Mathematical Symbols by Subject

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

Guide Set theory Definition symbols Set construction Set operations Set relations Number sets Cardinality Arithmetic Arithmetic operators Equality signs Comparison Divisibility Intervals Elementary functions Complex numbers Mathematical constants Calculus Sequences and series Functions Limits Asymptotic behaviour Differential calculus Integral calculus Vector calculus Topology Functional analysis Linear algebra and geometry Elementary geometry Vectors and matrices Vector calculus Matrix calculus Vector spaces Algebra Relations Group theory Field theory Ring theory Combinatorics Stochastics Probability theory Statistics Logic Operators Quantifiers Deduction symbols See also References External links

Guide

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 and . 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

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

Set construction

Symbol Usage Interpretation Article LaTeX HTML Unicode

\varnothing, Empty set Empty set ∅ U+2205 \emptyset Set consisting of the elements \{ \} U+007B/D and so on Set Set of elements , that satisfy (mathematics) \mid U+007C the condition \colon U+003A

Set operations

Symbol Usage Interpretation Article LaTeX HTML Unicode Union (set Union of the sets and \cup ∪ U+222A theory) Intersection Intersection of the sets and \cap ∩ U+2229 (set theory) Difference Difference of sets and \setminus U+2216 (set theory) Symmetric difference of sets and Symmetric \triangle Δ U+2206 difference Cartesian Cartesian product of sets and \times × U+2A2F product

Disjoint union of sets and \dot\cup U+228D Disjoint union Disjoint union of sets and \sqcup U+2294

\mathrm{C} U+2201 Complement Complement of the set (set theory) \bar U+0305

\mathcal{P} U+1D4AB

Power set of the set Power set \mathfrak{P} U+1D513

\wp U+2118

This is the least upper bound, Infimum and supremum, or join of all elements \bigvee U+22C1 supremum operated on. [1]

Set relations Symbol Usage Interpretation Article LaTeX HTML Unicode

\subset ⊂ U+2282 is a proper subset of Subset \subsetneq U+228A

is a subset of \subseteq ⊆ U+2286

\supset ⊃ U+2283 is a proper superset of Superset \supsetneq U+228B

is a superset of \supseteq ⊇ U+2287

\in ∈ U+2208 Element is in the set \ni, \owns ∋ U+220B Element (mathematics) \notin ∉ U+2209 Element is not in the set \not\ni U+220C

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

Number sets

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

Symbol Usage Interpretation Article LaTeX HTML Unicode

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

Arithmetic

Arithmetic operators Symbol Usage Interpretation Article LaTeX HTML Unicode

added to Addition + U+002B

subtracted from Subtraction - U+2212

\cdot · U+22C5 multiplied by Multiplication \times × U+2A2F

: U+003A

/ ⁄ U+2215 Division divided by (mathematics) \div ÷ U+00F7

\frac U+2044

Negative of the number or the Unary minus - − U+2212 additive inverse of

Plus or minus Plus or minus \pm ± U+00B1 sign Minus or plus \mp U+2213

( ) U+0028/9 Term is evaluated first Bracket [ ] U+005B/D

Equality signs

Symbol Usage Interpretation Article LaTeX HTML Unicode

equals Equality (mathematics) = U+003D

does not equal Inequality (mathematics) \neq ≠ U+2260

is identical to Identity (mathematics) \equiv ≡ U+2261 is approximately Approximation \approx ≈ U+2248 equal to

Proportionality \sim ∼ U+223C is proportional to (mathematics) \propto ∝ U+221D Correspondence corresponds to \widehat{=} U+2259 (mathematics)

Comparison

Symbol Usage Interpretation Article LaTeX HTML Unicode

is less than < < U+003C

is greater than > > U+003E

\le, \leq ≤ U+2264 is less than or equal to \leqq U+2266 Comparison (mathematics) \ge, \geq ≥ U+2265 is greater than or equal to \geqq U+2267

is much smaller than \ll U+226A

is much bigger than \gg U+226B Divisibility

Symbol Usage Interpretation Article LaTeX HTML Unicode

divides \mid U+2223 Divisibility does not divide \nmid U+2224

and are relatively prime Relatively prime \perp ⊥ U+22A5

Greatest common divisor of Greatest common \sqcap U+2293 and divisor \wedge U+2227

Least common multiple of Least common \sqcup U+2294 and multiple \vee U+2228 and are congruent Modular arithmetic \equiv ≡ U+2261 modulo

Intervals

Symbol Usage Interpretation Article LaTeX HTML Unicode

Closed interval between and

Open interval between and

( ) U+0028/9 Interval (mathematics) Right-open interval between and [ ] U+005B/D

Left-open interval between and

Elementary functions

Symbol Usage Interpretation Article LaTeX HTML Unicode

Absolute value of Absolute value \vert U+007C

[ ] U+005B/D Biggest whole number less than or equal to \lfloor ⌊ Floor and U+230A/B ceiling functions \rfloor ⌋ Smallest whole number \lceil ⌈ U+2308/9 greater than or equal to \rceil ⌉

Square root of Square root \sqrt √ 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 superscripta .

