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\begin_body
\begin_layout Title
Mathematical Writing: Proof Style Guide
\end_layout
\begin_layout Author
Brian Sherson
\end_layout
\begin_layout Section
Introduction
\end_layout
\begin_layout Standard
In this class, you will be
\emph on
writing
\emph default
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 mathemati
cal expressions.
Indeed, a mathematical proof is intended to
\emph on
convince
\emph default
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.
\end_layout
\begin_layout Standard
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.
\end_layout
\begin_layout Section
Organize your thoughts/Use paragraphs
\end_layout
\begin_layout Standard
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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.
\end_layout
\begin_layout Standard
Instead, a composition, unless very short, should be split up into multiple
paragraphs.
A paragraph is a
\series bold
section of writing that focuses on a single idea
\series default
.
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.
\end_layout
\begin_layout Section
Know your audience
\end_layout
\begin_layout Standard
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.
\end_layout
\begin_layout Standard
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.
\end_layout
\begin_layout Section
Know what you are writing
\end_layout
\begin_layout Standard
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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
\begin_inset Quotes eld
\end_inset
\begin_inset Formula $d$
\end_inset
divides
\begin_inset Formula $n$
\end_inset
\begin_inset Quotes erd
\end_inset
means, only to receive a response of
\begin_inset Quotes eld
\end_inset
\begin_inset Formula $n$
\end_inset
is divisible by
\begin_inset Formula $d$
\end_inset
.
\begin_inset Quotes erd
\end_inset
Such a response is only a restatement of
\begin_inset Quotes eld
\end_inset
\begin_inset Formula $d$
\end_inset
divides
\begin_inset Formula $n$
\end_inset
\begin_inset Quotes erd
\end_inset
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,
\series bold
do not use it
\series default
.
\end_layout
\begin_layout Standard
Another common problem is the abuse of the word
\begin_inset Quotes eld
\end_inset
equation
\begin_inset Quotes erd
\end_inset
and equal sign (
\begin_inset Formula $=$
\end_inset
).
This comes in the form of students calling everything that resembles mathematic
al notation an equation.
However, recall that an equation is a mathematical sentence in which
\begin_inset Quotes eld
\end_inset
\begin_inset Formula $=$
\end_inset
\begin_inset Quotes erd
\end_inset
is the verb.
A related abuse, of course, is using the equal sign as a generic verb.
\end_layout
\begin_layout Section
Properly introduce all variables
\end_layout
\begin_layout Standard
In almost all cases, any variable that is used should be given a formal
introduction.
This is generally done in one of two ways:
\end_layout
\begin_layout Enumerate
When choosing an arbitrary representative of a set:
\end_layout
\begin_deeper
\begin_layout Enumerate
Let
\begin_inset Formula $\varepsilon>0$
\end_inset
...
\end_layout
\begin_layout Enumerate
Let
\begin_inset Formula $n$
\end_inset
be an even integer...
\end_layout
\begin_layout Enumerate
Let
\begin_inset Formula $j$
\end_inset
and
\begin_inset Formula $k$
\end_inset
be odd integers...
\end_layout
\end_deeper
\begin_layout Enumerate
When chosen to fulfill a known quality:
\end_layout
\begin_deeper
\begin_layout Enumerate
Since
\begin_inset Formula $k$
\end_inset
is odd,
\series bold
there exists an integer
\begin_inset Formula $m$
\end_inset
such that
\series default
\begin_inset Formula $k=2m+1$
\end_inset
...
\end_layout
\begin_layout Enumerate
And so
\begin_inset Formula $2=x^{2}$
\end_inset
,
\series bold
for some positive real number
\begin_inset Formula $x$
\end_inset
\series default
.
\end_layout
\begin_layout Enumerate
Since the function
\begin_inset Formula $f$
\end_inset
is blurglecruncheon,
\series bold
there exists a gabbleblotchit function
\begin_inset Formula $g$
\end_inset
such that
\series default
...
\end_layout
\end_deeper
\begin_layout Standard
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:
\end_layout
\begin_layout Enumerate
Variables of summation and integration:
\begin_inset Formula
\[
\sum_{k=1}^{10}k^{2},\quad\int_{0}^{1}x^{2}\, dx.
\]
\end_inset
\end_layout
\begin_layout Enumerate
References to a real vector space of arbitrary dimension:
\begin_inset Quotes eld
\end_inset
Let
\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
\end_inset
...,
\begin_inset Quotes erd
\end_inset
in which while
\begin_inset Formula $G$
\end_inset
is introduced explicitly as an arbitrary subset of
\begin_inset Formula $\mathbb{R}^{n}$
\end_inset
,
\begin_inset Formula $n$
\end_inset
is understood as being an arbitrary positive integer.
