Mathematical Symbols Guide

Comprehensive guide to mathematical symbols in LaTeX. Learn operators, relations, delimiters, and special notation for professional mathematical documents.

This guide provides detailed coverage of mathematical symbols in LaTeX, organized by category with examples and best practices.

Quick tip: For a searchable symbol reference table, see our Symbol Reference page.

Quick Symbol Reference

Common Mathematical Symbols

Symbol	LaTeX	Symbol	LaTeX
+	+	-	-
×	\times	÷	\div
±	\pm	∓	\mp
·	\cdot	∗	\ast
⋆	\star	∘	\circ
⊕	\oplus	⊖	\ominus
⊗	\otimes	⊘	\oslash
=	=	≠	\neq
<	<	>	>
≤	\leq	≥	\geq
≪	\ll	≫	\gg
≈	\approx	∼	\sim
≃	\simeq	≅	\cong
≡	\equiv	∝	\propto

Set Theory and Logic Symbols

Symbol	LaTeX	Symbol	LaTeX
∈	\in	∉	\notin
⊂	\subset	⊆	\subseteq
⊃	\supset	⊇	\supseteq
∪	\cup	∩	\cap
∅	\emptyset	∅	\varnothing
∀	\forall	∃	\exists
∄	\nexists	∴	\therefore
∵	\because	⇒	\implies
⇔	\iff	¬	\neg
∧	\land	∨	\lor

Calculus and Analysis

Symbol	LaTeX	Symbol	LaTeX
∑	\sum	∏	\prod
∫	\int	∮	\oint
∂	\partial	∇	\nabla
∞	\infty	lim	\lim
→	\to	→	\rightarrow
⇒	\Rightarrow	⇔	\Leftrightarrow

Mathematical Operators

Basic Arithmetic Operators

basic-operators.tex

\documentclass{article}
\usepackage{amsmath}
\begin{document}

% Standard operators
$a + b - c \times d \div e$

% Alternative multiplication
$a \cdot b$ or $a \ast b$ or $a \star b$

% Plus/minus and minus/plus
$x = a \pm b$, $y = c \mp d$

% Advanced operators
$a \oplus b \ominus c \otimes d \oslash e$

% Fractions and ratios
$\frac{a}{b}$, $a/b$, $a:b$

\end{document}

Binary Operators

binary-operators.tex

% Set operations
$A \cup B$ (union)
$A \cap B$ (intersection)
$A \setminus B$ (set difference)
$A \triangle B$ (symmetric difference)

% Logic operations
$p \land q$ or $p \wedge q$ (and)
$p \lor q$ or $p \vee q$ (or)
$\neg p$ or $\lnot p$ (not)
$p \oplus q$ (exclusive or)

% Other binary operators
$a \circ b$ (composition)
$a \bullet b$ (bullet)
$a \diamond b$ (diamond)
$a \Box b$ (box)

Rendered output:

A \cup B → A ∪ B (union)

A \cap B → A ∩ B (intersection)

p \land q → p ∧ q (logical and)

Large Operators

large-operators.tex

% Summation
$\sum_{i=1}^{n} a_i$
$\displaystyle\sum_{i=1}^{n} a_i$ % Larger in inline

% Product
$\prod_{i=1}^{n} a_i$

% Integrals
$\int_a^b f(x)\,dx$
$\iint_D f(x,y)\,dA$
$\iiint_V f(x,y,z)\,dV$
$\oint_C F \cdot dr$

% Unions and intersections
$\bigcup_{i=1}^{n} A_i$
$\bigcap_{i=1}^{n} A_i$

% Other large operators
$\coprod$ (coproduct)
$\bigoplus$ (direct sum)
$\bigotimes$ (tensor product)
$\bigvee$ (join)
$\bigwedge$ (meet)

Rendered output:

\sum_{i=1}^{n} a_i → Σ(i=1 to n) aᵢ

\int_a^b f(x)\,dx → ∫ᵃᵇ f(x) dx

\bigcup_{i=1}^{n} A_i → ⋃(i=1 to n) Aᵢ

Relations and Comparisons

Basic Relations

relations.tex

\documentclass{article}
\usepackage{amssymb}
\begin{document}

% Equality and inequality
$a = b$, $a \neq b$ or $a \ne b$

% Comparisons
$a < b$, $a > b$
$a \leq b$ or $a \le b$
$a \geq b$ or $a \ge b$

% Much less/greater
$a \ll b$, $a \gg b$

% Approximately equal
$a \approx b$ (approximately)
$a \simeq b$ (similar equal)
$a \sim b$ (similar)
$a \cong b$ (congruent)

% Equivalence
$a \equiv b$ (equivalent)
$a \triangleq b$ (defined as)

\end{document}

Set Relations

set-relations.tex

% Membership
$x \in A$ (element of)
$x \notin A$ (not element of)
$A \ni x$ (contains)

