T he language and vocabulary of mathematics contain a large amount of symbols — some being more technical than others. Like letters in the alphabet, they can be used to form words, phrases and sentences that would constitute a larger part of the mathematical lexicon. \[ \begin{gather*}x \longrightarrow x+1 \longrightarrow (x+1)^2 \longrightarrow (x+1)^2 \ge 0 \\ \longrightarrow \forall x \in \mathbb{R} [ (x+1)^2 \ge 0 ] \end{gather*} \] A math symbol can be used for different purposes from one mathematical subfield to another (e.g., $\sim$ as logical negation and similarity of triangle), just as multiple symbols can be used to delineate the same concept or relation (e.g., $\times$ and $\cdot$ in multiplication).

A basic understanding about mathematical terminology is essential to a solid foundation in higher mathematics. To that end, the following is a compilation of some of the most well-adapted, commonly-used symbols in mathematics.

Moreover, these symbols are further categorized by their function into tables. More comprehensive lists of symbols — as categorized by subject and type — can be also found in the relevant pages below (or in the navigational panel).

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ConstantsIn mathematics, constants are symbols that are used to refer to non-varying objects . These can include key numbers, key mathematical sets, key mathematical infinities and other key mathematical objects (such as the identity matrix $I$).

Mathematical constants often take form of an alphabet letter — or a derivative of it. In some occasions, a constant might be regarded as a variable in the larger context. The following tables feature some of the most commonly-used constants, along with their name, meaning and usage.

Key Mathematical NumbersSymbol Name Explanation Example $0$ (Zero ) Additive identity of common numbers $3 + 0 =3$ $1$ (One ) Multiplicative identity of common numbers $5 \times 1 = 5$ $\sqrt{2}$ (Square root of $2$ ) Positive number whose square is $2$. Approximately $1.41421$. $(\sqrt{2} + 1)^2 = 3 + 2\sqrt{2}$ $e$ (Euler’s constant ) Base of the natural logarithm. Limit of the sequence $(1+\frac{1}{n})^n$. Approximately $2.71828$. $\ln (e^2) = 2 $ $\pi$ (Pi , Archimedes’ constant) Ratio of a circle’s circumference to its diameter. Half-circumference of a unit circle. Approximately $3.14159$. $\dfrac{\pi^2}{6} = \dfrac{1}{1^2} + \dfrac{1}{2^2} + \cdots$ $\varphi$ (Phi , golden ratio ) Ratio between a larger number $a$ and a smaller number $b$ when $\frac{a+b}{a} = \frac{a}{b}$. Positive solution to the equation $x^2-x-1 = 0$. $\varphi = \dfrac{1+\sqrt{5}}{2} \approx 1.61803$ $i$ (Imaginary unit ) The principal root of $-1$. Foundational component of a complex number. $(1+i)^2 = 2i$

Key Mathematical SetsFor a more comprehensive list, see key mathematical sets in algebra .

Symbol Name Explanation Example $\varnothing$ (Empty set ) Set with no element $|\varnothing| = 0$ $\mathbb{N}$ (N ) Set of natural numbers $\forall x, y \in \mathbb{N}$, $x+y \in \mathbb{N}$ $\mathbb{Z}$ (Z ) Set of integers (Z stands for zahlen, number in German) $ \mathbb{N} \subseteq \mathbb{Z}$ $\mathbb{Z}_+$ (Z-plus ) Set of positive integers $3 \in \mathbb{Z}_+$ $\mathbb{Q}$ (Q ) Set of rational numbers (Q stands for quotient) $\sqrt{2} \notin \mathbb{Q}$ $\mathbb{R}$ (R ) Set of real numbers $\forall x \in \mathbb{R}, x^2 \ge 0$ $\mathbb{R}_+$ (R-plus ) Set of positive real numbers $\forall x,y \in \mathbb{R}_+$, $xy \in \mathbb{R}_+$ $\mathbb{C}$ (C ) Set of complex numbers $\exists z \in \mathbb{C}\, (z^2 + 1 =0)$ $\mathbb{Z}_n$ (Z-n ) Set of integers modulo $n$ In the world of $\mathbb{Z}_2$, $1+1=0$. $\mathbb{R}^3$ (R-three ) Three-dimensional Euclidean space $(5, 1, 2) \in \mathbb{R}^3$

Key Mathematical InfinitiesIn mathematics, many different types of infinity exist. These include the purely notational use of the lemniscate symbol ($\infty$), and the use of the following symbols in the context of cardinal/ordinal infinities:

Symbol Name Explanation Example $\aleph_0$ (Aleph-naught ) Cardinality of the set of natural numbers $\aleph_0 + 5 = \aleph_0$ $\mathfrak{c}$ (Continuum ) Cardinality of the set of real numbers $\mathfrak{c}=2^{\aleph_0}$ $\omega$ (Omega ) Smallest infinite ordinal number $\forall n \in \mathbb{N}, n < \omega$

For a more comprehensive list, see cardinality-related symbols .

