Mathematics

Last updated on 2025-05-05 | Edit this page

Overview

Questions

  • How do I add mathematical expressions to a LaTeX document?

Objectives

  • Create Inline and Display Math Mode expressions in LaTeX

Math Mode

Typesetting mathematical expressions in LaTeX is one of its greatest strengths. We can mark up mathematical content in a logical way in what is known as math mode.

There are two kinds of math mode in LaTeX:

  • Inline math mode: for typesetting math within a line of text
  • Display math mode: for typesetting math on its own line

Inline Math Mode

Inline math mode is marked using a pair of dollar sign symbols ($ ... $). It is also possible to use the notation \(...\) to mark inline math mode. Simple expressions are entered without any special markup, and you’ll see that the math is spaced out nicely and has letters in italic.

LATEX 

\section{Mathematics}

There are two kinds of math mode in LaTeX: inline and display. Inline math mode is marked with
a pair of dollar signs, whereas display math mode is marked with a pair of square brackets.

\subsection{Inline Math Mode}

The Pythagorean theorem is $a^2 + b^2 = c^2$.

There are a lot of symbols and specialist math mode commands available in LaTeX:

  • Superscripts: x^2
  • Subscripts: x_1
  • Greek letters: \alpha, \beta, \gamma, etc.
  • Operators: \times, \div, \sin, \log, etc.
  • Fractions: \frac{numerator}{denominator}
  • Roots: \sqrt{expression}
  • Sums and integrals: \sum, \int, \oint, etc.
  • Brackets: (), [], {}, \langle, \rangle, etc.

Display Math Mode

Display math mode uses the exact same commands as inline math mode, but it is centered by default and is intended for larger equations that are “part of a paragraph”. It is also started and ended with a pair of square brackets (\[ ... \]).

Callout

Remember that [ and ] are special characters in LaTeX, which is why we have to “escape” it with a backslash(\) here.

LATEX 

\subsection{Display Math Mode}

The Fourier Transform is defined as:

\[
\hat{f}(\xi) = \int_{-\infty}^\infty f(x) e^{-2\pi i \xi x} \, dx
\]

Where:

\begin{itemize}
  \item \( f(x) \) is the function we are transforming,
  \item \( \hat{f}(\xi) \) is the Fourier Transform of \( f(x) \),
  \item \( \xi \) is the frequency variable,
  \item \( i \) is the imaginary unit.
\end{itemize}

Math Mode in LaTeX is not just for LaTeX users! It is also used other tools like Jupyter Notebooks, R Markdown and many markdown processors. There are even extensions for Google Docs and Microsoft Word that allow you to use LaTeX math mode.

Math in Enviroments

We can include math in an environment called equation to number the equations:

LATEX 

\subsection{Math in Environments}

The quadratic formula is:

\begin{equation}
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\label{eq:quadratic}
\end{equation}

This will allow us to refer to the equation later in the document with \cmd{ref} like this:
Refer to Equation \ref{eq:quadratic}.

We can now refer to this equation using the \ref command, just like we did in the previous section:

LATEX 

The quadratic formula is given in Equation~\ref{eq:quadratic}.

The amsmath Package

Mathematical notation is very rich, and the tools in the LaTeX kernel are sometimes not enough to cover everything. The amsmath package extends the capabilities of LaTeX for more complex mathematical typesetting. Let’s try it out:

Add the following to your document:

(In the preamble)

LATEX 

\usepackage{amsmath}

(In the body)

LATEX 

\subsection{The `amsmath` Package}

Solve the following recurrence for $ n,k\geq 0 $:
\[
Q_{n,0} = 1   \quad Q_{0,k} = [k=0];
\]

\[
Q_{n,k} = Q_{n-1,k}+Q_{n-1,k-1}+\binom{n}{k}, \quad\text{for $n$, $k>0$.}
\]

That look ok, but we really want the equations to be aligned vertically, not centered on the page. We can use the align environment from the amsmath package to do this:

LATEX 

\usepackage{amsmath}

LATEX 

\subsection{The `amsmath` Package}

Solve the following recurrence for $ n,k\geq 0 $:
\begin{align*}
  Q_{n,0} &= 1   \quad Q_{0,k} = [k=0];  \\
  Q_{n,k} &= Q_{n-1,k}+Q_{n-1,k-1}+\binom{n}{k}, \quad\text{for $n$, $k>0$.}
\end{align*}

The align* environment makes the equations line up on the amperstand (&)

There’s far too many options to cover here, but the amsmath User Guide contains many more examples.

Challenges

Challenge 1: Can you do it?

See if you can write the following mathematical expression in LaTeX:

  • Special Relativity: t’ = t / sqrt(1 - v^2 / c^2)

LATEX 

\[
t' = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}}
\]

Challenge 2: What’s wrong with this?

What’s wrong with the following LaTeX code?

LATEX 

\(F = G * (m_1 m_2) / r^2 \)

The code is missing the \frac command to create a fraction. The correct code should be:

LATEX 

\(F = G \frac{m_1 m_2}{r^2} \)

Challenge 3: The amsmath package and referencing equations

Add the following equation to your document: E = mc^2. For this, use the amsmath package and a numbered align environment. Label the equation using \label. Then, use the \ref command to refer to the equation in the text. You may find the following LaTeX template helpful:

LATEX 

\documentclass{article}
\usepackage{amsmath} % For better equation formatting

\begin{document}

\section{Referencing Equations}






\end{document}

LATEX 

\documentclass{article}
\usepackage{amsmath} % For better equation formatting

\begin{document}

\section{Referencing Equations}

We can include a numbered equation as follows:

\begin{equation}
  E = mc^2
  \label{eq:energy}
\end{equation}

Now, we refer to the equation in the text. The famous equation is \ref{eq:energy}.

\end{document}

Key Points

  • Inline math mode is marked with $ ... $ or \(...\)
  • Display math mode is marked with \[ ... \]