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\usepackage[latin1]{inputenc}
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\usepackage{amsmath}
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\begin{document}

\begin{center}
{\Huge
\LaTeX{} class 2
}\\
Clive Newstead, Tuesday 11th July 2017
\end{center}

\section{Math mode}

By default, \LaTeX{} is in \textit{text mode}, which is used for typing words and sentences. If you want to type mathematical notation, you have to enter \textit{math mode}. This can be done in a few ways.

\subsection{In-line equations}

The easiest is by using dollar sign: one dollar sign enters math mode, and the next exits math mode---this gives an in-line equation. For example, you could write $E=mc^2$ or $a+b=c$. Notice how the variables appear italicised and the spacing is nice. Compare with: a+b=c.

All mathematical variables, symbols, equations and so on, should be entered in math mode. For example, three times two is $3 \times 2$, three plus two is $3 + 2$, and three divided by two is $\frac{3}{2}$ or $\dfrac{3}{2}$. You can do square roots $\sqrt{x^2+1}$ and cube roots $\sqrt[3]{x^3+1}$ and $n$th roots $\sqrt[n]{x^n+1}$. Exponents are easy, but make sure you use braces if needed: compare $a^b+1$ and $a^{b+1}$.

Notice that spaces are ignored in math mode do nothing: $abcdef$ versus $a b c d e f$. To make a space appear, put a backslash before it: $a\ b\ c\ d\ e\ f$. There are smaller and larger spaces too: $a\, b\; c\ d \quad e \qquad f$.

\subsection{Displayed equations}

Sometimes, either for emphasis or for readability, equations are put on their own line. This can be done using double dollar signs instead of single dollar signs. For example
$$\forall n \in \mathbb{N},\ \sum_{k=0}^n k = \frac{n(n+1)}{2}$$
If you want to enter text mode after you've already entered math mode, you can do using the `text' command, for example
$$\sum_{k=0}^n k = \frac{n(n+1)}{2} \text{ for all } n \in \mathbb{N}$$
If you want it to be enumerated, you can use the `equation' environment:

\subsection{Aligned equations}

If you have a chain of equations, you can make them line up nicely using the `align*' environment, which is another way of entering math mode. For example:
\begin{align*}
(n+1)!-n! & = (n+1) \cdot n! - n! && \text{by definition of factorial} \\
          & = (n+1-1) \cdot n! && \text{common factor of $n!$} \\
          & = n \cdot n! && \text{cancelling the $1$s}
\end{align*}

\end{document}