Complex numbers Symbol Usage Interpretation Article LaTeX HTML Unicode

Real part of complex number \Re U+211C Complex number Imaginary part of complex number \Im U+2111

\bar U+0305 Complex conjugate of Complex conjugate \ast ∗ U+002A

Absolute value of complex number Absolute value \vert U+007C

Remark: real and imaginary parts of a complex number are often also denoted by and .

Mathematical constants

Symbol Usage Interpretation Article LaTeX HTML Unicode Pi, or Archimedes' Pi \pi {{pi}} U+03C0 constant

e e or {{mvar|e}} or e or e Euler's constant U+0065 (mathematics) \rm{e} {{math|e}}

Golden ratio Golden ratio \varphi φ U+03C6

Imaginary unit (square root i or {{mvar|i}} or i or i Imaginary unit U+0069 of −1) \rm{i} {{math|i}}

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

Calculus

Sequences and series

Symbol Usage Interpretation Article LaTeX HTML Unicode

Sum from to or over all Summation \sum ∑ U+2211 elements in set

Product from to or over all Product \prod ∏ U+220F elements in set (mathematics)

Coproduct from to or over all Coproduct \coprod U+2210 elements in set

Sequence of elements Sequence ( ) U+0028/9

Limit of a Sequence tends to limit \to → U+2192 sequence

tends to infinity Infinity \infty ∞ U+221E

Functions Symbol Usage Interpretation Article LaTeX HTML Unicode

Function maps from set to \to → U+2192 set Function (mathematics) Function maps element to \mapsto U+21A6 element

Image of element under function Image ( ) U+0028/9 (mathematics) Image of set under function [ ] U+005B/D

Restriction of function to set Restriction \vert U+007C (mathematics)

Placeholder for a variable as Free variable \cdot U+22C5 argument of function

Inverse function of Inverse function -1 U+207B

Composition of functions and Function \circ ∘ U+2218 ; composition

Convolution of functions and Convolution \ast ∗ U+2217

Fourier transform of function Fourier transform \hat U+0302

Limits

Symbol Usage Interpretation Article LaTeX HTML Unicode

\uparrow ↑ U+2191 Limit of function as approaches from below \nearrow U+2197

Limit of function as approaches \to → U+2192

Limit of a \searrow U+2198 Limit of function as approaches function from above \downarrow ↓ U+2193

Limit of a function as ^+ ⁺ U+207A approaches from the right Limit of a function as ^- ⁻ U+207B approaches from the left

Asymptotic behaviour Symbol Usage Interpretation Article LaTeX HTML Unicode

Function is asymptotically Asymptotic \sim ∼ U+223C equal to function analysis

Function grows slower than o U+006F

Function grows not \mathcal{O} U+1D4AA substantially faster than

Function grows as fast as Big O notation \Theta Θ U+0398

Function grows not \Omega Ω U+03A9 substantially slower than

Function grows faster than \omega ω U+03C9

Differential calculus

Symbol Usage Interpretation Article LaTeX HTML Unicode First or second derivative of function \prime ′ U+2032,U+2033

Alternative notation for fourth, fifth, or ^{Ⅳ Lagrange's ^{V} sixth derivative of} notation function

Alternative notation for fourth, fifth, or - ^{( ) ( ) U+0028/9 th derivative of} function

First or second derivative of function Newton's \dot, U+0307 with respect to time notation \ddot (in physics) An infinitesimally small change in

Derivative of function with respect to variable Leibniz's d d U+0064 notation Second derivative of function with respect to variable Total differential of function

Partial derivative of Partial function with \partial ∂ U+2202 derivative respect to variable

Integral calculus Symbol Usage Interpretation Article LaTeX HTML Unicode

, Definite integral between and or over set Integral \int ∫ U+222B

Curve integral along curve Curve integral \oint U+222E

Surface Surface integral over surface \iint U+222C integral

Volume Volume integral over volume \iiint U+222D integral

Vector calculus

Symbol Usage Interpretation Article LaTeX HTML Unicode Gradient of function Gradient

Divergence of vector field Divergence \nabla ∇ U+2207 Curl of vector field Curl (mathematics)