Other Key Mathematical ObjectsSymbol Name Explanation Example $\mathbf{0}$ (Zero ) Zero vector of a vector space $\forall \mathbb{v} \in V$, $\mathbf{v} + \mathbf{0} = \mathbf{v}$ $e$ (E ) Identity element of a group$e \circ e = e$ $I$ (I ) Identity matrix $AI = IA =I$ $C$ (C ) Constant of integration $\displaystyle \int 1 \, \mathrm{d}x =$ $x + C$ $\top$ (Tautology ) A sentence in formal logic which is unconditionally true For each proposition $P$, $P \land \top \equiv P$. $\bot$ (Contradiction ) A sentence in formal logic which is unconditionally false For each proposition $P$, $P \land \lnot P \equiv \bot.$ $Z$ (Z ) Standard normal distribution $Z \sim N(0,1)$

VariablesA mathematical variable is a symbol that functions as a placeholder for varying expressions or quantities . The same variable can be used on a repeated basis to refer to the same thing — or quantified to form sentences that have a more definite meaning: \begin{gather*}x, y \longrightarrow x + e^x = y \longrightarrow \exists y \in \mathbb{R}\, (x + e^x = y) \\ \longrightarrow \forall x \in \mathbb{R} \, \exists y \in \mathbb{R}\, (x + e^x = y) \end{gather*} In some cases, variables can be thought of as constants in narrower contexts (e.g., as parameters), while in other cases, variables are used in conjunction with subscripts to make up for the lack of letters (e.g., $x_3$).

While variables in mathematics are often used to represent numbers , they can also be used to represent other objects such as vectors, functions and matrices. The following tables document some of the most common conventions for variables — along with the context where they are adopted and used.

Variables for NumbersSymbol(s) Used For Example $m, n, p, q$ Integers and natural numbers If $mn$ is odd, then both $m$ and $n$ are odd. $a, b, c$ Coefficients of functions and equationsA line of the form $ax+by=0$ passes through the origin. $x, y, z$ Unknowns in functions and equationsIf $2x + 5= 3$, then $x=-1$. $\Delta$ Discriminant $\Delta = b^2 – 4ac$ for quadratic polynomials $i, j, k$ Index variables in summations and products$\sum _{i=1}^{10} i = 55$ $t$ Time At $t=5$, the velocity is $v(5)=32$. $z$ Complex numbers $z \overline{z} = |z|^2$

Variables in GeometryFor more symbols in geometry and trigonometry, see geometry and trigonometry symbols .

Symbol(s) Used For Example $P, Q, R, S$ Vertices $\overline{PQ} \perp \overline{QR}$ $\ell$ Lines $\ell_1 \parallel \ell_2$ $\alpha, \beta, \gamma, \theta$ Angles $\alpha + \beta + \theta = 180^{\circ}$

Variables in CalculusFor a more comprehensive list, see constants and variables in calculus .

Symbol(s) Used For Example $f(x), g(x,y),h(z)$ Functions $f(2) = g(3,1) + 5$ $a_n, b_n, c_n$ Sequences $\displaystyle a_ n = \frac{3}{n+2} $ $h, \Delta x$ Limiting variables in derivatives $\displaystyle \lim_{h \to 0} \frac{e^{h}-e^{0}}{h} = 1$ $\delta, \varepsilon$ Small quantities in proofs involving limits For all $\varepsilon >0$, there is a $\delta >0$ such that $|x|<\delta$ implies $|2x|<\varepsilon$. $F(x), G(x)$ Antiderivatives $F(x)’ = f(x)$

Variables in Linear AlgebraFor a more comprehensive list, see variables in algebra .

