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\usepackage{amsmath, amssymb, amsthm}
\usepackage{geometry}
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%Lines starting with a % are comments and won't show up in your document.
\title{TeX Template for Math 418 HW} %Your title goes here!
\author{author} %You put your name here.
%\date{} %to get rid of the date just delete the % in front of date{}. You can also fill it in with a different date if you like. For example, if you want to make it seem like you did the assignment ages ago you could write in a date from last week instead of the due date. Not that I've ever done such a thing myself ...
\begin{document}
\maketitle
\section{Section Title: Organizing your work.}
\subsection{Subsection Title.}
\subsubsection{Even smaller subsections etc.}
You can use {\it Section} and {\it Subsection} to organize your work. This is useful when writing a long paper, but less so when writing your average-lengthed homework assignment (unless, I suppose, you are planning to compile your own Dummit and Foote solution manual). Instead, for an assignment you could manually write in the numbering. For example, you could do what I do and write for exercise 3 in 8.1:
\noindent {\bf 8.1.3.} Claim: Whatever the claim is.
\vspace{.2in}
\noindent Proof: Proof goes here.
The noindent tells it not to indent that line of text (not necessary, just a preference of mine visually). I chose to use {\bf 8.1.3} to make the number bold, but other options are \emph{8.1.3} or {\it 8.1.3}, just something to make it stand out. The vspace tells it to leave vertical space between the claim and the proof. I told it to put in .2 inches of space by writing .2in in the parentheses. It also can use cm and pt if you prefer.
There is also a way to put in horizontal space, if you're curious. It looks \hspace {1in} like this.
Another option is to use the list environment, called enumerate. It works like this:
\begin{enumerate}
\item First item
\item Second item
\item Third Item
\begin{enumerate}
\item If you want a sublist.
\begin{enumerate}
\item If you want an even sub-er list . . .
\end{enumerate}
\item Then you might carry on with your old list . . .
\end{enumerate}
\item Or return to the original list . . .
\end{enumerate}
\section{Writing math.}
\subsection{Basics.}
To tell LaTeX you are writing math, you put dollar signs around the math parts of what you are writing. Use just one on either side for math in a paragraph, and two on each side to separate it onto its own lines. There are special commands for certain symbols, like square roots, fractions, and integrals, which can always be googled if need be. Here is an example:
\noindent Claim: The equation $x^2+x+1$ has no real solutions.
\vspace{.2in}
\noindent Proof: The complex solutions to the quadratic polynomial $$ax^2+bx+c$$ are given by the formula $$x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$$ when $a \neq 0$.
In this equation, $a=1$, $b=1$, and $c=1$, so the solutions are $$x=\frac{-1 \pm \sqrt{-3}}{2}.$$ These are not real numbers.
\qed %this adds the box for a bit of punctuation to my proof
\subsection{A non-exhaustive list of math symbols you may want to use.}
\begin{enumerate}
\item Fractions: $\frac{a}{b}$.
\item Square Roots: $\sqrt{3}$.
\item $N$th Roots: $\sqrt[n]{3}$.
\item Pi: $\pi$ (no surpise there, and the other Greek letters work the same way).
\item Important Sets: $\mathbb{Z}$, $\mathbb{N}$, $\mathbb{Q}$, $\mathbb{C}$, you get the idea. . .
\item Functions: $f \colon A \rightarrow B$, or for elements $x \mapsto y$.
\item Injective Arrow: $\hookrightarrow$, Surjective arrow: $\twoheadrightarrow$.
\item Modular arithmetic: $a \equiv b \mod n$.
\item If you want to write a set and have the curly braces show up: $S=\{z \in \mathbb{C} \colon z^5=1 \}.$
\item Just google any other symbols you want to write, to be honest. Or go to Detexify. You can draw in the symbol, and it will tell you how to TeX it.
\end{enumerate}
\end{document}