List of mathematical symbols by subject

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This list of mathematical symbols by subject shows a selection of the most common symbols that are used in modern mathematical notation within formulas, grouped by mathematical topic. As it is virtually impossible to list all the symbols ever used in mathematics, only those symbols which occur often in mathematics or mathematics education are included. Many of the characters are standardized, for example in DIN 1302 General mathematical symbols or DIN EN ISO 80000-2 Quantities and units – Part 2: Mathematical signs for science and technology.

The following list is largely limited to non-alphanumeric characters. It is divided by areas of mathematics and grouped within sub-regions. Some symbols have a different meaning depending on the context and appear accordingly several times in the list. Further information on the symbols and their meaning can be found in the respective linked articles.

Guide[edit]

The following information is provided for each mathematical symbol:

Symbol
The symbol as it is represented by LaTeX. If there are several typographic variants, only one of the variants is shown.
Usage
An exemplary use of the symbol in a formula. Letters here stand as a placeholder for numbers, variables or complex expressions. Different possible applications are listed separately.
Interpretation
A short textual description of the meaning of the formula in the previous column.
Article
The Wikipedia article that discusses the meaning (semantics) of the symbol.
LaTeX
The LaTeX command that creates the icon. Characters from the ASCII character set can be used directly, with a few exceptions (pound sign #, backslash \, braces {}, and percent sign %). High-and low-position is indicated via the characters ^ and _ and is not explicitly specified.
HTML
The icon in HTML, if it is defined as a named mark. Non-named characters can be indicated in the form can &#xnnnn by specifying the Unicode code point of the next column. High-and low-position can be indicated via <sup></sup> and <sub></sub>.
Unicode
The code point of the corresponding Unicode character. Some characters are combining and require the entry of additional characters. For brackets, the code points of the opening and the closing forms are specified.

Set theory[edit]

Definition symbols[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
is defined by Definition : U+003A
is defined as equal to
is defined as equivalent to

Set construction[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Empty set Empty set \varnothing,
\emptyset
&empty; U+2205
Set consisting of the elements and so on Set (mathematics) \{ \} U+007B/D
Set of elements , that satisfy the condition \mid U+007C
\colon U+003A

Set operations[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Union of the sets and Union (set theory) \cup &cup; U+222A
Intersection of the sets and Intersection (set theory) \cap &cap; U+2229
Difference of sets and Difference (set theory) \setminus U+2216
Symmetric difference of sets and Symmetric difference \triangle &Delta; U+2206
Cartesian product of sets and Cartesian product \times &times; U+2A2F
Disjoint union of sets and Disjoint union \dot\cup U+228D
Disjoint intersection of sets and \sqcup U+2294
Complement of the set Complement (set theory) \mathrm{C} U+2201
\bar U+0305
Power set of the set Power set \mathcal{P} U+1D4AB
\mathfrak{P} U+1D513
\wp U+2118

Set relations[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
is a proper subset of Subset \subset &sub; U+2282
\subsetneq U+228A
is a subset of \subseteq &sube; U+2286
is a proper superset of Superset \supset &sup; U+2283
\supsetneq U+228B
is a superset of \supseteq &supe; U+2287
Element is in the set Element (mathematics) \in &isin; U+2208
\ni, \owns &ni; U+220B
Element is not in the set \notin &notin; U+2209
\not\ni U+220C

Note: The symbols and are used inconsistently and often do not exclude the equality of the two quantities.

Number sets[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Natural numbers Natural number \mathbb{N} U+2115
Integers Integer \mathbb{Z} U+2124
Rational numbers Rational number \mathbb{Q} U+211A
Algebraic numbers Algebraic number \mathbb{A} U+1D538
Real numbers Real number \mathbb{R} U+211D
Complex numbers Complex number \mathbb{C} U+2102
Quaternions Quaternion \mathbb{H} U+210D

Cardinality[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Cardinality of the set Cardinality \vert U+007C
\# U+0023
Cardinality of the continuum Cardinality of the continuum \mathfrak{c} U+1D520
, , ... Infinite cardinals Aleph number \aleph U+2135
, , ... Beth numbers Beth number \beth U+2136

Arithmetic[edit]

Arithmetic operators[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
added to Addition + U+002B
subtracted from Subtraction - U+2212
multiplied by Multiplication \cdot &middot; U+22C5
\times &times; U+2A2F
divided by Division (mathematics) : U+003A
/ &frasl; U+2215
\div &divide; U+00F7
\frac U+2044
Negative of the number or the additive inverse of Unary minus - &minus; U+2212
Plus or minus Plus or minus sign \pm &plusmn; U+00B1
Minus or plus \mp U+2213
Term is evaluated first Bracket ( ) U+0028/9
[ ] U+005B/D

