amsmath

amsmath + amssymb + mathtools — Math Typesetting

CTAN: https://ctan.org/pkg/amsmath | https://ctan.org/pkg/mathtools
Manual: texdoc amsmath, texdoc mathtools

Setup

\usepackage{amsmath}    % core math environments
\usepackage{amssymb}    % extra symbols (loads amsfonts)
\usepackage{mathtools}  % fixes + extensions for amsmath (loads amsmath)
\usepackage{amsthm}     % theorem environments

% mathtools loads amsmath, so you can just use:
\usepackage{mathtools}
\usepackage{amssymb}
\usepackage{amsthm}

Equation Environments

Single Equations

% Numbered
\begin{equation}
  E = mc^2
  \label{eq:einstein}
\end{equation}

% Unnumbered
\begin{equation*}
  E = mc^2
\end{equation*}
% or: \[ E = mc^2 \]

align — Multiple Aligned Equations

\begin{align}
  f(x) &= x^2 + 2x + 1 \label{eq:f} \\
       &= (x+1)^2       \label{eq:f2}
\end{align}

% Unnumbered: align*
% Suppress single number: \nonumber or \notag before \\

gather — Centered, No Alignment

\begin{gather}
  x + y = 1 \\
  x - y = 3
\end{gather}

multline — Long Single Equation

\begin{multline}
  f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 \\
    + a_4 x^4 + a_5 x^5 + a_6 x^6
\end{multline}

First line left-aligned, last line right-aligned, middle centered.

split — Sub-alignment Inside equation

\begin{equation}
\begin{split}
  f(x) &= a + b + c \\
       &\quad + d + e
\end{split}
\end{equation}

Single equation number for the group.

cases

\begin{equation}
  |x| = \begin{cases}
    x  & \text{if } x \geq 0 \\
    -x & \text{if } x < 0
  \end{cases}
\end{equation}

% mathtools: dcases (displaystyle), rcases (right brace)
\begin{dcases}
  \frac{x}{2} & x > 0 \\
  0            & x = 0
\end{dcases}

aligned / gathered — Inline Sub-environments

% Use inside equation or \[ \] for sub-alignment
\begin{equation}
  \left\{
  \begin{aligned}
    2x + 3y &= 7 \\
    x - y   &= 1
  \end{aligned}
  \right.
\end{equation}

subequations

\begin{subequations} \label{eq:system}
\begin{align}
  x + y &= 1 \label{eq:system_a} \\
  x - y &= 3 \label{eq:system_b}
\end{align}
\end{subequations}
% Produces (1a), (1b)

Environment Summary

Environment	Alignment	Numbering	Use for
equation	centered	one number	single equation
align	& columns	per line	aligned equations
gather	centered	per line	unrelated equations
multline	first L, last R	one number	long equation
split	& columns	one number (parent)	sub-align in equation
cases	& columns	none (in equation)	piecewise functions
aligned	& columns	none (in equation)	inline sub-alignment
flalign	& columns	per line	full-width alignment

Add * for unnumbered variants (except split, aligned, gathered).

Text in Math

\text{if }           % normal text (respects surrounding font)
\mathrm{const}       % upright roman
\textit{word}        % italic text
\mathit{diff}        % math italic (different spacing than default)
\mathbf{v}           % bold (not for Greek — use \boldsymbol)
\boldsymbol{\alpha}  % bold Greek
\mathbb{R}           % blackboard bold: ℝ (amssymb)
\mathcal{L}          % calligraphic: ℒ
\mathfrak{g}         % Fraktur (amssymb)
\mathsf{X}           % sans-serif
\mathtt{code}        % monospace
\operatorname{span}  % upright operator (one-off)

Spacing in Math

Command	Width	Example
\,	3/18 em (thin)	\int f(x)\, dx
\: or \>	4/18 em (medium)
\;	5/18 em (thick)
\!	−3/18 em (negative thin)	\!\! to tighten
\quad	1 em	x \quad y
\qquad	2 em
\phantom{x}	space of "x"	alignment trick
\hphantom{x}	horizontal only
\vphantom{x}	vertical only

Operators

% Built-in: \sin, \cos, \tan, \log, \ln, \exp, \min, \max,
%   \sup, \inf, \lim, \det, \dim, \ker, \gcd, \Pr, \hom, ...

