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\begin{document}
\title{Mathematical Writing: Proof Style Guide}
\author{Brian Sherson}
\maketitle
\section{Introduction}
In this class, you will be \emph{writing} compositions that communicate
mathematical ideas. Most of these compositions are mathematical proofs.
Such compositions are so much more than just writing down a series
of mathematical expressions. Indeed, a mathematical proof is intended
to \emph{convince} the reader of the correctness of a claim, and as
such, includes explanations. Like any other composition, the rules
of writing apply, and therefore, when writing a proof, one must abide
by proper grammar, spelling, punctuation, and sentence structure.
However, it is unlikely you have had to write a composition for an
English class that involved communicating mathematical ideas beyond
citing statistics. Nevertheless, mathematical proofs should be written
not only with mathematical correctness in mind, but also as though
you were submitting one in a writing class.
\section{Organize your thoughts/Use paragraphs}
\begin{wrapfigure}{o}{1.5in}%
\includegraphics[height=1.75in]{wall_of_text_1664}\end{wrapfigure}%
Short of not doing the assignment at all, the quickest way to a low
score is to make your writing very difficult to read. In particular,
writing a massive wall of text is the quickest way to keep someone
from reading your writing. In the case of your TA and professor, writing
an unorganized composition will result in a low grade. Keep in mind
your TA and professors are human, and as such, fall victim to impatience.
Instead, a composition, unless very short, should be split up into
multiple paragraphs. A paragraph is a \textbf{section of writing that
focuses on a single idea}. When paragraphs are used properly, it is
easier for a reader to find a place to take a break, take a sip of
tea, come back to your composition, and easily identify where to resume
reading.
\section{Know your audience}
When writing a composition is knowing your audience. If you are writing
a composition with a math professor as your audience, you can leave
out many details such as mundane algebra and computations. However,
if you were writing a composition for a peer, or even someone with
slightly less mathematical background than you, you may need to include
such details. You want to include just enough details so that your
audience can follow along, but not so much that it bores them to tears.
However, for the purpose of this class, while you will ultimately
submit your compositions to your TA or professor, your audience will
be your peers. The goal is that we want you to be able to effectively
communicate mathematical ideas to your peers, not just mathematicians.
Incidentally, it will be highly recommended that you submit your writing
to your peers for feedback, and make revisions as necessary, before
submitting to your TA or professor.
\section{Know what you are writing}
\begin{wrapfigure}{o}{2in}%
\includegraphics[width=2in]{Do-not-think-it-means.jpeg}
\end{wrapfigure}%
Mathematical writing requires precision. As a general rule, you should
know the precise definition of every term you use in a proof. A rather
deflating experience is when a student is asked what ``$d$ divides
$n$'' means, only to receive a response of ``$n$ is divisible
by $d$.'' Such a response is only a restatement of ``$d$ divides
$n$'' into passive voice, and does not actually explain what it
means. Thus, a very important rule to live by when writing mathematics
is that if you do not know the precise meaning of a term, \textbf{do
not use it}.
Another common problem is the abuse of the word ``equation'' and
equal sign ($=$). This comes in the form of students calling everything
that resembles mathematical notation an equation. However, recall
that an equation is a mathematical sentence in which ``$=$'' is
the verb. A related abuse, of course, is using the equal sign as a
generic verb.
\section{Properly introduce all variables}
In almost all cases, any variable that is used should be given a formal
introduction. This is generally done in one of two ways:
\begin{enumerate}
\item When choosing an arbitrary representative of a set:
\begin{enumerate}
\item Let $\varepsilon>0$...
\item Let $n$ be an even integer...
\item Let $j$ and $k$ be odd integers...
\end{enumerate}
\item When chosen to fulfill a known quality:
\begin{enumerate}
\item Since $k$ is odd, \textbf{there exists an integer $m$ such that}
$k=2m+1$...
\item And so $2=x^{2}$, \textbf{for some positive real number $x$}.
\item Since the function $f$ is blurglecruncheon, \textbf{there exists
a gabbleblotchit function $g$ such that}...
