Maths notation explorer

94 symbols you will meet in machine learning papers and code: how to say each one, what it means, a tiny example, and the NumPy or PyTorch line that does the same thing.

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Paste its LaTeX (or the Unicode symbols). Each symbol the explorer knows is listed with its card.

94 symbols

Arithmetic and relations

$=$ equals“equals”Beginner
$\coloneqq$ is defined as“is defined as”Beginner
$\approx$ approximately equal“is approximately”Beginner
$\neq$ not equal“is not equal to”Beginner
$\leq$ less than or equal“is at most”Beginner
$\ll$ much less than“is much smaller than”Intermediate
$\propto$ proportional to“is proportional to”Intermediate
$\equiv$ identical to“is identically equal to”Intermediate
$\pm$ plus or minus“plus or minus”Beginner
$\sqrt{x}$ square root“root of”Beginner
$x^{2}$ exponent“x squared”Beginner
$a \times b$ times / dimension“times / by”Beginner

Number systems

$\mathbb{R}$ real numbers“R / the reals”Beginner
$\mathbb{R}^{n}, \mathbb{R}^{m\times n}$ real vector / matrix space“R n / R m by n”Beginner
$\mathbb{N}$ natural numbers“N / the naturals”Beginner
$\mathbb{Z}$ integers“Z / the integers”Beginner
$\mathbb{C}$ complex numbers“C / the complex numbers”Advanced

Sets

$\in$ element of“is in / belongs to”Beginner
$\notin$ not an element of“is not in”Beginner
$\subseteq$ subset“is a subset of”Beginner
$\subset$ proper subset (usually)“is a strict subset of”Intermediate
$\cup$ union“union”Beginner
$\cap$ intersection“intersect”Beginner
$\setminus$ set difference“minus / without”Beginner
$\emptyset$ empty set“the empty set”Beginner
$\{x \mid P(x)\}$ set-builder notation“the set of x such that…”Beginner
$A \times B$ Cartesian product“A cross B”Intermediate

Logic

$\forall$ for all“for all / for every”Beginner
$\exists$ there exists“there exists”Beginner
$\Rightarrow,\ \iff$ implies / if and only if“implies / if and only if”Beginner
$\therefore$ therefore“therefore”Beginner

Functions

$f: \mathbb{R}^{n} \to \mathbb{R}$ function arrow (type)“f maps R n to R”Beginner
$x \mapsto x^2$ maps to“x goes to x squared”Intermediate
$f \circ g$ composition“f after g”Intermediate
$\hat{y}$ hat“y hat”Beginner
$\bar{x}$ bar“x bar”Beginner
$\theta^{*}$ star“theta star”Intermediate
$e^{x},\ \exp(x)$ exponential“e to the x”Beginner
$\log x,\ \ln x$ logarithm“log of x”Beginner
$\operatorname{softmax}(\mathbf{z})_i = \frac{e^{z_i}}{\sum_j e^{z_j}}$ softmax“softmax of z”Beginner

Sums and products

$\sum_{i=1}^{n} x_i$ summation“sum from i equals 1 to n of x i”Beginner
$\prod_{i=1}^{n} x_i$ product“product from i equals 1 to n of x i”Beginner
$n!$ factorial“n factorial”Beginner
$\binom{n}{k}$ binomial coefficient“n choose k”Intermediate

Brackets and norms

$|x|$ absolute value“absolute value of x”Beginner
$\lfloor x \rfloor,\ \lceil x \rceil$ floor / ceiling“floor of x / ceiling of x”Beginner
$[a,b],\ (a,b)$ intervals“closed / open interval a to b”Beginner
$\langle \mathbf{a}, \mathbf{b} \rangle$ inner product“inner product of a and b”Intermediate
$\lVert \mathbf{x} \rVert_2$ norm“the norm of x / the 2-norm of x”Beginner

Greek letters and conventions

$\mu$ mu (mean)“mu”Beginner
$\sigma,\ \sigma^2,\ \sigma(z)$ sigma“sigma / sigma squared / sigmoid of z”Beginner
$\eta$ eta (learning rate)“eta”Beginner
$\alpha$ alpha“alpha”Beginner
$\lambda$ lambda“lambda”Beginner
$\theta$ theta (parameters)“theta”Beginner
$\epsilon$ epsilon“epsilon”Beginner
$\pi$ pi“pi”Beginner

