
|-
|align=center|xor
|-
|align=right|propositional logic, Boolean algebra
|-
||direct sum
|rowspan=3|The direct sum is a special way of combining several modules into one general module (the symbol ⊕ is used, is only for logic).


|rowspan=3|Most commonly, for vector spaces ''U'', ''V'', and ''W'', the following consequence is used:
''U'' = ''V'' ⊕ ''W'' ⇔ (''U'' = ''V'' + ''W'') ∧ (''V'' ∩ ''W'' = )
|-
|align=center|direct sum of
|-
|align=right|Abstract algebra
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|∀
||universal quantification
| rowspan=3|∀ ''x'': ''P''(''x'') means ''P''(''x'') is true for all ''x''.
| rowspan=3|∀ ''n'' ∈ : ''n''² ≥ ''n''.
|-
|align=center|for all; for any; for each
|-
|align=right|predicate logic
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|∃
||existential quantification
| rowspan=3|∃ ''x'': ''P''(''x'') means there is at least one ''x'' such that ''P''(''x'') is true.
| rowspan=3|∃ ''n'' ∈ : ''n'' is even.
|-
|align=center|there exists
|-
|align=right|predicate logic
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|∃!
||uniqueness quantification
| rowspan=3|∃! ''x'': ''P''(''x'') means there is exactly one ''x'' such that ''P''(''x'') is true.
| rowspan=3|∃! ''n'' ∈ : ''n'' + 5 = 2''n''.
|-
|align=center|there exists exactly one
|-
|align=right|predicate logic
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|:=

≡

:⇔
||definition
| rowspan=3|''x'' := ''y'' or ''x'' ≡ ''y'' means ''x'' is defined to be another name for ''y''

(''Some writers use'' ≡ ''to mean congruence'').

''P'' :⇔ ''Q'' means ''P'' is defined to be logically equivalent to ''Q''.
| rowspan=3|cosh ''x'' := (1/2)(exp ''x'' + exp (−''x''))

''A'' 'xor' ''B'' :⇔ (''A'' ∨ ''B'') ∧ ¬(''A'' ∧ ''B'')
|-
|align=center|is defined as
|-
|align=right|everywhere
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|
||congruence
| rowspan=3|△ABC △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF.
| rowspan=3|
|-
|align=center|is congruent to
|-
|align=right|geometry
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|≡
||congruence relation
| rowspan=3|a ≡ b (mod n) means a − b is divisible by n
| rowspan=3|5 ≡ 11 (mod 3)
|-
|align=center|... is congruent to ... modulo ...
|-
|align=right|modular arithmetic
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|{ , }
||set brackets
| rowspan=3|{''a'',''b'',''c''} means the set consisting of ''a'', ''b'', and ''c''.
| rowspan=3| = { 1, 2, 3, …}
|-
|align=center|the set of …
|-
|align=right|set theory
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|{ : }

{ | }
||set builder notation
| rowspan=3|{''x'' : ''P''(''x'')} means the set of all ''x'' for which ''P''(''x'') is true. {''x'' | ''P''(''x'')} is the same as {''x'' : ''P''(''x'')}.
| rowspan=3|{''n'' ∈  : ''n''² < 20} = { 1, 2, 3, 4}
|-
|align=center|the set of … such that
|-
|align=right|set theory
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|

{ } ||empty set
| rowspan=3| means the set with no elements. { } means the same.
| rowspan=3|{''n'' ∈  : 1 < ''n''² < 4} =
|-
|align=center| the empty set
|-
|align=right|set theory
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|∈


||set membership
| rowspan=3|''a'' ∈ ''S'' means ''a'' is an element of the set ''S''; ''a''  ''S'' means ''a'' is not an element of ''S''.
| rowspan=3|(1/2)⁻¹ ∈ 

2⁻¹  
|-
|align=center|is an element of; is not an element of
|-
|align=right|everywhere, set theory
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|⊆

⊂
||subset
| rowspan=3|(subset) ''A'' ⊆ ''B'' means every element of ''A'' is also element of ''B''.

(proper subset) ''A'' ⊂ ''B'' means ''A'' ⊆ ''B'' but ''A'' ≠ ''B''.

(''Some writers use the symbol ''⊂'' as if it were the same as ''⊆.)
| rowspan=3|(''A'' ∩ ''B'') ⊆ ''A''

 ⊂ 

 ⊂ 
|-
|align=center|is a subset of
|-
|align=right|set theory
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|⊇

⊃
||superset
| rowspan=3|''A'' ⊇ ''B'' means every element of ''B'' is also element of ''A''.

