Common Mathematical Notation

Mort Yao
Symbols Semantics LaTeX
\(a\), \(b\) (known) variable or element of a set
\(c\) constant
\(e\) mathematical constant \(e\) (Euler’s number)
\(\pi\) mathematical constant \(\pi\) \pi
\(\varepsilon\) small positive infinitesimal quantity \varepsilon
\(i\) general integer; subscript or superscript; imaginary unit
\(j\), \(k\), \(l\) general integer; subscript or superscript
\(n\), \(m\) general natural number; total number
\(p\) general prime number; general probability
\(q\) general prime number
\(x\), \(y\) (unknown) variable or element of a set; coordinate
\(z\), \(w\) general complex number; coordinate
\(\vert x \vert\), \(\vert z \vert\) absolute value
\(\bar{z}\) complex conjugate \bar{z}
\(\Re(z)\); \(\operatorname{Re}(z)\) real part \Re{z}
\(\Im(z)\); \(\operatorname{Im}(z)\) imaginary part \Im(z)
\(\mathcal{E}\) experiment \mathcal{E}
\(A\), \(B\) (known) set; event
\(S\), \(\Omega\) set; class of sets; sample space
\(U\) universal set
\(A^\complement\); \(\overline{A}\); \(\complement_U A\) complement of set A^\complement
\(\varnothing\) empty set \varnothing
\(X\), \(Y\) (unknown) set; random variable
\(\mathcal{X}\), \(\mathcal{Y}\) set with special structure, space \mathcal{X}
\(\mathcal{P}(S)\); \(2^S\) power set \mathcal{P}(S)
\(\vert S \vert\); \(\operatorname{Card}(S)\) cardinality of set
\(\aleph_0\) cardinality of natural numbers (Aleph-naught) \aleph_0
\(\mathfrak{c}\) cardinality of continuum \mathfrak{c}
\(\mathbb{N}\) set of natural numbers \mathbb{N}
\(\mathbb{Z}^+\) set of positive integers \mathbb{Z}^+
\(\mathbb{Z}\) set of integers \mathbb{Z}
\(\mathbb{Q}\) set of rational numbers \mathbb{Q}
\(\mathbb{R}\) set of real numbers \mathbb{R}
\(\mathbb{R}^n\) \(n\)-dimensional real coordinate space \mathbb{R}^n
\(\mathbb{C}\) set of complex numbers \mathbb{C}
\(\mathbb{C}^n\) \(n\)-dimensional complex coordinate space \mathbb{C}^n
\(\mathbf{A}\), \(\mathbf{B}\), \(\mathbf{X}\), \(\mathbf{Y}\) matrix \mathbf{A}
\(\boldsymbol{a}\), \(\boldsymbol{b}\), \(\boldsymbol{x}\), \(\boldsymbol{y}\) vector \boldsymbol{a}
\(\boldsymbol{e}\), \(\boldsymbol{i}\), \(\boldsymbol{j}\), \(\boldsymbol{k}\) unit vector
\(\boldsymbol{v}\) eigenvector
\(\lambda\) eigenvalue \lambda
\(\vert\vert \boldsymbol{a} \vert\vert\), \(\vert\vert \mathbf{A} \vert\vert\) norm
\(\langle \boldsymbol{a},\boldsymbol{b} \rangle\) inner product of vectors
\(\mathbf{I}_n\) identity matrix of size \(n\)
\(\operatorname{diag}(a_1,\dots,a_n)\) diagonal matrix of size \(n\)
\(\delta_{ij}\) Kronecker delta \delta_{ij}
\(\mathbf{A}^{\rm T}\) transpose of matrix \mathbf{A}^{\rm T}
\(\mathbf{A}^{-1}\) inverse of matrix \mathbf{A}^{-1}
\(\operatorname{rank}(\mathbf{A})\) rank of matrix
\(\det(\mathbf{A})\) determinant of matrix \det(\mathbf{A})
\(f\), \(g\), \(h\) function or mapping
\(\varphi\) function or mapping with special property \varphi
\(\delta\) Dirac delta function \delta
\(\eta\) Dedekind eta function \eta
\(\psi\), \(\Psi\) wave function \psi
\(\wp\) Weierstrass’s elliptic function \wp
\(\operatorname{negl}(x)\) negligible function
\(\min(S)\) minimum \min(S)
\(\max(S)\) maximum \max(S)
\(\sup(S)\) supremum \sup(S)
\(\inf(S)\) infimum \inf(S)
\(\lim(f(x))\); \(\lim(a_n)\) limit \lim(f(x))
\(d{x}\) differential operator
\(\partial{x}\) partial differential operator \partial{x}
\(\Delta\) forward difference; Laplace operator \Delta
\(\nabla\) backward difference; gradient \nabla
\(e^{x}\); \(\exp(x)\) exponential function
\(\ln(x)\) natural logarithm \ln(x)
\(\log(x)\) logarithm \log(x)
\(\mathcal{A}\) algorithm \mathcal{A}
\(\mathcal{O}(f(n))\) big-O notation \mathcal{O}(f(n))
\(\Omega(f(n))\) big-Omega notation \Omega(f(n))
\(\Theta(f(n))\) big-Theta notation \Theta(f(n))
\(p_X \left({x}\right)\) probability mass function (pmf) p_X(x)
\(\Pr[X=x]\) probability measure \Pr[X=x]
\(\operatorname{E}[X]\); \(\mu\) expectation \operatorname{E}[X]
\(\operatorname{Cov}(X,Y)\) covariance \operatorname{Cov}(X,Y)
\(\operatorname{Var}(X)\) variance \operatorname{Var}(X)
\(\sigma\) standard deviation \sigma
\(s\) (unbiased) sample standard deviation
\(\bar{X}\) sample mean \bar{X}
\(X \sim \operatorname{Pois}(\lambda)\) random variable that satisfies a probability distribution \sim