% Custom operator (preamble)
\DeclareMathOperator{\tr}{tr}       % like \sin
\DeclareMathOperator*{\argmax}{arg\,max}  % limits below in display

% Usage
\tr(A) = \sum_i a_{ii} \qquad x^* = \argmax_x f(x)

Matrices

\begin{pmatrix} a & b \\ c & d \end{pmatrix}   % ( )
\begin{bmatrix} a & b \\ c & d \end{bmatrix}   % [ ]
\begin{Bmatrix} a & b \\ c & d \end{Bmatrix}   % { }
\begin{vmatrix} a & b \\ c & d \end{vmatrix}   % | |  (determinant)
\begin{Vmatrix} a & b \\ c & d \end{Vmatrix}   % ‖ ‖
\begin{matrix}  a & b \\ c & d \end{matrix}    % no delimiters

% Small inline matrix
$\bigl(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\bigr)$

Max columns default = 10. For more: \setcounter{MaxMatrixCols}{20}

Delimiters

% Auto-sizing (use sparingly — can oversize)
\left( \frac{a}{b} \right)
\left[ \sum_i x_i \right]
\left\{ x \in \mathbb{R} \mid x > 0 \right\}
\left. \frac{df}{dx} \right|_{x=0}   % \left. = invisible delimiter

% Manual sizing (preferred for control)
\bigl(  \bigr)    % slightly bigger
\Bigl(  \Bigr)    % bigger
\biggl( \biggr)   % even bigger
\Biggl( \Biggr)   % biggest

% mathtools paired delimiters (best approach)
\DeclarePairedDelimiter\abs{\lvert}{\rvert}
\DeclarePairedDelimiter\norm{\lVert}{\rVert}
\DeclarePairedDelimiter\ceil{\lceil}{\rceil}
\DeclarePairedDelimiter\floor{\lfloor}{\rfloor}
\DeclarePairedDelimiter\set{\{}{\}}

% Usage:
\abs{x}       % |x|  (no sizing)
\abs*{x}      % \left|x\right|  (auto-sizing)
\abs[\big]{x} % \bigl|x\bigr|  (manual)

mathtools Additions

% Colon equals
\coloneqq   % :=
\eqqcolon   % =:

% dcases (displaystyle fractions in cases)
\begin{dcases} ... \end{dcases}

% rcases (right brace)
\begin{rcases} ... \end{rcases}

% Short intertext (less vertical space than \intertext)
\begin{align}
  f(x) &= x^2 \\
  \shortintertext{where}
  x &> 0
\end{align}

% Prescript (left sub/superscript)
\prescript{14}{6}{\mathrm{C}}   % ¹⁴₆C

% Cramped styles
\cramped{x^{x^x}}  % reduces height

% Smashed operator limits
\smashoperator{\sum_{i=1}^{n}}

Symbol Reference

Greek Letters

Lower	Upper	Lower	Upper
\alpha α	—	\nu ν	—
\beta β	—	\xi ξ	\Xi Ξ
\gamma γ	\Gamma Γ	\pi π	\Pi Π
\delta δ	\Delta Δ	\rho ρ	—
\epsilon ε	—	\sigma σ	\Sigma Σ
\varepsilon ε	—	\tau τ	—
\zeta ζ	—	\upsilon υ	\Upsilon Υ
\eta η	—	\phi ϕ	\Phi Φ
\theta θ	\Theta Θ	\varphi φ	—
\iota ι	—	\chi χ	—
\kappa κ	—	\psi ψ	\Psi Ψ
\lambda λ	\Lambda Λ	\omega ω	\Omega Ω
\mu μ	—

Variants: \varepsilon, \vartheta, \varphi, \varrho, \varsigma

Relations

Symbol	Command	Symbol	Command
≤	\leq	≥	\geq
≪	\ll	≫	\gg
≠	\neq	≈	\approx
∼	\sim	≃	\simeq
≡	\equiv	≅	\cong
∝	\propto	∈	\in
∉	\notin	⊂	\subset
⊆	\subseteq	⊃	\supset
⊇	\supseteq	⊄	\not\subset
≺	\prec	≻	\succ
⊥	\perp	∥	\parallel
⊢	\vdash	⊣	\dashv
⊨	\models