''A'' ⊃ ''B'' means ''A'' ⊇ ''B'' but ''A'' ≠ ''B''.

(''Some writers use the symbol ''⊃'' as if it were the same as ''⊇''.'')
| rowspan=3|(''A'' ∪ ''B'') ⊇ ''B''

 ⊃ 
|-
|align=center|is a superset of
|-
|align=right|set theory
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|∪
||set-theoretic union
| rowspan=3|(exclusive) ''A'' ∪ ''B'' means the set that contains all the elements from ''A'', or all the elements from ''B'', but not both.
"''A'' or ''B'', but not both."

(inclusive) ''A'' ∪ ''B'' means the set that contains all the elements from ''A'', or all the elements from ''B'', or all the elements from both ''A'' and ''B''.
"''A'' or ''B'' or both".
| rowspan=3|''A'' ⊆ ''B''  ⇔  (''A'' ∪ ''B'') = ''B'' (inclusive)
|-
|align=center|the union of … and …

union
|-
|align=right|set theory
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|∩
||set-theoretic intersection
| rowspan=3|''A'' ∩ ''B'' means the set that contains all those elements that ''A'' and ''B'' have in common.
| rowspan=3|{''x'' ∈  : ''x''² = 1} ∩  = {1}
|-
|align=center|intersected with; intersect
|-
|align=right|set theory
|-
| rowspan=3 bgcolor=#d0f0d0 align=center| $Delta$
||symmetric difference
| rowspan=3|$ADelta\; B$ means the set of elements in exactly one of ''A'' or ''B''.
| rowspan=3|{1,5,6,8} $Delta$ {2,5,8} = {1,2,6}
|-
|align=center|symmetric difference
|-
|align=right|set theory
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|
||set-theoretic complement
| rowspan=3|''A''  ''B'' means the set that contains all those elements of ''A'' that are not in ''B''.

− can also be used for set-theoretic complement as described above.
| rowspan=3|{1,2,3,4}  {3,4,5,6} = {1,2}
|-
|align=center|minus; without
|-
|align=right|set theory
|-
| rowspan=6 bgcolor=#d0f0d0 align=center|( )
||function application
| rowspan=3|''f''(''x'') means the value of the function ''f'' at the element ''x''.
| rowspan=3|If ''f''(''x'') := ''x''², then ''f''(3) = 3² = 9.
|-
|align=center|of
|-
|align=right|set theory
|-
|precedence grouping
| rowspan=3|Perform the operations inside the parentheses first.
| rowspan=3|(8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.
|-
|align=center|parentheses
|-
|align=right|everywhere
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|''f'':''X''→''Y''
||function arrow
| rowspan=3|''f'': ''X'' → ''Y'' means the function ''f'' maps the set ''X'' into the set ''Y''.
| rowspan=3|Let ''f'':  →  be defined by ''f''(''x'') := ''x''².
|-
|align=center|from … to
|-
|align=right|set theory,type theory
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|o
||function composition
| rowspan=3|''f''o''g'' is the function, such that (''f''o''g'')(''x'') = ''f''(''g''(''x'')).
| rowspan=3|if ''f''(''x'') := 2''x'', and ''g''(''x'') := ''x'' + 3, then (''f''o''g'')(''x'') = 2(''x'' + 3).
|-
|align=center|composed with
|-
|align=right|set theory
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|

'N'
||natural numbers
| rowspan=3|'N' means { 1, 2, 3, ...}, but see the article on natural numbers for a different convention.
| rowspan=3| = {|''a''| : ''a'' ∈ , ''a'' ≠ 0}
|-
|align=center|N
|-
|align=right|numbers
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|

'Z' ||integers
| rowspan=3| means {..., −3, −2, −1, 0, 1, 2, 3, ...} and ⁺ means {1, 2, 3, ...} = .
| rowspan=3| = {''p'', -''p'' : ''p'' ∈  ∪ {0}
|-
|align=center|Z
|-
|align=right|numbers
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|

'Q' ||rational numbers
| rowspan=3| means {''p''/''q'' : ''p'' ∈ , ''q'' ∈ .
| rowspan=3|3.14000... ∈

π 
|-
|align=center|Q
|-
|align=right|numbers
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|

'R' ||real numbers
| rowspan=3|ℝ means the set of real numbers.
| rowspan=3|π ∈

√(−1) 
|-
|align=center|R
|-
|align=right|numbers
|-
| rowspan=6 bgcolor=#d0f0d0 align=center|