\end_layout
\begin_layout Section
Sentence Structure
\end_layout
\begin_layout Standard
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
\begin_inset Quotes eld
\end_inset
\begin_inset Formula $x>2$
\end_inset
\begin_inset Quotes erd
\end_inset
is in its own right a sentence, where
\begin_inset Quotes eld
\end_inset
\begin_inset Formula $x$
\end_inset
\begin_inset Quotes erd
\end_inset
is the subject,
\begin_inset Quotes eld
\end_inset
\begin_inset Formula $>$
\end_inset
\begin_inset Quotes erd
\end_inset
is the verb, and
\begin_inset Quotes eld
\end_inset
2
\begin_inset Quotes erd
\end_inset
is the object.
\end_layout
\begin_layout Standard
A very important part of doing mathematics is knowing when a symbolic expression
represents a sentence such as
\begin_inset Quotes eld
\end_inset
\begin_inset Formula $x>2$
\end_inset
,
\begin_inset Quotes erd
\end_inset
\begin_inset Quotes eld
\end_inset
\begin_inset Formula $a^{n}+b^{n}\ne c^{n}$
\end_inset
,
\begin_inset Quotes erd
\end_inset
and
\begin_inset Formula $2\mid6$
\end_inset
, and when it represents an object (a noun, if you will), such as
\begin_inset Quotes eld
\end_inset
\begin_inset Formula $x^{2}+2x+1$
\end_inset
.
\begin_inset Quotes erd
\end_inset
\end_layout
\begin_layout Standard
Other rules to consider:
\end_layout
\begin_layout Enumerate
Mathematical sentences are never treated as standalone sentences in a compositio
n, but rather as clauses, functioning as parts of sentences.
For example:
\end_layout
\begin_deeper
\begin_layout Quotation
Since
\begin_inset Formula $x+2\ge6$
\end_inset
, it follows that
\begin_inset Formula $x^{2}\ge16$
\end_inset
.
\end_layout
\end_deeper
\begin_layout Enumerate
While mathematical sentences can be rewritten entirely in words (e.g.
\begin_inset Quotes eld
\end_inset
\begin_inset Formula $x$
\end_inset
is greater than or equal to three plus
\begin_inset Formula $y$
\end_inset
,
\begin_inset Quotes erd
\end_inset
in place of
\begin_inset Quotes eld
\end_inset
\begin_inset Formula $x\ge3+y$
\end_inset
\begin_inset Quotes erd
\end_inset
), this only makes your composition more difficult to read, and so you should
use the mathematical notation to its fullest.
\end_layout
\begin_layout Enumerate
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
(
\begin_inset Quotes eld
\end_inset
Two,
\begin_inset Quotes erd
\end_inset
\begin_inset Quotes eld
\end_inset
Ninety-four,
\begin_inset Quotes erd
\end_inset
etc...) and is for the purpose of counting (used in much the same way you
would in any form of writing.)
\end_layout
\begin_layout Enumerate
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:
\end_layout
\begin_deeper
\begin_layout Quotation
If
\begin_inset Formula $x>3$
\end_inset
, then
\begin_inset Formula $x^{2}>9$
\end_inset
.
\end_layout
\end_deeper
\begin_layout Enumerate
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:
\end_layout
\begin_deeper
\begin_layout Quotation
If
\begin_inset Formula $ax^{2}+bx+c=0$
\end_inset
, and
\begin_inset Formula $a\ne0$
\end_inset
, then:
\begin_inset Formula
\[
x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}.
\]
\end_inset
\end_layout
\begin_layout Standard
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.
\end_layout
\end_deeper
\begin_layout Enumerate
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.
\end_layout
\begin_layout Enumerate
An inevitable part of writing mathematical proofs is a long string of computatio
ns 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_inset Formula
\begin{flalign*}
& \hfill & a & =b & \hfill\\
& & & =c\\
& & & =d.
\end{flalign*}
\end_inset
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
\begin_inset Quotes eld
\end_inset
\begin_inset Formula $\le$
\end_inset
\begin_inset Quotes erd
\end_inset
can be mixed in as follows:
\begin_inset Formula
\begin{flalign*}
& \hfill & a & =b & \hfill\\
& & & \le c\\
& & & =d\\
& & &