Symbol(s) Used For Example $\mathbf{u}, \mathbf{v}, \mathbf{w}$ Vectors $3\mathbf{u}+4\mathbf{v}=\mathbf{w}$ $A, B, C$ Matrices $AX = B$ $\lambda$ Eigenvalues $A\mathbf{v}=\lambda \mathbf{v}$

Variables in Set Theory and LogicFor more comprehensive lists on the topics, see variables in logic and variables in set theory .

Symbol(s) Used For Example $A, B, C$ Sets $A \subseteq B \cup C$ $a, b, c$ Elements $a \in A$ $P, Q, R$ Propositions $P \lor \lnot P \equiv \top$

Variables in Probability and StatisticsFor a more comprehensive list, see variables in probability and statistics .

Symbol(s) Used For Example $X, Y, Z$ Random variables $E(X + Y) =$ $E(X) + E(Y)$ $\mu$ Population means $H_0\!:\mu = 5$ $\sigma$ Population standard deviations $\sigma_1 = \sigma_2$ $s$ Sample standard deviations $s \ne \sigma$ $n$ Sample sizes If $n\ge 30$, use the normal distribution. $\rho$ Population correlations $H_a\!: \rho < 0$ $r$ Sample correlations If $r = 0.75$, then $r^2 = 0.5625$. $\pi$ Population proportions $\pi = 0.5$ $p$ Sample proportions $p = \dfrac{X}{n}$

DelimitersSimilar to punctuation marks in English, delimiters are a set of symbols which indicate the boundaries between independent mathematical expressions. They are often used to specify the scope for which an operation or rule would apply, and can occur both as an isolate symbol or as a pair of opposite-looking symbols.

In many scenarios, delimiters are used primarily for grouping purposes . The following table features some of the most commonly-used delimiters, along with their function and usage.

Symbol(s) Function Example $.$ Decimal separator $25.9703$ $:$ Ratio indicator $1:4:9 =$ $3:12:27$ $,$ Object separator $(3, 5, 12)$ $(), [], \{\}$ Order-of-operation indicators$(a + b) \times c$ $(), []$ Interval indicators $3\notin (3,4]$, $4 \in (3,4]$. $(), []$ Vector/matrix builder $\begin{pmatrix} 1 & 4 \\ 3 & 6 \end{pmatrix}$ $\{\}$ Set builder $\{ \pi, e, i\}$ $|$, $\, :$ “Such that” markers $\{ x \in \mathbb{R} \, |\, x^2 – 2 =0 \}$ $| |, \| \|$ Norm-related operators $\| (3, 4) \| = 5$ $\begin{cases}\end{cases}$ Piecewise-function marker $f(x) = \begin{cases} 1 & x \ge 0 \\ 0 & x < 0 \end{cases}$ $\langle\rangle$ Inner product operator$\langle ka, b\rangle = k\langle a, b \rangle$ $\lceil \rceil$ Ceiling operator $\lceil 2.476 \rceil = 3$ $\lfloor \rfloor$ Floor operator $\lfloor \pi \rfloor = 3$

OperatorsAn operator is a symbol used to denote an operation — a function which takes one or multiple objects to another similar object. Most of the operators are unary and binary in nature (i.e., taking one and two inputs to their intended target, respectively), with the most common ones being the arithmetic operators (e.g., $+$).

Much like the case in English, operators allow one to expand the lexicon of mathematics where only finitely many symbols exist. The following tables feature some of the most commonly-used operators in mathematics — along with their usage and intended meaning.

Common OperatorsSymbol(s) Explanation Example $x + y$ Sum of $x$ and $y$$2a + 3a =5a$ $x-y$ Difference of $x$ and $y$$11-5=6$ $-x$ Additive inverse of $x$$-3 + 3 =0$ $x \times y$, $x \cdot y$, $xy$ Product of $x$ and $y$$(m+1)n=mn+n$ $x \div y, x / y$ Quotient of $x$ over $y$$152 \div 3 = 50.\overline{6}$ $\dfrac{x}{y}$ Fraction ($x$ over $y$)$\dfrac{53 + 5}{6} =$ $\dfrac{53}{6}+\dfrac{5}{6}$ $x^y$ $x$ raised to the power of $y$ $3^4 = 81$ $x \pm y$ $x$ plus and minus $y$ $\dfrac{-b \pm \sqrt{\Delta}}{2a}$ $\sqrt{x}$ Positive square root of $x$$\sqrt{2} \approx 1.414$ $|x|$ Absolute value of $x$$|x-3|<5$ $x \%$ $x$ percent $x \% \doteq \dfrac{x}{100}$

Function-related OperatorsFor a more comprehensive list, see function-related symbols .

