LaTeX Math Equations — Complete Guide

Master mathematical typesetting, from simple formulas to complex multi-line derivations.

Inline vs Display Math

LaTeX offers two ways to typeset math. Inline math is embedded within a paragraph using dollar signs. Display math is set apart on its own line using backslash-bracket notation:

% Inline math sits within running text:
The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

% Display math is centered on its own line:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Inline math is compact and adjusts to fit the line height. Display math is full-size and centered, making it easier to read for complex expressions.

Fractions

Use \frac{numerator}{denominator} to create fractions. In inline mode they appear smaller; in display mode they render at full size:

% Simple fraction:
\[ \frac{a}{b} \]

% Nested fractions:
\[ \frac{1}{1 + \frac{1}{x}} \]

% Binomial coefficient using \binom (requires amsmath):
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]

Subscripts and Superscripts

Use ^ for superscripts and _ for subscripts. Wrap multi-character exponents or indices in braces:

\[ x^2 \quad a_n \quad e^{i\pi} \quad x_{n+1} \]

% Combined subscript and superscript:
\[ x_i^2 \quad a_{n}^{(k)} \]

% Limits on operators:
\[ \lim_{n \to \infty} a_n = L \]

Greek Letters

Greek letters are written as backslash commands. Lowercase letters use the lowercase name, uppercase letters capitalize the first letter:

% Lowercase Greek:
\[ \alpha \quad \beta \quad \gamma \quad \delta \quad \epsilon \quad \pi \quad \sigma \quad \omega \]

% Uppercase Greek:
\[ \Gamma \quad \Delta \quad \Theta \quad \Lambda \quad \Sigma \quad \Omega \]

% Variant forms:
\[ \varepsilon \quad \varphi \quad \vartheta \]

Sums and Integrals

Summation and integral signs use subscripts and superscripts for their limits. In display mode, the limits are placed above and below the symbol:

% Summation:
\[ \sum_{i=1}^{n} i = \frac{n(n+1)}{2} \]

% Product:
\[ \prod_{k=1}^{n} k = n! \]

% Definite integral:
\[ \int_{a}^{b} f(x) \, dx \]

% Double integral:
\[ \iint_{D} f(x, y) \, dA \]

Note the \, before dx -- this thin space is a standard convention in typesetting integrals.

Matrices

Matrices require the amsmath package. Use pmatrix for parentheses, bmatrix for brackets, or vmatrix for determinants:

\usepackage{amsmath}

% Parenthesized matrix:
\[
\begin{pmatrix}
    a & b \\
    c & d
\end{pmatrix}
\]

% Bracketed matrix:
\[
\begin{bmatrix}
    1 & 0 & 0 \\
    0 & 1 & 0 \\
    0 & 0 & 1
\end{bmatrix}
\]

% Determinant:
\[
\begin{vmatrix}
    a & b \\
    c & d
\end{vmatrix}
= ad - bc
\]

Aligned Equations

The align environment (from amsmath) lets you write multi-line equations aligned at a specific point, typically the equals sign. Use & to mark the alignment point:

\usepackage{amsmath}

\begin{align}
    (x + y)^2 &= (x + y)(x + y) \\
              &= x^2 + xy + yx + y^2 \\
              &= x^2 + 2xy + y^2
\end{align}

Each line is automatically numbered. To suppress numbering, use align* instead of align. To suppress numbering on a single line, add \nonumber before \\.

Common Symbols

Here is a reference table of frequently used mathematical symbols: