Simple, Inline Equations

You can do a simple inline equation by using $an equation here$.

For example, you might do

$\lim\limits_{n \to \infty} \frac{1}{2^n} i\bar z$ 

which, if we put a little fake text around it, gives

Numbered, Centered Equations

When writing papers or other documents, it is sometimes preferable to have your equations centered and numbered, as opposed to in-line. Then, use the \begin{equation} and \end{equation} commands.

For example, if we use the code

\begin{equation}
\lim\limits_{n \to \infty} \frac{1}{2^n} i\bar z
\end{equation}

And add a little text around it, we get

You can remove the numbering of the equation by using \begin{equation*} and \end{equation*}.

For example, if we use the code

\begin{equation*}
\lim\limits_{n \to \infty} \frac{1}{2^n} i\bar z
\end{equation*}

and add a little text around it, we get

(though it should be noted you have to use the amsmath package for this).

Sometimes, it can be difficult to find the mathematical symbol you need. There are several options here. The first (and quickest) is to use Detexify, where you draw the symbol you'd like, and it tries to find what you want, like as shown below:

Another option is to use the comprehensive LaTeX symbols list, which can be found here. If you are using the package unicode-math this list of all supported symbols can be helpful. Another option is this website, which has common math symbols.

While standard LaTeX is all that is needed for most simple mathematical formulae and equations, sometimes more symbols and tools are needed. There are multiple packages available that will enhance your equations and provide you with more to work with. Three of the main packages are described below. Remember, to load a package, type \usepackage{package} in your document preamble.

amsmath

The amsmath package is an incredibly useful package. It is used to allow your equations to be centered but not numbered, as in \begin{equation*}, it is used to create matrices (as described below) and it introduces many other useful commands, such as \overset and \underset, described below. The amsmath package documentation can be found here.

mathtools

The mathtools package builds off of the amsmath package, adding further useful symbols and tools. It automatically loads the amsmath package, so you do not need to load both in your document preamble. The mathtools documentation can be found here.

amssymb

The amssymb package provides many extra symbols that can be very handy for more complex equations. The amssymb documentation can be found here.

Font packages

There are also various fonts you can use for your equations, as described on this question on the TeX stack exchange, for TeX, LaTeX, and friends.

This paper is a concise explanation of the different features provided by some packages as well as standard LaTeX; it is very helpful.

Some of the most common commands include:

• Fractions and Square Roots: For fractions, use \frac {numerator}{denominator}. For square roots, use \sqrt[root]{number}.
• Greek letters: use the commands given in the table below:

• Operators: \leq gives the less than or equal to symbol, \geq gives the greater than or equal to symbol, \neq gives the not equal symbol, \sum gives the summation symbol, \partial gives the partial derivative symbol, \nabla gives the Laplacian operator, \times gives the cross product or multiplication symbol, \cdot gives the dot product or multiplication symbol, and \int gives the integral symbol.
• Arrows: \rightarrow and \leftarrow give right and left arrows, respectively.
• Percents: If typing % in LaTeX, it is important to include a backslash, \% as the percent symbol is normally used for comments.
• Superscripts and Subscripts: To do a superscript, you can type x^2, or, for longer superscripts, x^{2x}. To do a subscript, you can type x_a, or, for longer subscripts, x_{ab}.
• Bold: Use \boldmath{...} to make your math symbols bold. Other options are given at this TeX.SX question. Math symbols are automatically italicized; if you don't want this to be true, make your equation text as described below.
• Infinity: To write infinity, use the command \infty.
• Moving items over or under another: First, for math operators only, there is an alternate method. You can type the math operator, say \int, and then use the \limits command. An example is \int\limits_{\infty} or \int\limits^{\infty}. Then, for normal cases, you can do \overset{top}{normal} or \underset{bottom}{normal}. This can be very useful for doing vectors. For example, you might do \overset{\rightarrow}{x} The amsmath package is need for overset and underset.
• Curly Braces: Because curly braces are used in commands, it is necessary to type \{ or \} to get curly braces.
• Text: To include text in equations, type \usepackage{amsmath} in the preamble, and then type \text{...}.
• Space: To add space in your equations, type \quad between the two items you want to separate (for example, you might have $2x \quad cos).

Let's say you cannot find the symbol you need anywhere. You can create a custom symbol. For example, the code

\documentclass{article}
\usepackage{graphicx,amsmath,amssymb}
\DeclareRobustCommand{\diamondtimes}{%
  \mathbin{\text{\rotatebox[origin=c]{45}{$\boxplus$}}}%
}

\begin{document}
$a\diamondtimes b$
\end{document}

creates and calls a symbol, giving

This is a simpler example; it merely has to rotate an already existent symbol. However, you can create more complex symbols.

This section is in the process of being expanded.

Matrices

You must always use the amsmath package if you are going to use the following commands. There are four main types of matrix, as shown in the code below:

\begin{matrix} 
    a & b \\
    c & d 
\end{matrix}
\quad
\begin{pmatrix} 
   a & b \\
   c & d 
\end{pmatrix}
\quad
\begin{bmatrix} 
    a & b \\
    c & d 
\end{bmatrix}
\quad
\begin{vmatrix} 
    a & b \\
    c & d 
\end{vmatrix}
\quad
\begin{Vmatrix} 
    a & b \\
    c & d 
\end{Vmatrix}

This code produces

There are a couple important things to note about this:

1. It is important you put your matrix within the equation, equation*, or $...$ environment - the bmatrix command is not a math environment on its own.
2. The construction of the matrix is actually fairly simple. For each row, you create each element (say x_{11}), then put a &, and then write the next element. For multiple rows, at the end of each row put \\ (you do not have to do this for the last row). It is fairly similar to a table in this.