Symbol Explanation Example $\text{dom}(f)$ Domain of function $f$If $g(x)=\ln{x}$, then $\text{dom}(g) = \mathbb{R}_+$. $\text{ran}(f)$ Range of function $f$If $h(y)=\sin y$, then it follows that $\text{ran}(h) = [-1,1]$. $f(x)$ Image of element $x$ under function $f$$g(5)=g(4)+3$ $f(X)$ Image of set $X$ under function $f$$f(A \cap B) \subseteq$ $f(A) \cap f(B)$ $f \circ g$ Composite function ($f$ of $g$)If $g(3)=5$ and $f(5)=8$, then $(f \circ g)(3) =8$.

Elementary FunctionsFor a more comprehensive list, see key functions in algebra .

Symbol(s) Explanation Example $k_n x^n + \cdots + k_0 x^0$ Polynomial with coefficients $k_0, \ldots, k_n$Polynomial $x^3+2x^2+3$ has a root in $(-3, -2)$. $e^x, \exp{x}$ Natural exponential function $e^{x+y}=e^x \cdot e^y$ $b^x$ Exponential function with base $b$$2^x > x^2$ for large $x$. $\ln x$ Natural logarithmic function $\ln (x^2) = 2 \ln{x}$ $\log x$ Logarithm function of base 10 (or base $e$)$\log 10000 = 4$ $\log_b x$ Logarithm function of base $b$$\log_2 x = \dfrac{\ln x}{\ln 2}$ $\sin x$ Sine function $\sin \pi = 0$ $\cos x$ Cosine function $\cos \dfrac{\pi}{4} = \dfrac{\sqrt{2}}{2}$ $\tan x$ Tangent function $\tan x = \dfrac{\sin x}{\cos x}$

Algebra-related OperatorsFor a more comprehensive list, see operators in algebra .

Symbol(s) Explanation Example $\gcd (x,y)$ Greatest common factor of $x$ and $y$${\gcd (35,14)=7}$ $\lfloor x \rfloor$ Floor of $x$ (largest integer smaller or equal to $x$)$\lfloor 3.6 \rfloor = 3$ $\lceil x \rceil$ Ceiling of $x$ (smallest integer larger or equal to $x$)$\lceil \pi \rceil = 4$ $\min (A)$ Minimum of set $A$If $\min (A) = 3$, then $\min(A+5)=8$. $\max (A)$ Maximum of set $A$$\max(A \cup B) \ge \max (A)$ $x\text{ mod } y$ Remainder of $x$ under modulus $y$$36\text{ mod } 5 = 1$ $\sum_{i=m}^n a_i$ Sum of $a_i$ (where $i$ runs from $m$ to $n$)$\sum_{i=1}^5 i^2 = 55$ $\prod_{i=m}^n a_i$ Product of $a_i$ (where $i$ runs from $m$ to $n$)$\prod_{i=1}^n i = n!$ $[a]$ Equivalence class of element $a$$[a]\doteq \{x \,|\, xRa\}$ $\deg f$ Degree of polynomial $f$$\deg (2x^2 +3x+5)=2$ $\overline{z}$ Conjugate of complex number $z$$\overline{5-8i}=5+8i$ $|z|$ Absolute value of complex number $z$$|e^{\pi i}|=1$ $\text{arg}(z)$ Arguments of complex number $z$$\text{arg}(1+i)= \\[0.3em] \dfrac{\pi}{4} + 2\pi n$

Geometry-related OperatorsFor more symbols in geometry and trigonometry, see geometry and trigonometry symbols .

Symbol(s) Explanation Example $\angle ABC$ Angle formed by vertices $A$, $B$ and $C$$\angle ABC = \angle CBA$ $\measuredangle ABC$, $m\angle ABC$ Measure of angle formed by vertices $A$, $B$ and $C$$\measuredangle ABC = \measuredangle A’B’C’$ $\overleftrightarrow{AB}$ Infinite line formed by points $A$ and $B$$\overleftrightarrow{AB}= \\ \overleftrightarrow{BA}$ $\overline{AB}$ Line segment between points $A$ and $B$If $B \ne B’$, then $\overline{AB} \ne \\ \overline{AB’}.$ $\overrightarrow{AB}$ Ray from point $A$ to point $B$$\overrightarrow{AB} \cong \\ \overrightarrow{CD}$ $|AB|$ Distance between point $A$ and point $B$$|AB|<{|A’B’|}$ $\triangle ABC$ Triangle formed by vertices $A$, $B$ and $C$$\triangle ABC \cong \triangle A’B’C’$ $\square ABCD$ Quadrilateral formed by vertices $A$, $B$, $C$ and $D$$\square ABCD = \square DCBA$

Logic-related OperatorsFor a more comprehensive list, see operators in logic .

