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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Principles of Mathematical Writing">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 15:06:22 1997}
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\section{Principles of Mathematical Writing}

Writing mathematics well requires thought and effort. Over many years,
mathematicians have developed several principles that help them to
communicate effectively and precisely. The following list of hints provides
a solid foundation for mathematical writing. If you are a student, ask your
instructor for additional hints as you master these.\medskip

\begin{description}
\item  \textsf{Write in complete sentences. Avoid disjointed series of
formulas.\medskip }
\end{description}

\begin{itemize}
\begin{itemize}
\item  \emph{BAD}: $3x-4=5x+6$, $3x-5x=6+4$, $-2x=10$, $x=-5\medskip $

\item  \emph{GOOD:} To solve the equation $3x-4=5x+6,$ add $4-5x$ to each
side to get 
\[
3x-5x=6+4.
\]
Combine like terms to get 
\[
-2x=10,
\]
and divide both sides by $-2$ to get $x=-5$.

Substituting $-5$ for $x$ gives $3(-5)-4=-19$ and $5(-5)+6=-19,$ which means
that $x=-5$ is indeed a solution.\medskip 
\end{itemize}
\end{itemize}

\begin{description}
\item  \textsf{Never begin a sentence with a symbol.\medskip }
\end{description}

\begin{itemize}
\begin{itemize}
\item  \emph{BAD:} $x$ is bigger than $10$.

\item  \emph{GOOD:} The number $x$ is bigger than $10$.\medskip
\end{itemize}
\end{itemize}

\begin{description}
\item  \textsf{Separate formulas by words.\medskip }
\end{description}

\begin{itemize}
\begin{itemize}
\item  \emph{BAD:} If $x<3$, $2x<6$.

\item  \emph{GOOD:} If $x<3$, then $2x<6$.\medskip
\end{itemize}
\end{itemize}

\begin{description}
\item  \textsf{Display important or complicated formulas.\medskip }
\end{description}

\begin{itemize}
\begin{itemize}
\item  \emph{BAD:} The solution to $ax^{2}+bx+c=0$ is given by the formula $%
x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}$ .

\item  \emph{GOOD:} The solution to the equation 
\[
ax^{2}+bx+c=0 
\]
is given by the formula 
\[
x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}. 
\]
\medskip
\end{itemize}
\end{itemize}

\begin{description}
\item  \textsf{Avoid unnecessary symbolism when words will provide a clear
statement.\medskip }
\end{description}

\begin{itemize}
\begin{itemize}
\item  \emph{BAD:} If $\varepsilon >0\Longrightarrow \exists \delta >0\ni
0<\left| x-a\right| <\delta \Longrightarrow \left| f(x)-f(a)\right|
<\varepsilon ,$ $f$ is continuous at $a$, $\lim_{x\rightarrow a}f(x)=f(a)$.

\item  \emph{GOOD:} The function $f$ is continuous at $a$ if for every real
number $\varepsilon >0$ there exists a real number $\delta >0$ such that
whenever $0<\left| x-a\right| <\delta $ it follows that $\left|
f(x)-f(a)\right| <\varepsilon $. In this case we write $\lim_{x\rightarrow
a}f(x)=f(a).$\medskip
\end{itemize}
\end{itemize}

\begin{description}
\item  \textsf{Do not use the same notation to represent two different
things in the same expression.\medskip }
\end{description}

\begin{itemize}
\item  \emph{BAD:} Define $h$ by the equation 
\[
h(x)=\frac{f(x+h)-f(x)}{h} 
\]

\item  \emph{GOOD:} Define $g$ by the equation 
\[
g(x)=\frac{f(x+h)-f(x)}{h} 
\]
\medskip
\end{itemize}

\begin{description}
\item  \textsf{Commas and periods should be placed inside quotation marks. }%
Colons, semicolons, question marks, and exclamation points stay outside
quotation marks unless they are part of the quotation.\medskip
\end{description}

\begin{itemize}
\begin{itemize}
\item  \emph{BAD:} She said, ``I like mathematics''.

\item  \emph{BAD:} Why did you say, ``I like mathematics?''

\item  \emph{GOOD:} Why did you say, ``I like mathematics''?

\item  \emph{GOOD:} She said, ``I like mathematics.''

\item  \emph{GOOD:} He said, ``Why do you like mathematics?''\medskip
\end{itemize}
\end{itemize}

\begin{description}
\item  \textsf{Parentheses and brackets should be strictly logical.}
Sentences may be entirely enclosed inside parentheses. Ending punctuation
for parentheses that include only a partial sentence should appear outside
the parentheses.\medskip
\end{description}

\begin{itemize}
\begin{itemize}
\item  \emph{BAD:} I can read, (but I cannot write.)

\item  \emph{GOOD:} I can listen, (but I cannot sing).

\item  \emph{GOOD:} I can do mathematics. (However, I cannot write about
it.)\medskip
\end{itemize}
\end{itemize}

\begin{description}
\item  \textsf{Use the spelling checker.\medskip }
\end{description}

\begin{itemize}
\begin{itemize}
\item  \emph{BAD:} If I find a mispelt werd, I shud fix it.

\item  \emph{GOOD:} Misspelled words are a distraction to the reader.\medskip
\end{itemize}
\end{itemize}

\QTR{NoteLeadin}{With the} spelling checker available on the \textsf{Tools}%
\textsf{\ }submenu, there are no excuses for misspelled words in reports or
on homework assignments! You can apply the spelling checker to selected
parts of your document or to your entire document.\medskip

\begin{description}
\item  \textsf{Be consistent.\medskip }
\end{description}

Many rules are still open to debate. For example many authors use ``$x$%
-axis'' and ``$x$-coordinate'' whereas others prefer ``$x$ axis'' and ``$x$
coordinate'' to describe axes and coordinates. Some authors include ending
punctuation for sentences that end in displayed mathematics, while others
insist that a display is itself a form of punctuation and that an additional
period is not necessary. Adopt a set of standards that are generally
acceptable and stick with these standards.\medskip

Those who read your writing will love you for helping them to read and
understand what you have written. If you are a student, it can also improve
your course grade.\noindent

\noindent \hrulefill

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\noindent \hrulefill \bigskip

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