An important reason for using LaTeX is that it is fairly easy to typeset mathematical expressions. This lesson we will discuss some of the most used commands for typesetting mathematics.

For basic mathematics, no packages are needed.
We will use `mathtools`

, an extension of `amsmath`

, since we will
typeset more complex expressions:

```
\usepackage{mathtools}
```

Please consent the main reference Wikibooks/LaTeX/Mathematics, whenever you are in doubt. For more details you can look at this Short Math Guide for LaTeX, or the documentation of amsmath or mathtools.

Mathematics within text like $1+1=2$ is called **inline** mathematics.
A line that consists of a single mathematical expression like
\[
1+1=2
\]
is called a **display**. Inline mathematics can be typeset between `$`

s.

We know that $1+1=2$.

```
We know that $1+1=2$.
```

Mathematics can be typeset in a display by putting it between `\[`

and
`\]`

.

Examine the curve given by
\[
y = x^2 + 2.
\]
Does it attain a minimum?

```
Examine the curve given by
\[
y = x^2 + 2.
\]
Does it attain a minimum?
```

Recall that an empty line creates a new paragraph. Therefore there are no empty lines before or after the display in this example.

To **number** displays automatically one uses the `equation`

environment. We will see later how to reference displays.

Since both $a$ and $b$ are odd we have
\begin{equation}
a + b = 2x
\end{equation}
and
\begin{equation}
a-b = 2y
\end{equation}
for some $x$ and $y$.

```
Since both $a$ and $b$ are odd we have
\begin{equation}
a + b = 2x
\end{equation}
and
\begin{equation}
a-b = 2y
\end{equation}
for some $x$ and $y$.
```

LaTeX switches to **math mode** in every mathematical environment. The
most important differences compared to *text mode* are:

- Spaces and line breaks will be ignored; whitespace will be determined by the operators used or by special commands.
- Blank lines are not permitted.
- A single letter will be considered the name of a variable and will
be typeset accordingly. If you wish to use normal text in
*math mode*you have to use the special command`\text{...}`

.

In the rest of this lesson we will assume that you are working in *math
mode*.

Powers and indices can be inserted using the hat `^`

and the underscore
`_`

. For example $

\[
a^2\quad a_i\quad a_i^2
\]

```
a^2 a_i a_i^2
```

The `_`

and `^`

bind themselves to the next *group*. Recall that groups
can be made with `{`

and `}`

. Hence

\[
a^{n+1}\quad a^n+1\quad a_{i/2}
\]

```
a^{n+1} a^n+1 a_{i/2}
```

Typeset the powers

\[
(a^b)^c,\quad a^{b^c}
\]

A **fraction** can be created using `\frac{2}{3}`

. It is possible to create
fractions inside fractions.

\[
\frac{\frac{1}{x}+\frac{1}{y}}{y-z}
\]

```
\frac{\frac{1}{x}+\frac{1}{y}}{y-z}
```

We can use `\tfrac`

for small fractions, it is up to the user to decide when
to use this.

**Roots** can be inserted as `\sqrt{2}`

or `\sqrt[n]{2}`

for
$\sqrt[n]{2}$.

A **sum** or **integral** is given by `\sum`

or `\int`

, use `^`

and `_`

to specify the boundaries. For example

\[
\sum_{i=1}^{10} t_i\quad \int_0^\infty e^{-x}\, dx
\]

```
\sum_{i=1}^{10} t_i
\int_0^\infty e^{-x}\, dx
```

There are more operators that use `^`

and `_`

for specifying boundaries,
for example `\prod`

$\prod$ and `\bigcup`

$\bigcup$. A more
detailed list is shown at the end of this page.

You can scale *brackets* and *delimiters*, e.g. `|`

or `(`

, automatically
using `\left`

and `\right`

.

\[
\left| \sum_{n=1}^\infty a_n \right| \leq \sum_{n=1}^\infty \left| a_n \right|
\]

```
\left| \sum_{n=1}^\infty a_n \right| \leq \sum_{n=1}^\infty
\left| a_n \right|
```

With `\ldots`

$\ldots$ and `\cdots`

$\cdots$ one makes dots. The
context determines which dots to use. For example, we write
`1,2,\ldots,n`

$1,2,\ldots,n$ and `1\cdot 2 \cdots n`

$1\cdot
2\cdots n$. In general we try to choose the dots in such a way that
they are at the same height as the surrounding operators. The command
`\dots`

tries to accomplish this.