Laplace operator of function Laplace operator \Delta Δ U+2206

D'Alembert operator of function D'Alembert operator \square U+25A1

Topology

Symbol Usage Interpretation Article LaTeX HTML Unicode

Boundary of set Boundary (topology) \partial ∂ U+2202

Interior of set Interior (topology) \circ ° U+02DA

Closure of set Closure (topology) \overline U+0305

Punctured neighbourhood of Punctured — \dot point neighbourhood U+0307

Functional analysis

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

Linear algebra and geometry Elementary geometry

Symbol Usage Interpretation Article LaTeX HTML Unicode Line segment between points [ ] U+005B/D and

Line segment \vert U+007C Length of line segment between points and \overline U+0305

Euclidean vector Vector between points and \vec U+20D7 and Affine space Angle between line segments Angle \angle ∠ U+2220 and

Triangle with vertices , and Triangle \triangle U+25B3 Quadrilateral with vertices , , Quadrilateral \square U+25A1 and Lines and are parallel Parallel \parallel U+2225 (geometry) Lines and are not parallel \nparallel U+2226

Lines and are orthogonal Orthogonality \perp ⊥ U+27C2

Vectors and matrices

Symbol Usage Article LaTeX

Row vector comprising elements \begin{pmatrix} through ...

Vector (mathematics and \end{pmatrix} physics) Column vector comprising elements or through \left( \begin{array} {...} Matrix comprising elements through Matrix (mathematics) ... \end{array} \right)

Vector calculus Symbol Usage Interpretation Article LaTeX HTML Unicode

\cdot · U+22C5

Dot product of vectors ( ) U+0028/9 Dot product and \langle ⟨ U+27E8/9 \rangle ⟩

Cross product of vectors \times × U+2A2F Cross product and [ ] U+005B/D

Triple product of vectors , Triple product ( ) U+0028/9 and Dyadic product of vectors Dyadic product \otimes ⊗ U+2297 and Wedge product of vectors Wedge product \wedge U+2227 and

Length of vector Euclidean norm \vert U+007C

Norm Norm of vector \Vert, \| U+2016 (mathematics) Normalized vector of Unit vector \hat{} U+0302 vector

Matrix calculus

Symbol Usage Interpretation Article LaTeX HTML Unicode

Product of matrices and Matrix multiplication \cdot · U+22C5 Hadamard product of matrices Hadamard product \circ U+2218 and (matrices) Kronecker product of matrices Kronecker product \otimes ⊗ U+2297 and

Transposed matrix of matrix Transposed matrix T U+0054

H U+0048 Conjugate transpose of matrix Conjugate transpose \ast ∗ U+002A

\dagger † U+2020

Inverse matrix of matrix Inverse matrix -1 U+207B

Moore–Penrose pseudoinverse Moore–Penrose + U+002B of matrix pseudoinverse

Determinant of Matrix Determinant \vert U+007C

\Vert, Norm of matrix Matrix norm U+2016 \|

Vector spaces Symbol Usage Interpretation Article LaTeX HTML Unicode Sum of vector spaces and + U+002B Direct sum of Direct sum of vector spaces modules \oplus ⊕ U+2295 and Direct product of vector Direct product \times × U+2A2F spaces and Tensor product of vector Tensor product \otimes ⊗ U+2297 spaces and Quotient space of vector Quotient space / ⁄ U+002F space by subspace (linear algebra) Orthogonal complement of Orthogonal \perp ⊥ U+27C2 subspace complement

Dual space of vector space \ast ∗ U+002A Annihilator space of the set of Dual space 0 U+0030 vectors

Linear hull of the set of \langle ⟨ Linear hull U+27E8/9 vectors \rangle ⟩

Algebra

Relations

Symbol Usage Interpretation Article LaTeX HTML Unicode Composition of Composition of relations and relations \circ U+2218

Operation of elements and Operation \bullet • U+2219 (general) (mathematics) \ast ∗ U+2217 Order relation between elements Order relation \leq ≤ U+2264 and Element is a predecessor of \prec U+227A element Successor ordinal Element is a successor of \succ U+227B element Equivalence relation between Equivalence \sim ∼ U+223C elements and relation

Equivalence class of element Equivalence class [ ] U+005B/D

Quotient set of set by Quotient set / ⁄ U+002F equivalence relation

Inverse relation of relation Inverse relation -1 U+207B

Transitive closure of relation Transitive closure + U+002B

Reflexive closure of relation Reflexive closure \ast ∗ U+002A

Group theory Symbol Usage Interpretation Article LaTeX HTML Unicode

Groups and are Group \simeq U+2243 isomorphic isomorphism \cong ≅ U+2245 Direct product of groups Direct product \times × U+2A2F and Semidirect product of groups Semidirect \rtimes U+22CA and product Wreath product of groups Wreath product \wr U+2240 and

is a subgroup of group \leq ≤ U+2264 is a proper subgroup of Subgroup \lt < U+003C group is a normal subgroup of Normal \vartriangleleft U+22B2 group subgroup Quotient group of group Quotient group / ⁄ U+002F by normal subgroup Index of subgroup in Index of a \colon U+003A group subgroup