Equality signs[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
equals Equality (mathematics) = U+003D
does not equal Inequality (mathematics) \neq &ne; U+2260
is identical to Identity (mathematics) \equiv &equiv; U+2261
is approximately equal to Approximation \approx &asymp; U+2248
is proportional to Proportionality (mathematics) \sim &sim; U+223C
\propto &prop; U+221D
corresponds to Correspondence (mathematics) \widehat{=} U+2259

Comparison[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
is less than Comparison (mathematics) < &lt; U+003C
is greater than > &gt; U+003E
is less than or equal to \le, \leq &le; U+2264
\leqq U+2266
is greater than or equal to \ge, \geq &ge; U+2265
\geqq U+2267
is much smaller than \ll U+226A
is much bigger than \gg U+226B

Divisibility[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
divides Divisibility \mid U+2223
does not divide \nmid U+2224
and are relatively prime Relatively prime \perp &perp; U+22A5
Greatest common divisor of and Greatest common divisor \sqcap U+2293
\wedge U+2227
Least common multiple of and Least common multiple \sqcup U+2294
\vee U+2228
and are congruent modulo Modular arithmetic \equiv &equiv; U+2261

Intervals[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Closed interval between and Interval (mathematics) ( )
[ ]
U+0028/9
U+005B/D
Open interval between and
Right-open interval between and
Left-open interval between and

Elementary functions[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Absolute value of Absolute value \vert U+007C
Biggest whole number less than or equal to Floor and ceiling functions [ ] U+005B/D
\lfloor \rfloor &lfloor; &rfloor; U+230A/B
Smallest whole number greater than or equal to \lceil \rceil &lceil; &rceil; U+2308/9
Square root of Square root \sqrt &radic; U+221A
-th root of nth root
percent Percent \% U+0025

Note: the power function is not represented by its own icon, but by the positioning of the exponent as a superscript.

Complex numbers[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Real part of complex number Complex number \Re U+211C
Imaginary part of complex number \Im U+2111
Complex conjugate of Complex conjugate \bar U+0305
\ast &lowast; U+002A
Absolute value of complex number Absolute value \vert U+007C
Remark: real and complex parts of a complex number are often also denoted by and .

Mathematical constants[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Pi, or Archimedes' constant Pi \pi &pi; U+03C0
Euler's constant e (mathematics) \rm{e} U+0065
Golden ratio Golden ratio \varphi &phi; U+03C6
Imaginary unit (square root of −1) Imaginary unit \rm{i} U+0069

See also: mathematical constant for symbols of additional mathematical constants.

Calculus[edit]

Sequences and series[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Sum from to or over all elements in set Summation \sum &sum; U+2211
Product from to or over all elements in set Product (mathematics) \prod &prod; U+220F
Coproduct from to or over all elements in set Coproduct \coprod U+2210
Sequence of elements Sequence ( ) U+0028/9
Sequence tends to limit Limit of a sequence \to &rarr; U+2192
tends to infinity Infinity \infty &infin; U+221E

Functions[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Function maps from set to set Function (mathematics) \to &rarr; U+2192
Function maps element to element \mapsto U+21A6
Image of element under function Image (mathematics) ( ) U+0028/9
Image of set under function
[ ] U+005B/D
Restriction of function to set Restriction (mathematics) \vert U+007C
Placeholder for a variable as argument of function Free variable \cdot U+22C5
Inverse function of Inverse function -1 U+207B
Composition of functions and Function composition \circ U+2218
Convolution of functions and Convolution \ast &lowast; U+2217
Fourier transform of function Fourier transform \hat U+0302

Limits[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Limit of function as approaches from below Limit of a function \uparrow &uarr; U+2191
\nearrow U+2197
Limit of function as approaches \to &rarr; U+2192
Limit of function as approaches from above \searrow U+2198
\downarrow &darr; U+2193

Asymptotic behaviour[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Function is asymptotically equal to function Asymptotic analysis \sim &sim; U+223C
Function grows slower than Big O notation o U+006F
Function grows not substantially faster than \mathcal{O} U+1D4AA
Function grows as fast as \Theta &Theta; U+0398
Function grows not substantially slower than \Omega &Omega; U+03A9
Function grows faster than \omega &omega; U+03C9

Differential calculus[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
First or second derivative of function Differentiation (mathematics) \prime &prime; U+2032
First or second derivative of function with respect to time (in physics) \dot, \ddot U+0307
-th derivative of function ( ) U+0028/9
Derivative of function with respect to variable d U+0064
Total differential of function Total differential
Partial derivative of function with respect to variable Partial derivative \partial &part; U+2202

Integral calculus[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
, Definite integral between and or over set Integral \int &int; U+222B
Curve integral along curve Curve integral \oint U+222E
Surface integral over surface Surface integral \iint U+222C
Volume integral over volume Volume integral \iiint U+222D