Typeset the display.

\[
\left | \sum_{i=1}^n a_ib_i \right| \leq \left( \sum_{i=1}^n a_i^2 \right)^{\frac{1}{2}} \cdot \sqrt{\sum_{i=1}^n b^2_i}
\]

In *math mode* all spaces and line breaks are ignored; whitespace is
determined by the operators or commands used. However, sometimes it is
necessary to create some horizontal whitespace. The command
`\quad`

inserts whitespace the size of the font type, so 10pt for a 10pt font.
A `\qquad`

inserts twice as much whitespace. These commands also work in
*text mode*. In *math mode* we have the following specific commands:

command | length |
---|---|

`\,` |
3/18 quad |

`\:` |
4/18 quad |

`\;` |
5/18 quad |

`\!` |
$-$3/18 quad |

\[
\int y\, dx
\]

```
\int y\, dx
```

With the following commands we can choose different fonts. For the last
two the package `amssymb`

is required.

command | example | usage | |
---|---|---|---|

`\mathrm` |
$\mathrm{ABCDEF}$ | $\mathrm{abcdef}$ | |

`\mathit` |
$\mathit{ABCDEF}$ | $\mathit{abcdef}$ | |

`\mathbf` |
$\mathbf{ABCDEF}$ | $\mathbf{abcdef}$ | sets, vectors |

`\mathcal` |
$\mathcal{ABCDEF}$ | $\mathcal{abcde}$ | categories, big-oh notation |

`\mathbb` |
$\mathbb{ABCDEF}$ | sets |

We prefer not to use these commands directly in the code, but in a new command to seperate the content and layout. For example:

The complex numbers $\mathbb{C}$.

```
\documentclass[a4paper]{article}
\usepackage[dutch]{babel}
\usepackage{mathtools}
\usepackage{amssymb}
\newcommand{\field}[1]{\mathbb{#1}}
\newcommand{\CC}{\field{C}}
\begin{document}
The complex numbers $\CC$.
\end{document}
```

Typeset the display.

\[
\forall \epsilon > 0 \quad \exists \delta \quad \text{such that} \quad \forall x,y \in \mathbb{R}:
|x-y|< \delta \implies | f(x+\delta) - f(x)| < \epsilon.
\]

Hint: text inside *math mode* must be inserted using `\text{...}`

. You can
use Wikibooks/LaTeX for
a list of symbols and LaTeX commands.

An **operator** is a function written as a word, like $\sin$ and
$\log$. LaTeX has many predefined operators, but sometimes it is
necessary to define one yourself. We create operators using
`\DeclareMathOperator`

in the preamble.

The next two exercises are about typesetting theorems.

- Create a new LaTeX file containing the following code:
`\documentclass[a4paper,12pt]{article} \usepackage[english]{babel} \usepackage{mathtools} \usepackage{amssymb} \usepackage{amsthm} \theoremstyle{plain} \newtheorem{theorem}{Theorem} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \begin{document} \begin{definition} \end{definition} \begin{theorem}[Cauchy-Schwarz inequality] \end{theorem} \end{document}`

- Typeset the following definition:
**Definition 1.**If $P(X\in S)=1$ for a finite $S$ then the*expectation*\[ \mathbb{E}X=\sum_{x\in S} xP(X=x). \] - Typeset the following theorem. You may or may not use
`\DeclareMathOperator`

, whatever you find convenient.**Theorem 2**(Cauchy-Schwarz inequality)**.***If $X$ and $Y$ are random variables with $\mathbb{E}X^2 \lt \infty$ and $\mathbb{E}Y^2 \lt \infty$, then*\[ \DeclareMathOperator{\Var}{Var} \DeclareMathOperator{\Cov}{Cov} |\Cov(X,Y)|\le\sqrt{\Var(X)}\sqrt{\Var (Y)}. \]

- Choose one of the following statements:
- There are infinitely many prime numbers.
- The number $\sqrt2$ is not rational.

- Create a new LaTeX file containing:
`\documentclass[a4paper,12pt]{article} \usepackage[english]{babel} \usepackage{mathtools} \usepackage{amssymb} \usepackage{amsthm} \theoremstyle{plain} \newtheorem{theorem}{Theorem} \begin{document} \begin{theorem} \end{theorem} \begin{proof} \end{proof} \end{document}`

- Typeset your chosen statement using the
`theorem`

environment defined in the preamble. - Prove the theorem and write your proof down in a
`proof`

environment.

You don’t need to remember all LaTeX commands. There are lot of references on the internet:

- Wikibooks/LaTeX/Mathematics and Wikibooks/LaTeX/Advanced Mathematics: The LaTeX wikibook contains well maintained material for (learning) LaTeX.
- LaTeX Cheat Sheet: A comprehensive document with a lot of commands. Save it and print it.

- What is a math environment? What rules apply in math mode?
- How to insert special characters?
- What do
`$`

,`[`

`]`

,`_`

,`^`

,`\alpha`

,`\sin`

,`\log`

,`\frac`

,`\binom`

,`\sqrt[n]`

,`\sum`

,`\int`

,`()`

,`[]`

,`\{ \}`

and`| |`

do? - How to create your own operator?
- How to insert horizontal whitespace?