% from sarah-marie belcastro's website
\documentclass[12pt]{article}   % 10 and 11 point also available
\usepackage{amsmath, amssymb}  % these packages should be included in your TeX installation

\begin{document}                % end of the preamble

% This document was edited from one Tom Hull gave me, which he yanked from somewhere else a couple of decades ago.
% It incorporates some 2020 suggestions from Murray Eisenberg.

\title{My Awesome Paper}        % defines the title info
\author{Minnie Maximallian}

\maketitle                      % this actually does the title.  Omit this if you don't want a title.

Introduction blah blah. Just type in text normally!  \LaTeX\ will format normal text very professionally, just like a real publication.

\section{Our very first section}

To start things off I'll blah blah blah    % LaTeX ignores
more text   blah    blah blah   blahblah.  % extra spaces

But then I'll blah blah blah blah blab. Notice that by leaving a blank line, we started a new paragraph! 
Yippity-dee!

\section{Our Second Section!}

\subsection{Math formulae in text}

Use dollar signs before and after a mathematical
expression. One formula is $x-3y=7$. Another
is $A=\pi r^2$.

\subsection{Important-looking math formulae}

Use double dollar signs to put a formula/equation 
on a line by itself!

$${d\over dx}\arcsin x=
        (1-x^2)^{-\frac12} =
          \int_0^x\theta(1-\theta^2)^{-\frac32}d\theta $$

There are easily-remembered keywords for all
the mathematical symbols you want to use. (In fact, often guessing at the name for a command will end up being correct\dots)
Begin each with a backslash.

$$\alpha+\beta < \gamma \leq \Gamma
       \approx \lim_{n \rightarrow \infty} x_n =
       \prod_{i=1}^{\infty} \sqrt{y_i} $$

\subsection{Magical parentheses and things!}

\LaTeX\ keeps track of how big things are very nicely!
$$\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{x}}}}}}}}$$
$$\left\{ {n\choose 0}+{n\choose 1}+{n\choose 2}+\cdots
+{n\choose n}\right\} = 2^n$$   % Every \left must have a \right following it.
$$\left(\left(\left(\left(x^2\right)^2\right)^2\right)^2\right)^2=x^{32}$$ 

Notice how these work: Every time we have a \verb+\left+ command we need a \verb+\right+ command to go with it.  Using these commands with parentheses () and brackets \{\} will make them as big as they need to be.

\subsection{Neat formulae}
To print a list of equations with the equal signs aligned:

\begin{align*} % the * suppresses numbering for the lines
  x_{i+1} &= N^{i+1}(x_0) \\           % \\ means newline in
              &= N(x_i) \\                         % this environment
              &= x_i- \frac{f(x_i)}{f'(x_i)}       % But no \\ on the
\end{align*}                         

Here are fancy things:
\begin{equation*}
  D(x)=\begin{cases}
    1 &\text{if $x$ is rational,}
    \\
    (x+1)^2 &\text{otherwise}  
  \end{cases}
  \quad\text{ or }\quad                         
  M= \begin{pmatrix}
    1 & 0 & 1 \\
    2 & -1 & 0 \\
    3 & 3 & 17
  \end{pmatrix}
\end{equation*}

Notice how I use the \verb+\text+ command to do normal text inside of ``math mode."  (Also notice how I just got the quotation marks to work  properly there!)

\subsection{Lists}

A bulleted list
\begin{itemize}
\item of words
\item in the rest
\item of this sentence.
\end{itemize}

\noindent A numbered list
\begin{enumerate}
\item of words
\item in the rest
\item of this sentence.
\end{enumerate}

\subsection{Other stuff}

\LaTeX\ has lots of of commands to make {\normalsize text} {\large large}, {\Large larger},
{\Huge huge}, {\small small}, {\footnotesize footnote size}, {\scriptsize script size}, and {\tiny tiny}!

\newcommand{\qed}{\raise1truemm\hbox{$\infty\hskip-.165in\lower6truept\hbox{$\smile$}$}}

We can also define new commands. For example, the above (see the source code!)
produces a command \verb+\qed+ which can be used as an alternative to qed,
namely \qed.

Or suppose we needed a falling power command, which \LaTeX\ doesn't provide. To
do this, we need to define a command with an argument.

\newcommand{\fall}[1]{\underline{#1}}

This gives us $n^{\fall k}=n(n-1)(n-2)\cdots (n-k+1)$. 

\section{Typesetting things like Homework!}

On the next page I'll do an example of how you might want to format homework and things like that.  There are fun commands like \verb+\hline+ and \verb+\hfill+ that can be used to do make fun-looking headers and such.  I'll also include an example of how to make a TABLE of things in \LaTeX.  But first I'll use the \verb+\clearpage+ command to make a new page and then I'll do the sample homework!  Let's go!


\clearpage

\noindent Thomas Hull \hfill 3/29/04

\centerline{\bf A goofy homework assignment!} % some people prefer \textbf{blah blah} 

\smallskip

\hrule

\vspace{.05in}       % the \vspace command makes some vertical space.  Very useful!

\noindent {\bf (1)} Find the airspeed velocity of an unladen swallow.

\vspace{.1in}

{\bf Solution:} This problem is unclear because it does not specify whether it is an African or European swallow.  Therefore it is impossible to give an accurate answer.  Phooey!

\vspace{.2in}

\noindent{\bf (2)} Make a table.  Any table.  Just for fun.

\vspace{.1in}

{\bf Solution:} Here is a table, just for fun:

\begin{tabular}{||l||c|c|c|c|c|c||} % | (pipe) gives a vertical divider. c indicates a centered column. l (ell) gives a left-justified column.  To make a text column that wraps, use p{.5cm}
\hline\hline
Module angle & $30^\circ$ & $45^\circ$ & $54.7^\circ$ & $60^\circ$ & $67.5^\circ$ &
$75.5^\circ$\\
\hline\hline
3-gon (triangle) & $\checkmark$ & $\checkmark$ & $\checkmark$ & $\checkmark$ &
$\checkmark$ & $\checkmark$\\
\hline
4-gon (square) & & $\checkmark$ & $\checkmark$ & $\checkmark$ & $\checkmark$ &
$\checkmark$\\
\hline
5-gon & & & $\checkmark$ & $\checkmark$ & $\checkmark$ & $\checkmark$\\
\hline
6-gon & & & & $\checkmark$ & $\checkmark$ & $\checkmark$\\
\hline
8-gon & & & & & $\checkmark$ & $\checkmark$\\
\hline
10-gon & & & & & & $\checkmark$\\
\hline
12-gon & & & & & & $\checkmark$\\
\hline\hline
\end{tabular}

\end{document}