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List of mathematical symbols by subject

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.

Contents

GuideEdit

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 theoryEdit

Definition symbolsEdit

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

Set constructionEdit

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 operationsEdit

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 union of sets   and   \sqcup U+2294
    Complement of the set   Complement (set theory) \mathrm{C} U+2201
    \overline U+0305
    Power set of the set   Power set \mathcal{P} U+1D4AB
    \mathfrak{P} U+1D513
    \wp U+2118
    This is the least upper bound, supremum, or join of all elements operated on. [1] Infimum and supremum \bigvee U+22C1

Set relationsEdit

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 setsEdit

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

CardinalityEdit

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

ArithmeticEdit

Arithmetic operatorsEdit

Symbol Usage Interpretation Article LaTeX HTML Unicode
      added to   Addition + U+002B
      subtracted to   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 signsEdit

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 similar to   Equivalence class \sim &sim; U+223C
      is proportional to   Proportionality (mathematics) \propto &prop; U+221D
      corresponds to   Correspondence (mathematics) \widehat{=} U+2259

ComparisonEdit

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

DivisibilityEdit

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

IntervalsEdit

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 functionsEdit

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 numbersEdit

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
    Argument of   Argument (complex analysis) \arg
Remark: real and imaginary parts of a complex number are often also denoted by   and  .

Mathematical constantsEdit

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

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

CalculusEdit

Sequences and seriesEdit

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

FunctionsEdit

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 &#8728; U+2218
    Convolution of functions   and   Convolution \ast &lowast; U+2217
    Fourier transform of function   Fourier transform \hat U+0302

LimitsEdit

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
    Limit of a function   as   approaches   from the right ^+ &#8314; U+207A
    Limit of a function   as   approaches   from the left ^- &#8315; U+207B

Asymptotic behaviourEdit

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 calculusEdit

Symbol Usage Interpretation Article LaTeX HTML Unicode
    First or second derivative of function   Lagrange's notation \prime &prime; U+2032,U+2033
    Alternative notation for fourth, fifth, or sixth derivative of function   ^{V} <sup>&#8547;</sup>
    Alternative notation for fourth, fifth, or  -th derivative of function   ( ) <sup>( )</sup> U+0028/9
    First or second derivative of function   with respect to time (in physics) Newton's notation \dot, \ddot U+0307
    An infinitesimally small change in   Leibniz's notation d d U+0064
  Derivative of function   with respect to variable  
 
  Second derivative of function   with respect to variable  
  Total differential of function  
    Partial derivative of function   with respect to variable   Partial derivative \partial &part; U+2202

Integral calculusEdit

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 calculusEdit

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

TopologyEdit

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 analysisEdit

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 geometryEdit

Elementary geometryEdit

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
and Affine space
\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 matricesEdit

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

or

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

Vector calculusEdit

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 calculusEdit

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
    Hadamard division of matrices   and   Hadamard product (matrices) \oslash U+2298
    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 spacesEdit

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

AlgebraEdit

RelationsEdit

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 theoryEdit

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 theoryEdit

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 \overline U+0305
    Extension of a field   by adding an algebraic element   Field extension, Algebraic number field ( ) U+0028/9
  Field of real or complex numbers Field (mathematics) \mathbb{K} U+1D542
  Finite field Finite field \mathbb{F} U+1D53D

Ring theoryEdit

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  
(Uncommon, needs to be defined before the first use)
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 [[ ]] U+005B/D

CombinatoricsEdit

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

StochasticsEdit

Probability theoryEdit

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

StatisticsEdit

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

LogicEdit

OperatorsEdit

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
    If B then A, or not B without A. It is not to be confused with the assignment operator in computer science. Converse implication \leftarrow U+2190

Quantifiers