Subgroup generated by set Generating set \langle ⟨ U+27E8/9 of a group \rangle ⟩ Commutator of elements Commutator [ ] U+005B/D and

Field theory

Symbol Usage Interpretation Article LaTeX HTML Unicode

/ ⁄ U+002F Extension of field over field Field extension \mid U+007C

Degree of field extension Degree of a field \colon U+003A over extension

Algebraic closure of field Algebraic closure \overline U+0305

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

Finite field Finite field \mathbb{F} U+1D53D

Ring theory Symbol Usage Interpretation Article LaTeX HTML Unicode

Group of \ast ∗ U+2217 Group of units of ring units \times × U+2A2F is an ideal of ring (Uncommon, needs to Ideal (ring \vartriangleleft U+22B2 be defined before the theory) first use) Quotient ring of ring Quotient / ⁄ U+002F by ideal ring Polynomial ring over Polynomial [ ] U+005B/D ring with variable ring Ring of formal power Formal series and ring of power [[ ]] U+005B/D formal Laurent series series

Combinatorics

Symbol Usage Interpretation Article LaTeX HTML Unicode Number of permutations of Factorial elements Number of derangements of elements (permutations without fixed Derangement ! U+0021 points) Number of involutions without fixed Double points ( odd) factorial Number of -combinations of Combination elements without repetition Number of permutations of \binom U+0028/9 Multinomial elements of which are coefficient identical Number of -combinations of Multiset (( )) U+0028/9 elements with repetition

Rising factorial from with factors Pochhammer \overline U+0305 symbol Falling factorial from with factors \underline U+0332

Product of all primes up to Primorial \# U+0023

Stochastics

Probability theory Symbol Usage Interpretation Article LaTeX HTML Unicode

Probability of event Probability measure P U+2119

Probability of event given Conditional \mid U+007C event probability Expected value of the random Expected value E U+1D53C variable Variance of the random Variance V U+1D54D variable Standard deviation of the Standard deviation random variable \sigma σ U+03C3 Covariance of random Covariance variables and Correlation of random variables Correlation \rho ρ U+03C1 and Random variable has \sim ∼ U+223C distribution Probability Random variable has distribution \approx ≈ U+2248 distribution approximately Event is independent from Independence \perp ⊥ U+22A5 event (probability theory)

Remark: for operators there are several notational variants; instead of round brackets also square brackets are used

Statistics

Symbol Usage Interpretation Article LaTeX HTML Unicode

Average of the values \bar U+0305

Average over all values in the set Average \langle ⟨ U+27E8/9 (in physics) \rangle ⟩

Estimator for parameter Estimator \hat U+0302

Logic

Operators Symbol Usage Interpretation Article LaTeX HTML Unicode Logical Proposition and proposition \land ∧ U+2227 conjunction Proposition or proposition (or Logical \lor ∨ U+2228 both) disjunction

Proposition follows from Logical \Leftrightarrow ⇔ U+21D4 proposition and vice versa equivalence \leftrightarrow ↔ U+2194

From proposition follows Logical \Rightarrow ⇒ U+21D2 proposition consequence \rightarrow → U+2192

\oplus ⊕ U+2295 Either proposition or proposition Exclusive or \veebar U+22BB

\dot\lor U+2A52

\lnot ¬ U+00AC Logical Not proposition negation \bar U+0305

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

Quantifiers

Symbol Usage Interpretation Article LaTeX HTML Unicode

\forall ∀ U+2200 For all elements Universal quantification \bigwedge U+22C0

\exists ∃ U+2203 At least one element Existential exists quantification \bigvee U+22C1

\exists! ∃! U+2203 Exactly one element Uniqueness exists quantification \dot\bigvee U+2A52

Existential No element exists \nexists U+2204 quantification

Deduction symbols Symbol Usage Interpretation Article LaTeX HTML Unicode Proposition can be syntactically Propositional \vdash U+22A2 A\vdashderived from proposition calculus Proposition follows semantically A\models Inference from proposition B \models U+22A8 \models Tautology Proposition is universally true A (logic) A\top \top U+22A4

A\botProposition is contradictory Contradiction \bot ⊥ U+22A5 Proposition is true, therefore A\therefore \therefore U+2234 proposition is true Deductive Proposition is true, because is reasoning \because U+2235 A\becausetrue

\blacksquare U+220E End of proof Q.E.D. \Box U+25A1