Symbol Explanation Example $\lnot P$ Negation (not $P$)$\lnot (1=2)$ $P \land Q$ Conjunction ($P$ and $Q$)$P \land Q \equiv$ $Q \land P$ $P \lor Q$ Disjunction ($P$ or $Q$)$\pi^e \in \mathbb{Q} \, \lor$ $\pi^e \notin \mathbb{Q}$ $P \to Q$ Conditional (if $P$, then $Q$)$P \to Q \equiv$ $(\lnot P \lor Q)$ $P \leftrightarrow Q$ Biconditional ($P$ if and only if $Q$)$P \leftrightarrow Q \implies \\ P \to Q$ $\forall \mathbf{x} P(\mathbf{x})$ Universal statement (for all $\mathbf{x}$, $P(\mathbf{x})$)$\forall y \in \mathbb{N},$ $y+1 \in \mathbb{N}$ $\exists \mathbf{x} P(\mathbf{x})$ Existential statement (there exists $\mathbf{x}$ such that $P(\mathbf{x})$)$\exists z \, {(z^2 = -\pi)}$

Set-related OperatorsFor a more comprehensive list, see operators in set theory .

Symbol Explanation Example $\overline{A} $, $A^c$ Complement of set $A$$\overline{\smash{\overline{A}}\vphantom{\bar{A}}}=A$ $A\cap B$ Intersection of sets $A$ and $B$$\{2,5\}\cap {\{1,3\}} = \varnothing$ $A\cup B$ Union of sets $A$ and $B$$\mathbb{Z}\cup\mathbb{N}=\mathbb{Z}$ $A/B$, $A-B$ Difference of sets $A$ and $B$In general, $A-B \ne$ $B-A.$ $A \times B$ Cartesian product of sets $A$ and $B$$(11, -35) \in \mathbb{N} \times \mathbb{Z}$ $\mathcal{P}(A)$ Power set of set $A$$\mathcal{P}(\varnothing)= \{\varnothing \}$ $|A|$ Cardinality of set $A$$|\mathbb{N}|=\aleph_0$

Vector-related OperatorsFor a more comprehensive list, see operators in linear algebra .

Symbol Explanation Example $\| \mathbf{v} \|$ Norm of vector $\mathbf{v}$$\| (3,4)\|=5$ $\mathbf{u} \cdot \mathbf{v}$ Dot product of vectors $\mathbf{u}$ and $\mathbf{v}$$\mathbf{u} \cdot \mathbf{u} = \| \mathbf{u}\|^2$ $\mathbf{u} \times \mathbf{v}$ Cross product of vectors $\mathbf{u}$ and $\mathbf{v}$$\mathbf{u} \times \mathbf{u}=\mathbf{0}$ $\text{proj}_{\mathbf{v}}\mathbf{u}$ Projection of vector $\mathbf{u}$ onto vector$\mathbf{v}$$\text{proj}_{(0,1)}(5,4)=$ $(0,4)$ $\text{span}(S)$ Span of set of vectors $S$$\text{span}(\{\mathbf{i},\mathbf{j}\})=\mathbb{R}^2$ $\text{dim}(V)$ Dimension of vector space $V$$\text{dim}(\mathbb{R}^3)=3$

Matrix-related OperatorsFor a more comprehensive list, see operators in linear algebra .

Symbol(s) Explanation Example $A+B$ Sum of matrices $A$ and $B$$A + X = B$ $A-B$ Difference of matrices $A$ and $B$In general, $A-B \ne B-A.$ $-A$ Additive inverse of matrix $A$$B+ (-B)=0$ $kA$ Scalar product of matrix $A$ and number $k$$(-1)A=-A$ $AB$ Product of matrices $A$ and $B$$AI=IA=A$ $A^T$ Transpose of matrix $A$$I^T=I$ $A^{-1}$ Multiplicative inverse of matrix $A$$(AB)^{-1}= \\ B^{-1}A^{-1}$ $\text{tr}(A)$ Trace of square matrix $A$$\text{tr}(A^T)=\text{tr}(A)$ $\det(A), |A|$ Determinant of square matrix $A$$\left| \begin{pmatrix} 1 & 4 \\ 3 & 2 \end{pmatrix} \right| = \\ 2-12 = -10$

Probability-related OperatorsFor a more comprehensive list, see operators in probability and statistics .