\end{enumerate}
\end{enumerate}
There are notable instances when variables do not need to be explicitly
introduced, but their meaning is understood from their context. Such
notable instances are:
\begin{enumerate}
\item Variables of summation and integration:
\[
\sum_{k=1}^{10}k^{2},\quad\int_{0}^{1}x^{2}\, dx.
\]
\item References to a real vector space of arbitrary dimension: ``Let $G\subseteq\mathbb{R}^{n}$...,''
in which while $G$ is introduced explicitly as an arbitrary subset
of $\mathbb{R}^{n}$, $n$ is understood as being an arbitrary positive
integer.
\end{enumerate}
\section{Sentence Structure}
You may already know from English class that a sentence must contain
a subject and a verb, and in some instances, an object. However, mathematics
is also a language of its own, and has its parts of speech, and sentences.
Consider that ``$x>2$'' is in its own right a sentence, where ``$x$''
is the subject, ``$>$'' is the verb, and ``2'' is the object.
A very important part of doing mathematics is knowing when a symbolic
expression represents a sentence such as ``$x>2$,'' ``$a^{n}+b^{n}\ne c^{n}$,''
and $2\mid6$, and when it represents an object (a noun, if you will),
such as ``$x^{2}+2x+1$.''
Other rules to consider:
\begin{enumerate}
\item Mathematical sentences are never treated as standalone sentences in
a composition, but rather as clauses, functioning as parts of sentences.
For example:
\begin{quotation}
Since $x+2\ge6$, it follows that $x^{2}\ge16$.
\end{quotation}
\item While mathematical sentences can be rewritten entirely in words (e.g.
``$x$ is greater than or equal to three plus $y$,'' in place of
``$x\ge3+y$''), this only makes your composition more difficult
to read, and so you should use the mathematical notation to its fullest.
\item Sentences should never begin with a numeral, nor should they begin
with mathematical notation. Beginning a sentence with a number is
permissible if it is spelled out (``Two,'' ``Ninety-four,'' etc...)
and is for the purpose of counting (used in much the same way you
would in any form of writing.)
\item Simple mathematical notation should be displayed inside of the paragraph
as though it were just another word. Such presentation is called an
in-line formula. For example:
\begin{quotation}
If $x>3$, then $x^{2}>9$.
\end{quotation}
\item Some mathematial notation is more complex, and as such, may be difficult
to display as an in-line formula. To remedy this, such notation should
be displayed on its own line, in the center of the page as follows:
\begin{quotation}
If $ax^{2}+bx+c=0$, and $a\ne0$, then:
\[
x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}.
\]
\end{quotation}
Notice that the quadratic formula is set in what is called a display
formula. Furthermore, it is essential to remember that this formula
is still considered to be part of a sentence. In this case, the quadratic
formula is the end of the sentence, and so a period must be placed
at the end of the formula.
\item Some formulas can also be set as a display formula if they are generally
regarded as an important formula. Furthermore, it may also be convenient
to number such formulas so as to be able to make references to them
elsewhere in your composition. Such formula numbers can be placed
in either the right or left margins, so long as you maintain a logical
and easy to follow numbering system, and remain consistent.
\item An inevitable part of writing mathematical proofs is a long string
of computations and algebraic manipulations. Such computations and
manipulations should be set as displayed formulas, and each step is
placed on its own separate line, as in:
\begin{flalign*}
& \hfill & a & =b & \hfill\\
& & & =c\\
& & & =d.
\end{flalign*}
Notice that all the work is done on the right-hand side, and we only
right down the left-hand side on the top line. DO NOT write the left-hand
side on subsequent lines; it will only serve as visual clutter. Also,
notice the equal signs are lined up vertically. However, such manipulations
are not limited to equalities, and in fact, other relations, such
as ``$\le$'' can be mixed in as follows:
\begin{flalign*}
& \hfill & a & =b & \hfill\\
& & & \le c\\
& & & =d\\
& & &