% Subset relations
$A \subset B$ (proper subset)
$A \subseteq B$ (subset or equal)
$A \supset B$ (proper superset)
$A \supseteq B$ (superset or equal)

% Special subsets
$A \sqsubset B$ (square subset)
$A \sqsubseteq B$
$A \subsetneq B$ (subset not equal)

% Parallel and perpendicular
$a \parallel b$
$a \perp b$

Advanced Relations

advanced-relations.tex

% Proportional and asymptotic
$y \propto x$ (proportional to)
$f(x) \asymp g(x)$ (asymptotic)

% Order relations
$a \prec b$ (precedes)
$a \preceq b$ (precedes or equal)
$a \succ b$ (succeeds)
$a \succeq b$ (succeeds or equal)

% Other relations
$a \models b$ (models)
$a \vdash b$ (proves)
$a \dashv b$ (reverse proves)
$a \smile b$ (smile)
$a \frown b$ (frown)

Arrows

Basic Arrows

basic-arrows.tex

% Single arrows
$\rightarrow$ or $\to$
$\leftarrow$ or $\gets$
$\leftrightarrow$
$\uparrow$, $\downarrow$
$\updownarrow$

% Double arrows
$\Rightarrow$ (implies)
$\Leftarrow$ (implied by)
$\Leftrightarrow$ or $\iff$ (if and only if)
$\Uparrow$, $\Downarrow$
$\Updownarrow$

% Long arrows
$\longrightarrow$
$\longleftarrow$
$\longleftrightarrow$
$\Longrightarrow$
$\Longleftarrow$
$\Longleftrightarrow$

Special Arrows

special-arrows.tex

% Maps to
$f: A \to B$ or $f: A \rightarrow B$
$x \mapsto f(x)$
$x \longmapsto f(x)$

% Hooked arrows
$\hookrightarrow$ (injection)
$\hookleftarrow$

% Two-headed arrows
$\twoheadrightarrow$ (surjection)
$\twoheadleftarrow$

% Harpoons
$\rightharpoonup$, $\rightharpoondown$
$\leftharpoonup$, $\leftharpoondown$
$\rightleftharpoons$

% Diagonal arrows
$\nearrow$ (northeast)
$\searrow$ (southeast)
$\swarrow$ (southwest)
$\nwarrow$ (northwest)

Delimiters

Brackets and Parentheses

delimiters.tex

\documentclass{article}
\usepackage{amsmath}
\begin{document}

% Basic delimiters
$(a + b)$
$[a + b]$
$\{a + b\}$
$\langle a, b \rangle$

% Floor and ceiling
$\lfloor x \rfloor$ (floor)
$\lceil x \rceil$ (ceiling)

% Absolute value and norm
$|x|$ or $\lvert x \rvert$
$\|x\|$ or $\lVert x \rVert$

% Automatic sizing
$\left( \frac{a}{b} \right)$
$\left[ \sum_{i=1}^{n} a_i \right]$
$\left\{ x : x > 0 \right\}$

% Manual sizing
$\big( \Big( \bigg( \Bigg($
$\big] \Big] \bigg] \Bigg]$

\end{document}

Advanced Delimiters

advanced-delimiters.tex

% Mixed delimiters
$\left( a, b \right]$ (half-open interval)
$\left[ a, b \right)$

% Invisible delimiters
$\left. \frac{df}{dx} \right|_{x=0}$

% Multiple sizes
\begin{align}
&\text{Small: } (x) \\
&\text{big: } \big(x\big) \\
&\text{Big: } \Big(x\Big) \\
&\text{bigg: } \bigg(x\bigg) \\
&\text{Bigg: } \Bigg(x\Bigg)
\end{align}

% Angle brackets for inner products
$\langle x, y \rangle$
$\langle x \mid y \rangle$ % Quantum mechanics

Greek Letters Extended

Variant Forms

greek-variants.tex

% Standard vs variant forms
$\epsilon$ vs $\varepsilon$
$\theta$ vs $\vartheta$
$\pi$ vs $\varpi$
$\rho$ vs $\varrho$
$\sigma$ vs $\varsigma$
$\phi$ vs $\varphi$

% Bold Greek (requires bm package)
\usepackage{bm}
$\bm{\alpha}$, $\bm{\beta}$, $\bm{\Omega}$

% Upright Greek (requires upgreek)
\usepackage{upgreek}
$\upalpha$, $\upbeta$, $\upgamma$

Special Notation

Dots and Accents

dots-accents.tex

% Dots
$a_1 + a_2 + \cdots + a_n$ (centered dots)
$a_1, a_2, \ldots, a_n$ (low dots)
$\vdots$ (vertical dots)
$\ddots$ (diagonal dots)