Binary Operators

Symbol	Command	Symbol	Command
±	\pm	∓	\mp
×	\times	÷	\div
·	\cdot	∗	\ast
⊕	\oplus	⊗	\otimes
∪	\cup	∩	\cap
∨	\vee	∧	\wedge
∘	\circ	•	\bullet
†	\dagger	‡	\ddagger

Arrows

Symbol	Command	Symbol	Command
→	\to / \rightarrow	←	\leftarrow
⇒	\Rightarrow	⇐	\Leftarrow
↔	\leftrightarrow	⇔	\Leftrightarrow
↦	\mapsto	⟶	\longrightarrow
↑	\uparrow	↓	\downarrow
⇑	\Uparrow	⇓	\Downarrow
↗	\nearrow	↘	\searrow
⟹	\implies	⟸	\impliedby
⟺	\iff
↪	\hookrightarrow	↠	\twoheadrightarrow

Big Operators

Symbol	Command	Symbol	Command
∑	\sum	∏	\prod
∫	\int	∮	\oint
∬	\iint	∭	\iiint
⋃	\bigcup	⋂	\bigcap
⨁	\bigoplus	⨂	\bigotimes
⋁	\bigvee	⋀	\bigwedge
∐	\coprod

Dots

Command	Output	Use
\cdots	⋯	between operators: $a + \cdots + z$
\ldots	…	in lists: $a_1, \ldots, a_n$
\vdots	⋮	vertical
\ddots	⋱	diagonal (matrices)

Misc Symbols (amssymb)

Symbol	Command	Symbol	Command
ℝ	\mathbb{R}	ℂ	\mathbb{C}
ℤ	\mathbb{Z}	ℕ	\mathbb{N}
ℚ	\mathbb{Q}	∅	\emptyset / \varnothing
∞	\infty	∂	\partial
∇	\nabla	ℓ	\ell
ℏ	\hbar	∀	\forall
∃	\exists	¬	\neg
√	\sqrt{x}	ⁿ√	\sqrt[n]{x}
∠	\angle	△	\triangle
⊤	\top	⊥	\bot
♠	\spadesuit	♣	\clubsuit

Accents

Command	Result	Use
\hat{a}	â	unit vectors
\bar{a}	ā	averages, conjugates
\vec{a}	a⃗	vectors
\dot{a}	ȧ	time derivative
\ddot{a}	ä	second derivative
\tilde{a}	ã	approximations
\widehat{abc}	wide hat	spanning
\widetilde{abc}	wide tilde	spanning
\overline{abc}	overline	sets, conjugates
\underline{abc}	underline
\overbrace{a+b}^{n}	over-brace	grouping
\underbrace{a+b}_{n}	under-brace	grouping

Theorem Environments (amsthm)

\usepackage{amsthm}

% Define theorem types (preamble)
\newtheorem{theorem}{Theorem}[section]     % numbered by section
\newtheorem{lemma}[theorem]{Lemma}         % shares theorem counter
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

% Usage
\begin{theorem}[Pythagoras]
  For a right triangle: $a^2 + b^2 = c^2$.
\end{theorem}

\begin{proof}
  Obvious. % ends with □ automatically
  % Use \qedhere to place □ at end of equation:
  % \[ a^2 + b^2 = c^2 \qedhere \]
\end{proof}

Theorem Styles

Style	Body font	Header	Use for
plain (default)	italic	bold	theorems, lemmas
definition	upright	bold	definitions, examples
remark	upright	italic	remarks, notes

Common Pitfalls

Problem	Cause	Fix
$$...$$	TeX primitive, bad spacing	Use \[...\] or equation*
eqnarray	Bad spacing around =	Use align instead (always)
Equation number on wrong line	\\ after last line	Remove trailing \\
\left...\right across lines	Can't span \\	Use \bigl...\bigr manually
Bold Greek	\mathbf doesn't work	Use \boldsymbol{\alpha}
Missing \displaystyle	Fractions small inline	Use \dfrac (or \displaystyle\frac)
:= spacing wrong	Colon treated as relation	Use \coloneqq (mathtools)
\mid vs | vs `	`	Different spacing
Overfull hbox in equation	Equation too wide	Use multline, split, or manual breaks
Proof box missing	\qed inside environment	Use \qedhere at the end