Vector calculus[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Gradient of function Gradient \nabla &nabla; U+2207
Divergence of vector field Divergence
Curl of vector field Curl (mathematics)
Laplace operator of function Laplace operator \Delta &Delta; U+2206
D'Alembert operator of function D'Alembert operator \square U+25A1

Topology[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Boundary of set Boundary (topology) \partial &part; U+2202
Interior of set Interior (topology) \circ &deg; U+02DA
Closure of set Closure (topology) \overline U+0305
Punctured neighbourhood of point Punctured neighbourhood \dot U+0307

Functional analysis[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Topological dual space of topological vector space Dual space \prime &prime; U+2032
Bidual space of normed vector space
Completion of metric space Complete metric space \hat U+0302
Embedding of topological vector space into Embedding \hookrightarrow U+21AA

Linear algebra and geometry[edit]

Elementary geometry[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Line segment between points and Line segment [ ] U+005B/D
Length of line segment between points and \vert U+007C
\overline U+0305
Vector between points and Euclidean vector \vec U+20D7
Angle between line segments and Angle \angle &ang; U+2220
Triangle with vertices , and Triangle \triangle U+25B3
Quadrilateral with vertices , , and Quadrilateral \square U+25A1
Lines and are parallel Parallel (geometry) \parallel U+2225
Lines and are not parallel \nparallel U+2226
Lines and are orthogonal Orthogonality \perp &perp; U+27C2

Vectors and matrices[edit]

Symbol Interpretation Article LaTeX
Row vector comprising elements through Vector (mathematics and physics) \begin{pmatrix}
...
\end{pmatrix}

oder

\left(
\begin{array}{...}
...
\end{array}
\right)
Column vector comprising elements through
Matrix comprising elements through Matrix (mathematics)

Vector calculus[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Dot product of vectors and Dot product \cdot &middot; U+22C5
( ) U+0028/9

\langle \rangle &lang; &rang; U+27E8/9
Cross product of vectors and Cross product \times &times; U+2A2F
[ ] U+005B/D
Triple product of vectors , and Triple product ( ) U+0028/9
Dyadic product of vectors and Dyadic product \otimes &otimes; U+2297
Wedge product of vectors and Wedge product \wedge U+2227
Length of vector Euclidean norm \vert U+007C
Norm of vector Norm (mathematics) \Vert, \| U+2016
Normalized vector of vector Unit vector \hat{} U+0302

Matrix calculus[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Product of matrices and Matrix multiplication \cdot &middot; U+22C5
Hadamard product of matrices and Hadamard product (matrices) \circ U+2218
Kronecker product of matrices and Kronecker product \otimes &otimes; U+2297
Transposed matrix of matrix Transposed matrix T U+0054
Conjugate transpose of matrix Conjugate transpose H U+0048
\ast &lowast; U+002A
\dagger &dagger; U+2020
Inverse matrix of matrix Inverse matrix -1 U+207B
Moore–Penrose pseudoinverse of matrix Moore–Penrose pseudoinverse + U+002B
Determinant of Matrix Determinant \vert U+007C
Norm of matrix Matrix norm \Vert, \| U+2016

Vector spaces[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Sum of vector spaces and Direct sum of modules + U+002B
Direct sum of vector spaces and \oplus &oplus; U+2295
Direct product of vector spaces and Direct product \times &times; U+2A2F
Tensor product of vector spaces and Tensor product \otimes &otimes; U+2297
Quotient space of vector space by subspace Quotient space (linear algebra) / &frasl; U+002F
Orthogonal complement of subspace Orthogonal complement \perp &perp; U+27C2
Dual space of vector space Dual space \ast &lowast; U+002A
Annihilator space of the set of vectors 0 U+0030
Linear hull of the set of vectors Linear hull \langle \rangle &lang; &rang; U+27E8/9

Algebra[edit]

Relations[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Composition of relations and Composition of relations \circ U+2218
Operation of elements and (general) Operation (mathematics)
\bullet &bull; U+2219
\ast &lowast; U+2217
Order relation between elements and Order relation \leq &le; U+2264
Element is a predecessor of element Successor ordinal \prec U+227A
Element is a successor of element \succ U+227B
Equivalence relation between elements and Equivalence relation \sim &sim; U+223C
Equivalence class of element Equivalence class [ ] U+005B/D
Quotient set of set by equivalence relation Quotient set / &frasl; U+002F
Inverse relation of relation Inverse relation -1 U+207B
Transitive closure of relation Transitive closure + U+002B
Reflexive closure of relation Reflexive closure \ast &lowast; U+002A