'C' ||complex numbers
| rowspan=3| means {''a'' + ''b'' '''i''' : ''a'',''b'' ∈ .
| rowspan=3|'''i''' = √(−1) ∈
|-
|align=center|''C''
|-
|align=right|numbers
|-
||arbitrary constant
| rowspan=3| ''C'' can be any number, most likely unknown; usually occurs when calculating antiderivatives.
| rowspan=3|if ''f(x)'' = 6''x''² + 4''x'', then ''F(x)'' = 2''x''³ + 2''x''² + ''C'', where ''F'(x)'' = ''f(x)''
|-
|align=center|C
|-
|align=right|integral calculus
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|

'K'
||real or complex numbers
| rowspan=3|'K' means the statement holds substituting 'K' for 'R' and also for 'C'.
| rowspan=3|
:

because

:

and

:.
|-
|align=center|K
|-
|align=right|linear algebra
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|∞
||infinity
| rowspan=3|∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits.
| rowspan=3|
|-
|align=center|infinity
|-
|align=right|numbers
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|||…||
||norm
| rowspan=3| || ''x'' || is the norm of the element ''x'' of a normed vector space.
| rowspan=3| || ''x''  + ''y'' || ≤  || ''x'' ||  +  || ''y'' ||
|-
|align=center|norm of

length of
|-
|align=right| linear algebra
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|∑
||summation
| rowspan=3|
$sum\_\{k=1\}^\{n\}\{a\_k\}$ means ''a''₁ + ''a''₂ + … + ''a''_''n''.
| rowspan=3|
$sum\_\{k=1\}^\{4\}\{k^2\}$ = 1² + 2² + 3² + 4² 

::= 1 + 4 + 9 + 16 = 30
|-
|align=center|sum over … from … to … of
|-
|align=right|arithmetic
|-
| rowspan=6 bgcolor=#d0f0d0 align=center|∏
||product
| rowspan=3|
$prod\_\{k=1\}^na\_k$ means ''a''₁''a''₂···''a''_''n''.
| rowspan=3|
$prod\_\{k=1\}^4(k+2)$ = (1+2)(2+2)(3+2)(4+2)
::= 3 × 4 × 5 × 6 = 360
|-
|align=center|product over … from … to … of
|-
|align=right|arithmetic
|-
||Cartesian product
| rowspan=3|
$prod\_\{i=0\}^\{n\}\{Y\_i\}$ means the set of all (n+1)-tuples
::(''y''₀, …, ''y''_''n'').
| rowspan=3|
$prod\_\{n=1\}^\{3\}\{mathbb\{R\}\}\; =\; mathbb\{R\}\; imesmathbb\{R\}\; imesmathbb\{R\}\; =\; mathbb\{R\}^3$
|-
|align=center|the Cartesian product of; the direct product of
|-
|align=right|set theory
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|
||coproduct
| rowspan=3|
| rowspan=3|
|-
|align=center|coproduct over … from … to … of
|-
|align=right|category theory
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|′

^•
||derivative
| rowspan=3|''f'' ′(''x'') is the derivative of the function ''f'' at the point ''x'', i.e., the slope of the tangent to ''f'' at ''x''.

The dot notation indicates a time derivative. That is .
| rowspan=3|If ''f''(''x'') := ''x''², then ''f'' ′(''x'') = 2''x''
|-
|align=center|… prime

derivative of
|-
|align=right|calculus
|-
| rowspan=6 bgcolor=#d0f0d0 align=center|∫
||indefinite integral or antiderivative
| rowspan=3|∫ ''f''(''x'') d''x'' means a function whose derivative is ''f''.
| rowspan=3| ∫''x''² d''x'' = ''x''³/3 + C
|-
|align=center|indefinite integral of