Symbol(s) Explanation Exampe $n!$ Factorial of $n$$4! = 4 \cdot 3 \cdot 2 \cdot 1$ $n P\,r$ Permutation ($n$ permute $r$)$5 P\,3=5 \cdot 4 \cdot 3$ $\displaystyle nCr, \binom{n}{r}$ Combination ($n$ choose $r$)$\displaystyle \binom{5}{2} = \binom{5}{3}$ $P(E)$ Probability of event $E$$P(A\cup B\cup C) = 0.\overline{3}$ $P(A\,|\,B)$ Conditional probability of event $A$ given event $B$$P(A\,|\,B)=\dfrac{P(A\cap B)}{P(B)}$ $E(X)$ Expected value of random variable $X$$E(X+Y)=$ $E(X)+E(Y)$ $V(X)$ Variance of random variable $X$$V(5X) = 25V(X)$

Statistics-related OperatorsFor a more comprehensive list, see statistical operators .

Symbol Explanation Example $\overline{X}$ Sample mean of data set $X$$\overline{3X} = 3 \overline{X}$ $s^2$ Sample variance $s^2 = \dfrac{\sum (X – \overline{X})^2}{n -1}$ $\sigma^2$ Population variance $\sigma^2 = \dfrac{\sum (X – \mu)^2}{n}$

Key Probability Functions and DistributionsFor a more comprehensive list, see probability-distribution-related operators .

Symbol(s) Explanation Example ${\text{Bin}(n,p)}$ Binomial distribution with $n$ trials and probability of success $p$If $X$ stands for the number of heads in 10 coin tosses, then $X \sim \text{Bin}(10, 0.5)$. $\text{Geo}(p)$ Geometric distribution with probability of success $p$If $Y \sim \text{Geo}(1/5)$, then $E(Y)=5$. $U(a,b)$ Continuous uniform distribution from $a$ to $b$If $X \sim U(3,7)$, then $V(X)=\dfrac{(7-3)^2}{12}$. $N(\mu,\sigma^2)$ Normal distribution with mean $\mu$ and variance $\sigma^2$If $X \sim N(3, 5^2)$, then $\dfrac{X – 3}{5} \sim Z$. $z_{\alpha}$ Positive Z-score associated with significance level $\alpha$$z_{0.05} \approx 1.645$ $t_{\alpha, \nu}$ Positive t-score associated with significance level $\alpha$ and degree of freedom $\nu$$t_{0.05, 1000} \approx z_{0.05}$ $\chi^2_{\alpha, \nu}$ Chi-squared score associated with significance level $\alpha$ and degree of freedom $\nu$$\chi^2_{0.05, 30} \approx 43.77$ $F_{\alpha, \nu_1, \nu_2}$ F-score associated with significance level $\alpha$ and degrees of freedom $\nu_1$ and $\nu_2$$F_{0.05, 20, 20} \approx 2.1242$

Calculus-related OperatorsFor a more comprehensive list, see calculus and analysis symbols .

Symbol(s) Explanation Example $\displaystyle \lim_{n \to \infty} a_n$ Limit of sequence $a_n$$\displaystyle \lim_{n \to \infty} \dfrac{n+3}{2n} = \dfrac{1}{2}$ $\displaystyle \lim_{x \to c} f(x)$ Limit of function $f$ as $x$ tends to $c$$\displaystyle \lim_{x \to 3} \frac{\pi\sin x}{2} = \\[0.8em] \displaystyle \frac{\pi}{2}\lim_{x \to 3} \sin x$ $\text{sup}(A)$ Supremum (least upper bound) of set $A$$\text{sup}([ -3,5))=5$ $\text{inf}(A)$ Infimum (greatest lower bound) of set $A$If $\displaystyle B=\left\{\frac{1}{1}, \frac{1}{2}, \ldots\right\}$, then $\text{inf}(B)=0$. $f’, f^{\prime\prime}, f^{\prime\prime\prime}, f^{(n)}$ First, second, third and $n$th derivative of function $f$ $\left( \sin x \right)^{\prime\prime\prime} = -\cos (x)$ $\displaystyle \int_a^b f(x)\, \mathrm{d}x$ Definite integral of function $f$ from $a$ to $b$$ \displaystyle \int_0^1 \dfrac{1}{1+x^2}\, \mathrm{d}x = \dfrac{\pi}{4}$ $\displaystyle \int f(x)\,\mathrm{d}x$ Indefinite integral of function $f$$ \displaystyle \int \ln x\,\mathrm{d}x = \\ x\ln x-x+C$ $f_x$ Partial derivative of multivariate function $f$ with respect to $x$If $f(x,y)= x^2 y^3$, then $f_x (x,y) = 2xy^3$.