Group theory[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Groups and are isomorphic Group isomorphism \simeq U+2243
\cong &cong; U+2245
Direct product of groups and Direct product \times &times; U+2A2F
Semidirect product of groups and Semidirect product \rtimes U+22CA
Wreath product of groups and Wreath product \wr U+2240
is a subgroup of group Subgroup \leq &le; U+2264
is a proper subgroup of group \lt &lt; U+003C
is a normal subgroup of group Normal subgroup \vartriangleleft U+22B2
Quotient group of group by normal subgroup Quotient group / &frasl; U+002F
Index of subgroup in group Index of a subgroup \colon U+003A
Subgroup generated by set Generating set of a group \langle \rangle &lang; &rang; U+27E8/9
Commutator of elements and Commutator [ ] U+005B/D

Field theory[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Extension of field over field Field extension / &frasl; U+002F
\mid U+007C
\colon U+003A
Degree of field extension over Degree of a field extension
Algebraic closure of field Algebraic closure \bar U+0305
Extension of a field by adding an algebraic element Field extension, Algebraic number field K(\alpha)
Field of real or complex numbers Field (mathematics) \mathbb{K}
Finite field Finite field \mathbb{F} U+1D53D

Ring theory[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Group of units of ring Group of units \ast &lowast; U+2217
\times &times; U+2A2F
is an ideal of ring Ideal (ring theory) \vartriangleleft U+22B2
Quotient ring of ring by ideal Quotient ring / &frasl; U+002F
Polynomial ring over ring with variable Polynomial ring [ ] U+005B/D
Ring of formal power series and ring of formal Laurent series Formal power series [[ ]] (( ))

Combinatorics[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Number of permutations of elements Factorial ! U+0021
Number of derangements of elements (permutations without fixed points) Derangement
Number of involutions without fixed points ( odd) Double factorial
Number of -combinations of elements without repetition Combination \binom U+0028/9
Number of permutations of elements of which are identical Multinomial coefficient
Number of -combinations of elements with repetition Multiset U+0028/9
Rising factorial from with factors Pochhammer symbol \overline U+0305
Falling factorial from with factors \underline U+0332
Product of all primes up to Primorial \# U+0023

Stochastics[edit]

Probability theory[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Probability of event Probability measure P U+2119
Probability of event given event Conditional probability \mid U+007C
Expected value of the random variable Expected value E U+1D53C
Variance of the random variable Variance V U+1D54D
Standard deviation of the random variable Standard deviation \sigma &sigma; U+03C3
Covariance of random variables and Covariance
Correlation of random variables and Correlation \rho &rho; U+03C1
Random variable has distribution Probability distribution \sim &sim; U+223C
Random variable has distribution approximately \approx &asymp; U+2248
Event is independent from event Independence (probability theory) \perp &perp; U+22A5
Remark: for operators there are several notational variants; instead of round brackets also square brackets are used

Statistics[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Average of the values Average \bar U+0305
Average over all values in the set (in physics) \langle \rangle &lang; &rang; U+27E8/9
Estimator for parameter Estimator \hat U+0302

Logic[edit]

Operators[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Proposition and proposition Logical conjunction \land &and; U+2227
Proposition or proposition (or both) Logical disjunction \lor &or; U+2228
Proposition follows from proposition and vice versa Logical equivalence \Leftrightarrow &hArr; U+21D4
\leftrightarrow &harr; U+2194
From proposition follows proposition Logical consequence \Rightarrow &rArr; U+21D2
\rightarrow &rarr; U+2192
Either proposition or proposition Exclusive or \oplus &oplus; U+2295
\veebar U+22BB
\dot\lor U+2A52
Not proposition Logical negation \lnot &not; U+00AC
\bar U+0305

Quantifiers[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
For all elements Universal quantification \forall &forall; U+2200
\bigwedge U+22C0
At least one element exists Existential quantification \exists &exist; U+2203
\bigvee U+22C1
Exactly one element exists Uniqueness quantification \exists! &exist;! U+2203
\dot\bigvee U+2A52
No element exists Existential quantification \nexists U+2204

Deduction symbols[edit]

Symbol Usage Interpretation Article LaTeX HTML Unicode
Proposition can be syntactically derived from proposition Propositional calculus \vdash U+22A2
Proposition follows semantically from proposition Inference \models U+22A8
Proposition is universally true Tautology (logic)
\top U+22A4
Proposition is contradictory Contradiction \bot &perp; U+22A5
Proposition is true, therefore proposition is true Deductive reasoning \therefore U+2234
Proposition is true, because is true \because U+2235
End of proof Q.E.D. \blacksquare U+220E
\Box U+25A1

See also[edit]

References[edit]

Note: This article is a translation of the German Wikipedia article de:Liste mathematischer Symbole.

External links[edit]