the antiderivative of
|-
|align=right|calculus
|-
||definite integral
| rowspan=3|∫_''a''^''b'' ''f''(''x'') d''x'' means the signed area between the ''x''-axis and the graph of the function ''f'' between ''x'' = ''a'' and ''x'' = ''b''.
| rowspan=3|∫₀^''b'' x²  d''x'' = ''b''³/3;
|-
|align=center|integral from … to … of … with respect to
|-
|align=right|calculus
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|∮
||contour integral or closed line integral
| rowspan=3|Similar to the integral, but used to denote a single integration over a closed curve or loop. It is sometimes used in physics texts involving equations regarding Gauss's Law, and while these formulas involve a closed surface integral, the representations describe only the first integration of the volume over the enclosing surface. Instances where the latter requires simultaneous double integration, the symbol would be more appropriate. A third related symbol is the closed volume integral, denoted by the symbol .
The contour integral can also frequently be found with a subscript capital letter C, ∮_C, denoting that a closed loop integral is, in fact, around a contour C, or sometimes dually appropriately, a circle C. In representations of Gauss's Law, a subscript capital S, ∮_S, is used to denote that the integration is over a closed surface.
| rowspan=3|
|-
|align=center|contour integral of
|-
|align=right|calculus
|-
| rowspan=9 bgcolor=#d0f0d0 align=center|∇
| gradient
| rowspan=3|∇''f'' (x₁, …, x_''n'') is the vector of partial derivatives (''∂f'' / ''∂x''₁, …, ''∂f'' / ''∂x''_''n'').
| rowspan=3|If ''f'' (''x'',''y'',''z'') := 3''xy'' + ''z''², then ∇''f'' = (3''y'', 3''x'', 2''z'')
|-
|align=center|del, nabla, gradient of
|-
|align=right|vector calculus
|-
| divergence
| rowspan=3|$$
abla cdot ec v = {partial v_x over partial x} + {partial v_y over partial y} + {partial v_z over partial z}
| rowspan=3|If , then $$
abla cdot ec v = 3y + 2yz .
|-
|align=center|del dot, divergence of
|-
|align=right|vector calculus
|-
| curl
|rowspan=3|$$
abla imes ec v = left( {partial v_z over partial y} - {partial v_y over partial z}
ight) mathbf{i} + left( {partial v_x over partial z} - {partial v_z over partial x}
ight) mathbf{j} + left( {partial v_y over partial x} - {partial v_x over partial y}
ight) mathbf{k}
|rowspan=3|If , then $$
abla imesec v = -y^2mathbf{i} - 3xmathbf{k} .
|-
|align=center|curl of
|-
|align=right|vector calculus
|-
| rowspan=6 bgcolor=#d0f0d0 align=center|∂
||partial differential
| rowspan=3| With ''f'' (x₁, …, x_''n''), ∂f/∂x_i is the derivative of ''f'' with respect to x_i, with all other variables kept constant.
| rowspan=3| If ''f''(x,y) := x²y, then ∂''f''/∂x = 2xy
|-
|align=center|partial, d
|-
|align=right|calculus
|-
|boundary
| rowspan=3| ∂''M'' means the boundary of ''M''
| rowspan=3| ∂{x : ||x|| ≤ 2} = {x : ||x|| = 2}
|-
|align=center|boundary of
|-
|align=right|topology
|-
| rowspan=6 bgcolor=#d0f0d0 align=center|⊥
||perpendicular
| rowspan=3|''x'' ⊥ ''y'' means ''x'' is perpendicular to ''y''; or more generally ''x'' is orthogonal to ''y''.
| rowspan=3|If ''l'' ⊥ ''m'' and ''m'' ⊥ ''n'' then ''l'' || ''n''.
|-
|align=center|is perpendicular to
|-
|align=right|geometry
|-
||bottom element
| rowspan=3|''x'' = ⊥ means ''x'' is the smallest element.
| rowspan=3|∀''x'' : ''x'' ∧ ⊥ = ⊥
|-
|align=center|the bottom element
|-
|align=right|lattice theory
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|||
||parallel
| rowspan=3|''x'' || ''y'' means ''x'' is parallel to ''y''.
| rowspan=3|If ''l'' || ''m'' and ''m'' ⊥ ''n'' then ''l'' ⊥ ''n''.
|-
|align=center|is parallel to
|-
|align=right|geometry
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|
||entailment
| rowspan=3| ''A''  ''B'' means the sentence ''A'' entails the sentence ''B'', that is in every model in which ''A'' is true, ''B'' is also true.
| rowspan=3| ''A''  ''A'' ∨ ¬''A''
|-
|align=center|entails
|-
|align=right| model theory
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|
||inference
| rowspan=3|''x''  ''y'' means ''y'' is derived from ''x''.
| rowspan=3| ''A'' → ''B''  ¬''B'' → ¬''A''
|-
|align=center|infers or is derived from
|-
|align=right|propositional logic, predicate logic
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|
||normal subgroup
| rowspan=3| ''N''  ''G'' means that ''N'' is a normal subgroup of group ''G''.
| rowspan=3| ''Z''(''G'')  ''G''
|-
|align=center|is a normal subgroup of
|-
|align=right|group theory
|-
| rowspan=6 bgcolor=#d0f0d0 align=center|/
||quotient group
| rowspan=3| ''G'' / ''H'' means the quotient of group ''G'' modulo its subgroup ''H''.
| rowspan=3| {0, ''a'', 2''a'', ''b'', ''b''+''a'', ''b''+2''a''} / {0, ''b''} = {{0, ''b''}, {''a'', ''b''+''a''}, {2''a'', ''b''+2''a''}}
|-
|align=center| mod
|-
|align=right| group theory
|-
|quotient set
| rowspan=3| ''A''/~ means the set of all ~ equivalence classes in ''A''.
| rowspan=3| If we define ~ by x ~ y ⇔ x − y ∈ , then
/~ = {{''x'' + ''n'' : ''n'' ∈  : x ∈ (0,1]}
|-
|align=center| mod
|-
|align=right| set theory
|-
| rowspan=6 bgcolor=#d0f0d0 align=center|≈
|approximately equal
| rowspan=3|''x'' ≈ ''y'' means ''x'' is approximately equal to ''y''.
| rowspan=3|π ≈ 3.14159
|-
|align=center|is approximately equal to
|-
|align=right|everywhere
|-
||isomorphism
| rowspan=3| ''G'' ≈ ''H'' means that group ''G'' is isomorphic to group ''H''.
| rowspan=3| ''Q'' / {1, −1} ≈ ''V'',
where ''Q'' is the quaternion group and ''V'' is the Klein four-group.
|-
|align=center | is isomorphic to
|-
|align=right| group theory
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|~
||same order of magnitude
| rowspan=3| ''m'' ~ ''n'' means the quantities ''m'' and ''n'' have the same order of magnitude, or general size.

