1. Inline and Display Math

1.1. Inline Math

To embed math within text, enclose it in $...$:

The function $f(x) = x^2 + 1$ is always positive.

Result: The function $f(x)=x^2+1$ is always positive.

1.2. Display Math

To display math on its own line, use $$...$$:

$$\int_0^\infty e^{-x^2} dx = \frac{\sqrt{\pi}}{2}$$

Result:

2. Numbered Equation Environments

2.1. equation Environment

Assigns a number to a single equation:

\begin{equation}
e^{i\pi} + 1 = 0
\end{equation}

Result:

To suppress the number, use equation*:

\begin{equation*}
\sum_{k=0}^{n} \binom{n}{k} = 2^n
\end{equation*}

Result:

2.2. align Environment

Displays multiple equations with alignment. Use & to mark alignment points and \\ for line breaks:

\begin{align}
(a + b)^2 &= a^2 + 2ab + b^2 \\
(a - b)^2 &= a^2 - 2ab + b^2 \\
(a + b)(a - b) &= a^2 - b^2
\end{align}

Result:

2.3. gather Environment

Lists equations centered without alignment:

\begin{gather}
x^2 + y^2 = r^2 \\
x = r \cos\theta \\
y = r \sin\theta
\end{gather}

Result:

2.4. multline Environment

Splits a long equation across multiple lines. The first line is left-aligned and the last line is right-aligned:

\begin{multline}
(x_1 + x_2 + x_3 + x_4 + x_5)^2 \\
= x_1^2 + x_2^2 + ... + x_5^2 \\
\quad + 2(x_1 x_2 + x_1 x_3 + ...)
\end{multline}

Result:

3. Piecewise Definitions and Matrices

3.1. cases Environment

Write piecewise definitions:

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

Result:

Another example (the Fibonacci sequence):

$$
f(n) = \begin{cases}
1 & n = 0 \\
1 & n = 1 \\
f(n-1) + f(n-2) & n \geq 2
\end{cases}
$$

Result:

3.2. Matrix Environments

Choose the environment based on the bracket style:

$$
A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad
\det A = \begin{vmatrix} 1 & 2 \\ 3 & 4 \end{vmatrix} = -2
$$

Result:

Use smallmatrix for inline matrices:

The matrix $\left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right)$ like so.

Result: The matrix $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ like so.

4. Math Symbols and Commands

4.1. Greek Letters

Uppercase: $\Gamma ,\Delta ,\Theta ,\Lambda ,\Xi ,\Pi ,\Sigma ,\Phi ,\Psi ,\Omega$

4.2. Operators and Relations

4.3. Big Operators

4.4. Superscripts, Subscripts, and Decorations

4.5. Fractions and Roots

4.6. Delimiter Sizing

\left and \right auto-size delimiters:

4.7. Math Fonts

5. Custom Commands

You can define custom commands with \newcommand. Writing them at the top of an article makes them available within that article, but registering them in the Settings preamble editor makes them available across all articles automatically.

\newcommand{\R}{\mathbb{R}}
\newcommand{\norm}[1]{\left\| #1 \right\|}
\newcommand{\inner}[2]{\left\langle #1, #2 \right\rangle}

With the above definitions, \R expands to $\mathbb{R}$, \norm{v} to $\Vert v\Vert$, and \inner{u}{v} to $⟨u,v⟩$.

\DeclareMathOperator can be used to define math operators:

\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\rank}{rank}

This gives proper spacing: $\mathrm{tr}(AB)=\mathrm{tr}(BA)$, $\mathrm{rank}(A)\le \min (m,n)$.

6. Summary

On Folio, standard LaTeX math syntax works almost exactly as expected. All commands and environments supported by KaTeX are available. Next, we'll look at theorem environments in detail.