% Over and under dots
$\dot{x}$ (first derivative)
$\ddot{x}$ (second derivative)
$\dddot{x}$ (third derivative)
$\ddddot{x}$ (fourth derivative)

% Accents
$\hat{a}$ (hat)
$\check{a}$ (check)
$\tilde{a}$ (tilde)
$\acute{a}$ (acute)
$\grave{a}$ (grave)
$\bar{a}$ (bar)
$\vec{a}$ (vector)
$\breve{a}$ (breve)

% Wide accents
$\widehat{ABC}$
$\widetilde{xyz}$
$\overline{a + b}$
$\underline{a + b}$

Over and Under Operations

over-under.tex

% Overbraces and underbraces
$\overbrace{a + b + c}^{\text{sum}}$
$\underbrace{x \cdot x \cdot x}_{n \text{ times}}$

% Overlining and underlining
$\overline{AB}$ (line segment)
$\underline{important}$

% Stacking
$\overset{?}{=}$ (question over equals)
$\underset{n \to \infty}{\lim}$ (limit notation)
$\overset{def}{=}$ (defined as)

Mathematical Alphabets

Special Font Commands

math-alphabets.tex

% Blackboard bold (requires amssymb)
$\mathbb{N}$ (natural numbers)
$\mathbb{Z}$ (integers)
$\mathbb{Q}$ (rationals)
$\mathbb{R}$ (reals)
$\mathbb{C}$ (complex)

% Calligraphic
$\mathcal{A}$, $\mathcal{B}$, $\mathcal{L}$

% Fraktur (requires amssymb)
$\mathfrak{a}$, $\mathfrak{g}$, $\mathfrak{H}$

% Script (requires mathrsfs)
\usepackage{mathrsfs}
$\mathscr{A}$, $\mathscr{F}$, $\mathscr{L}$

% Bold
$\mathbf{x}$, $\mathbf{A}$
$\boldsymbol{\alpha}$ (bold Greek)

Spacing in Math Mode

Manual Spacing

math-spacing.tex

% Spacing commands
$ab$ (no space)
$a\,b$ (thin space)
$a\:b$ (medium space)
$a\;b$ (thick space)
$a\quad b$ (quad space)
$a\qquad b$ (double quad)

% Negative spacing
$a\!b$ (negative thin space)

% Common uses
$\int f(x)\,dx$ (before dx)
$n\text{-}th$ (hyphen in text)
$5\,\text{cm}$ (before units)

% Text in math
$x \in \mathbb{R} \text{ such that } x > 0$

Common Symbol Combinations

Physics Notation

physics-symbols.tex

% Derivatives
$\frac{d}{dx}$, $\frac{\partial}{\partial x}$
$\nabla$ (gradient)
$\nabla \cdot$ (divergence)
$\nabla \times$ (curl)
$\Box$ or $\square$ (d'Alembertian)

% Quantum mechanics
$\hbar$ (reduced Planck constant)
$\langle \psi | \phi \rangle$ (inner product)
$| \psi \rangle$ (ket)
$\langle \phi |$ (bra)

% Units
$^\circ$ (degree)
$\AA$ (Angstrom)

Statistics and Probability

statistics-symbols.tex

% Probability
$P(A \mid B)$ (conditional)
$\mathbb{E}[X]$ (expectation)
$\text{Var}(X)$ (variance)
$X \sim N(\mu, \sigma^2)$ (distribution)

% Statistics
$\bar{x}$ (mean)
$\hat{\theta}$ (estimator)
$s^2$ (sample variance)
$r$ (correlation)

Best Practices

Symbol usage guidelines:

1. Consistency: Use the same notation throughout your document
2. Standards: Follow field-specific conventions
3. Clarity: Define non-standard symbols
4. Spacing: Use proper spacing around operators
5. Size: Use \displaystyle in important inline formulas
6. Packages: Load necessary packages (amsmath, amssymb)

Troubleshooting

Common symbol issues:

1. Missing symbols: Add \usepackage{amssymb}
2. Wrong size: Use \displaystyle or display math
3. Spacing issues: Use manual spacing commands
4. Font issues: Check if special packages needed
5. Encoding: Use \usepackage[utf8]{inputenc}

Quick Reference Card

Essential Symbols

Category	Symbols
Greek	\alpha \beta \gamma \delta \epsilon \theta \lambda \mu \pi \sigma \phi \omega
Operators	\sum \prod \int \cup \cap \oplus \otimes
Relations	\leq \geq \neq \approx \equiv \sim \subset \in
Arrows	\to \gets \leftrightarrow \Rightarrow \mapsto
Delimiters	\{ \} \langle \rangle \lfloor \rfloor
Accents	\hat{} \tilde{} \bar{} \vec{} \dot{}

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Next: Master Matrices and arrays for structured mathematical layouts, or explore Scientific notation for physics and chemistry.