Scripts, Fractions, Brackets, and Matrices

Use ^ to write superscripts, _ to write subscripts, and \frac to write fractions. amsmath provides matrices, piecewise functions, and more reliable mathematical structures.

EasyLaTeX compiler setting: pdfLaTeX + article; amsmath needs to be loaded.

Superscripts and Subscripts#

Use ^ for superscripts and _ for subscripts:

\[
  x^2 + y_1 = z.
\]

When a superscript or subscript contains multiple characters, curly braces must be used to delimit its scope:

\[
  x^{n+1}, \qquad a_{i,j}, \qquad e^{-x^2}.
\]

x^n+1 only makes n become a superscript, while +1 remains on the baseline; this is one of the most common formula errors made by beginners.

Fractions and Radicals#

Use \frac{numerator}{denominator}:

\[
  \frac{a+b}{c+d}.
\]

Use \sqrt{...} for square roots; use an optional bracket parameter for nth roots:

\[
  \sqrt{x}, \qquad \sqrt[3]{x+y}.
\]

Nested fractions can appear too tall when used inline. If the formula itself requires explanation or contains multiple layers of structure, use a standalone equation or split the derivation.

Automatically Sized Brackets#

For short content, you can directly use (, ), [, and ]. When parentheses surround a fraction or matrix, use \left and \right to automatically match the height:

\[
  \left( \frac{a}{b} \right)^2.
\]

When only one side needs visible brackets, use the invisible delimiter \left. or \right. on the other side:

\[
  \left. \frac{d}{dx} x^n \right|_{x=1} = n.
\]

\left must be paired with \right in the same mathematical group. There is no need to overuse automatically scalable parentheses in short formulas; excessively large parentheses can impair readability.

Matrices#

After loading amsmath, environments such as pmatrix, bmatrix, etc. can be used. Columns are separated by &, and line endings use \\ for line breaks:

\[
  \begin{pmatrix}
    a & b \\
    c & d
  \end{pmatrix}
\]

Common environments are as follows:

Piecewise Functions#

cases The environment generates a large brace on the left, suitable for piecewise definitions:

\[
  f(x) =
  \begin{cases}
    x^2, & x \geq 0, \\
    -x,  & x < 0.
  \end{cases}
\]

& Align the expressions and conditions in separate columns; English explanations in the conditions may use \text{...}.

Common Misuses#

Put all content inside a pair of curly braces#

x^{n+1} requires curly braces, but x^2 does not. Curly braces are used only to delimit command arguments or the scope of superscripts and subscripts; they themselves do not appear in the PDF.

Forgetting the Double Backslash at the End of a Matrix Row#

In a matrix, a line break is \\, not a single backslash as in ordinary text. The last row usually does not need \\.

Enclose a tall matrix in ordinary parentheses#

( \begin{matrix} ... \end{matrix} ) brackets do not automatically grow taller. Use pmatrix, or use the paired \left( and \right).

Complete Example#

\documentclass[a4paper,12pt]{article}
\usepackage{amsmath}

\begin{document}

\[
  x^{n+1} = \frac{x^n \cdot x}{1}.
\]

\[
  A = \begin{pmatrix}
    1 & 2 \\
    3 & 4
  \end{pmatrix}.
\]

\[
  f(x) =
  \begin{cases}
    x^2, & x \geq 0, \\
    -x,  & x < 0.
  \end{cases}
\]

\end{document}