(''Note that'' ~ ''is used for an approximation that is poor, otherwise use '' ≈ .)
| rowspan=3|2 ~ 5

8 × 9 ~ 100

but π² ≈ 10
|-
|align=right|roughly similar

poorly approximates
|-
|align=right|Approximation theory
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|〈,〉

( | )

< , >

·

:
||inner product
| rowspan=3|〈''x'',''y''〉 means the inner product of ''x'' and ''y'' as defined in an inner product space.

For spatial vectors, the dot product notation, ''x''·''y'' is common.

For matricies, the colon notation may be used.
| rowspan=3|The standard inner product between two vectors ''x'' = (2, 3) and ''y'' = (−1, 5) is:
〈x, y〉 = 2 × −1 + 3 × 5 = 13


$A:B\; =\; sum\_\{i,j\}\; A\_\{ij\}B\_\{ij\}$
|-
|align=center|inner product of
|-
|align=right|linear algebra
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|
||tensor product
| rowspan=3| ''V''  ''U'' means the tensor product of ''V'' and ''U''.
| rowspan=3| {1, 2, 3, 4}  {1, 1, 2} =
{{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}}
|-
|align=center| tensor product of
|-
|align=right| linear algebra
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|
★
||convolution
| rowspan=3| ''f'' 
★  ''g'' means the convolution of ''f'' and ''g''.
| rowspan=3| $(f$
★ g )(t) = int f( au) g(t - au), d au
|-
|align=center| convolution, convoluted with
|-
|align=right| functional analysis
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|

||mean
| rowspan=3| (often read as "x bar") is the mean (average value of $x\_i$).
| rowspan=3|.
|-
|align=center|overbar, … bar
|-
|align=right|statistics
|-
|rowspan=3 bgcolor=#d0f0d0 align=center|$overline\{z\}$
||complex conjugate
| rowspan=3|$overline\{z\}$ is the complex conjugate of ''z''.
| rowspan=3|$overline\{3+4i\}\; =\; 3-4i$
|-
|align=center|conjugate
|-
|align=right|complex numbers
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|$riangleq$
||delta equal to
| rowspan=3|$riangleq$ means equal by definition. When $riangleq$ is used, equality is not true generally, but rather equality is true under certain assumptions that are taken in context. Some writers prefer ≡.
| rowspan=3|$p(x\_1,x\_2,...,x\_n)\; riangleq\; prod\_\{i=1\}^n\; p(x\_i\; |\; x\_\{pi\_i\})$.
|-
|align=center|equal by definition
|-
|align=right|everywhere
|}

See also

★ Mathematical alphanumeric symbols

★ Table of logic symbols

★ Mathematical notation

★ ISO 31-11

★ Roman letters used in mathematics

★ Greek letters used in mathematics

★ Notation in probability

★ Physical constants

External links

★ Jeff Miller: ''Earliest Uses of Various Mathematical Symbols''

★ TCAEP - Institute of Physics

★ GIF and PNG Images for Math Symbols

★ Mathematical Symbols in Unicode