Relational SymbolsRelational symbols are used to express mathematical relations between multiple objects. Many relational symbols are binary in nature, in that they take two objects as inputs and turn them into complete, meaningful sentences (as in the case of the inequality symbol $<$).

Since relational symbols form the building blocks of mathematical sentences , they are of foundational importance in mathematics. The following tables document some of the most commonly-used relational symbols — along with their usage and meaning.

Equality-based Relational SymbolsSymbol(s) Explanation Example $x = y$ $x$ and $y$ are equal $3x-x = 2x$ $x \ne y$ $x$ and $y$ are non-equal $2 \ne 3$ $x \approx y$ $x$ and $y$ are approximately equal $\pi \approx 3.1416$ $x \sim y$, $xRy$ $x$ is related to $y$ (as defined by the relation $R$) $xRy$ if and only if $|x| = |y|$. $a \equiv b$ $x$ is equivalent to $y$ $2 \equiv 101$ in mod $33$ $f \propto g$ Function $f$ is proportional to function $g$ $V \propto r^3$

Comparison-based Relational SymbolsFor a more comprehensive list, see comparison-based relational symbols in algebra .

Symbol Explanation Example $x < y$ $x$ is less than $y$ $\sin(x) < 3$ $x > y$ $x$ is greater than $y$ $\pi > e$ $x \le y$ $x$ is less than or equal to $y$ $n! \le n^n$ $x \ge y$ $x$ is greater than or equal to $y$ $x^2 \ge 0$

Number-related Relational SymbolsSymbol Explanation Example $m \mid n$ Integer $m$ divides integer $n$ $101 \mid 1111$ $m \perp n$ Integers $m$ and $n$ are coprime $31 \perp 97$

Geometry-related Relational SymbolsFor more symbols in geometry and trigonometry, see geometry and trigonometry symbols .

Symbol Explanation Example $\ell_1 \parallel \ell_2$ Lines $\ell_1$ and $\ell_2$ are parallel $\overline{PQ} \parallel \overline{RS}$ $\ell_1 \perp \ell_2$ Lines $\ell_1$ and $\ell_2$ are perpendicular $\overrightarrow{AB} \perp \overrightarrow{BC}$ $F \sim F’$ Figure $F$ is similar to figure $F’$ $\triangle ABC \sim \triangle DEF$ $F \cong F’$ Figure $F$ is congruent to figure $F’$ $\square ABCD \cong \square PQRS$

Set-related Relational SymbolsFor a more comprehensive list, see relational symbols in set theory .

Symbol Explanation Example $a \in A$ Element $a$ is a membe r of set $A$ $\dfrac{2}{3} \in \mathbb{R}$ $a \notin A$ Element $a$ is not a member of set $A$ $\pi \notin \mathbb{Q}$ $A \subseteq B$ Set $A$ is a subset of set $B$ $A \cap B \subseteq A$ $A = B$ Set $A$ is equal to set $B$ If $A = B$, then $A \subseteq B$.

Logic-related Relational SymbolsFor a more comprehensive list, see relational symbols in logic .

Symbol Explanation Example ${P\! \implies \! Q}$ Sentence $P$ implies sentence $Q$ $x$ is even $\implies$ $2$ divides $x$ $P\! \impliedby \! Q$ Sentence $P$ is implied by sentence $Q$ $x = 3 \impliedby$ $3x+ 2 =11$ $P\!\iff \!Q,$ $P \equiv Q$ Sentence $P$ if and only if sentence $Q$ $x \ne y \iff$ $(x-y)^2 > 0$ $P \therefore Q$ $P$, therefore $Q$ $i \in \mathbb{C} \therefore$ $\exists z\, (z \in \mathbb{C})$ $P \because Q$ $P$, because $Q$ $x=\dfrac{\pi}{2} \because$ $\sin x = 1$ and $\cos x =0$

Probability-related Relational SymbolsFor a more comprehensive list, see relational symbols in probability and statistics .