Linear algebra

$\mathbf{x}$ bold lowercase (vector)“vector x”Beginner
$\mathbf{X},\ A$ capital letter (matrix)“matrix X”Beginner
$x_i,\ x_{ij},\ x^{(i)}$ subscripts & superscript indices“x sub i / x i j / x superscript i”Beginner
$\mathbf{A}^{\top}$ transpose“A transpose”Beginner
$\mathbf{a}\cdot\mathbf{b}$ dot product“a dot b”Beginner
$\mathbf{a}\odot\mathbf{b}$ Hadamard (element-wise) product“a element-wise times b”Intermediate
$\mathbf{a}\times\mathbf{b}$ cross product (3-D)“a cross b”Advanced
$\mathbf{a}\otimes\mathbf{b}$ tensor / outer / Kronecker product“a tensor b”Advanced
$\mathbf{A}^{-1}$ matrix inverse“A inverse”Intermediate
$\mathbf{I}_n$ identity matrix“identity / I n”Beginner
$\det(\mathbf{A}),\ |\mathbf{A}|$ determinant“determinant of A”Intermediate
$\operatorname{tr}(\mathbf{A})$ trace“trace of A”Intermediate
$\mathbf{A}\mathbf{v} = \lambda\mathbf{v}$ eigenvalue / eigenvector“A v equals lambda v”Intermediate
$\operatorname{rank}(\mathbf{A})$ rank“rank of A”Intermediate
$\operatorname{diag}(\mathbf{v})$ diagonal matrix“diag of v”Intermediate

Calculus

$\frac{df}{dx},\ f'(x)$ derivative“d f d x / f prime of x”Beginner
$\frac{\partial f}{\partial x}$ partial derivative“partial f partial x / del f del x”Beginner
$\nabla_{\theta} L$ nabla / gradient“grad L / del L with respect to theta”Beginner
$\mathbf{J} = \frac{\partial \mathbf{f}}{\partial \mathbf{x}}$ Jacobian“the Jacobian of f”Advanced
$\mathbf{H} = \nabla^{2} f$ Hessian“the Hessian of f”Advanced
$\int_a^b f(x)\,dx$ integral“integral from a to b of f of x d x”Intermediate
$\lim_{x\to a} f(x)$ limit“the limit as x approaches a of f of x”Intermediate
$\infty$ infinity“infinity”Beginner
$\Delta x$ Delta (change)“delta x / change in x”Beginner
$\dot{x}$ Newton's dot (time derivative)“x dot”Advanced

Probability and statistics

$P(A),\ p(x)$ probability“P of A / p of x”Beginner
$P(A \mid B)$ conditional probability“probability of A given B”Beginner
$a \mid b$ vertical bar — many meanings“given / such that / divides”Intermediate
$\mathbb{E}_{x\sim p}[f(x)]$ expectation“expected value of f of x, x drawn from p”Beginner
$x \sim p(x)$ is distributed as / sampled from“x is drawn from p”Beginner
$\mathcal{N}(\mu, \sigma^2)$ normal (Gaussian) distribution“normal with mean mu and variance sigma squared”Beginner
$\boldsymbol{\Sigma} = \operatorname{Cov}(\mathbf{x})$ covariance matrix“Sigma / covariance of x”Intermediate
$\operatorname{Var}(X)$ variance“variance of X”Beginner
$D_{\mathrm{KL}}(P \,\Vert\, Q)$ KL divergence“K L divergence of P from Q”Advanced
$\mathbb{1}[y = k]$ indicator function“indicator of y equals k”Intermediate

Optimization

$\arg\min_{\theta} L(\theta)$ arg min / arg max“the theta that minimizes L”Beginner
$\min_{x} f(x),\ \max(a, b)$ min / max“min over x of f of x”Beginner

No symbol matches that. Try a name (gradient), a symbol (∇) or a LaTeX command (\nabla).

References and further reading

The topics in this section follow the standard mathematical foundations of machine learning. All explanations, examples, exercises and visualizations on Epoch Zero are original.

For deeper study we recommend Mathematics for Machine Learning by Marc Peter Deisenroth, A. Aldo Faisal and Cheng Soon Ong (Cambridge University Press), at mml-book.github.io; Basics of Linear Algebra for Machine Learning by Jason Brownlee; and Wikipedia’s Glossary of mathematical symbols (CC BY-SA 4.0).

The “read an equation” approach is inspired by a talk by Duncan MacFarlane (UT Dallas).