Symbol Explanation Example $A \perp B$ Events $A$ and $B$ are independent Since $A \perp B$, we have that $P(A \cap B) = P(A)P(B).$ $X \sim F$ Random variable $X$ follows distribution $F$ $Y \sim \text{Bin}(30, 0.4)$

Calculus-related Relational SymbolsFor a more comprehensive list, see relational symbols in asymptotic analysis .

Symbol Explanation Example $f(x) \sim g(x)$ Function $f$ is asymptotically equal to function $g$ $\pi(x) \sim \dfrac{x}{\ln x}$ $f(x) \in O(g(x))$ Function $f$ is in the big-O of $g$ ($f$ “grows at most as fast” as $g$) $2x^2 + 3x + 3 \in O(x^2)$

Notational SymbolsA notational symbol is a convention or shorthand whose role is different from that of a constant, variable, delimiter, operator or relational symbol. It often simply delineates the notational system being used, and might even refer to concepts that have little bearing to any definite mathematical object (e.g., $\infty$).

Common Notational SymbolsSymbol(s) Explanation Example $\ldots, \cdots$ Horizontal ellipsis symbols to indicate a definite, unlisted pattern$1^2 + 2^2 + \cdots + n^2$ $\vdots, \ddots$ Vertical ellipsis symbols to indicate a definite, unlisted pattern$\left( \begin{smallmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn}\end{smallmatrix}\right)$ $f\!: A \to B$, $A \overset{f}{\to} B$ Function $f$ with domain $A$ and codomain $B$A function $g: \mathbb{N} \to \mathbb{R}$ can be thought of as a sequence. $x \mapsto f(x)$ Function rule mapping element $x$ to $f(x)$The function $x \mapsto x^2$ is increasing in the interval $[0, \infty)$. ${Q.E.D.}, \square, \blacksquare$ End-of-the-proof symbol Thus the result is established as desired. $\,\blacksquare$ $Q.E.A.$, ⨳ , ※ Contradiction symbol Multiplying both sides of the equation yields that $1=2$. ※

Notational Symbols in Geometry and TrigonometryFor more symbols in geometry and trigonometry, see geometry and trigonometry symbols .

Symbol Explanation Example $^{\circ}$ Degree symbol $\cos(90^{\circ}) = 0$ $’$ Arcminute symbol$35′ = \dfrac{35}{60}$ degrees $^{\prime\prime}$ Arcsecond symbol$20^{\prime\prime} = \left( \dfrac{20}{60} \right) ^{\prime}$ $\text{rad}$ Radian $\pi \text{ rad} = 180^{\circ}$ $\text{grad}$ Gradian $100 \text{ grad} = 90^{\circ}$ ∟ Right angle $-$, $=$, $\equiv$ Equal angle /length

Notational Symbols in CalculusFor more symbols in calculus, see calculus and analysis symbols .

Symbol Explanation Example $+\infty$ Positive infinity $\left(\dfrac{n^2 +1}{n}\right) \to +\infty$ $-\infty$ Negative infinity $\displaystyle \lim_{x \to -\infty} e^x =0$ $\Delta \mathbf{x}$ Change in variable $\mathbf{x}$$m = \dfrac{\Delta y}{\Delta x}$ $\mathrm{d} \mathbf{x}$ Differential of variable $\mathbf{x}$$\mathrm{d}y = f'(x)\, \mathrm{d}x$ $\partial \mathbf{x}$ Partial differential of variable $\mathbf{x}$$\dfrac{\partial f}{\partial x}\,\mathrm{d} x$ $\mathrm{d} \mathbf{f}$ Total differential of multivariate function $\mathbf{f}$$\mathrm{d} g(x,y) = \\ \dfrac{\partial g}{\partial x}\,\mathrm{d}x + \dfrac{\partial g}{\partial y}\,\mathrm{d}y$

Notational Symbols in Probability and StatisticsFor a more comprehensive list, see notational symbols in probability and statistics .

For lists of symbols categorized by type and subject , refer to the relevant pages below for more.

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