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+\hyphenation{super-script}
+\title{Maths in \LaTeX: Part~1, Back to Basics}
+\author[R.~A. Bailey]{R.~A. Bailey\\Goldsmiths'
+College, University of London}
+
+\def\realrm{\rmfamily\upshape}
+\newtheorem{prequestion}{Question}
+\newtheorem{preanswer}{Answer}
+\renewcommand{\thepreanswer}{}
+\newenvironment{question}{\begin{prequestion}\realrm}{\end{prequestion}}
+\newenvironment{answer}{\begin{preanswer}\realrm}{\end{preanswer}}
+\newtheorem{preqn}{Exercise}
+\newenvironment{qn}{\begin{preqn}\realrm}{\end{preqn}}
+\newcommand{\latexword}[1]{{\ttfamily\upshape #1}}
+\newcommand{\theoremheadfont}[1]{{\textbf{#1}}}
+
+\newcommand{\lamport}{{\it \LaTeX: A Document Preparation System\/} by Leslie
+Lamport}
+\newcommand{\shortlamp}{\emph{The Manual}}
+
+\begin{Article}
+\section{Introduction}
+The book \lamport\ is rather coy about Mathematics. It simply does not reveal
+the full range of Mathematical expressions that can be correctly typeset
+without going outside \LaTeX. The result is that some Mathematical authors,
+while attracted to the generic mark-up of \LaTeX, believe that they need to use
+plain \TeX\ or \AmSTeX\
+to write their documents.
+
+This sequence of tutorials seeks to correct that impression, by
+explaining what Mathematical expressions can be typeset with \LaTeX\
+without the need for the \latexword{amstex} package. Perhaps this will
+provoke someone else to write a tutorial on that package. The first
+part is mostly, but not entirely, devoted to things which you can find
+in \shortlamp, even though you may have overlooked some of them.
+Succeeding parts (in the next and later issues of \BV) will be mostly
+about Mathematical goodies provided by \TeX\ but upon which
+\shortlamp\ is silent, even though they are necessary and quite easy
+to use. The final part will deal with arrays, concentrating on their
+use in Mathematics.
+
+These are tutorials, so I expect you, the reader, to do some work.
+Every so often comes a group of exercises, which you are supposed to
+do. Use \LaTeX\ to typeset everything in the exercise except sentences
+in italics, which are instructions. If you are not satisfied that you
+can do the exercise, then write to me with hard copy of your input and
+output (no email address before we go to press, I'm afraid): I will
+include a solution in the following issue of \BV.
+
+A word on fonts. Fonts in Mathematics are handled differently in
+\LaTeX\ 2.09, in NFSS, and in \LaTeXe. Rather than compare these
+systems every time that I mention fonts, I shall limit myself to
+\LaTeX\ 2.09. With any luck, this will enrage some knowledgeable
+person enough to write an article on handling of Maths fonts in
+different flavours of \LaTeX.
+
+\section{What does it look like?}
+
+\subsection{Maths Mode}
+\AllTeX\ has a special state, called {\em Maths mode}, which it must
+be in to recognize Mathematical expressions and typeset them properly.
+Maths mode in \LaTeX\ is everything between |\(| and~|\)|,
+or, alternatively, everything between |$| and~|$|. The
+parentheses are better for trapping errors, because it is obvious
+whether the left or right one is missing, if any. A missing~|$|
+causes \AllTeX\ to swap Maths mode and ordinary mode from then
+onwards, giving strange output but no errors until it eventually meets
+something like |x^2| that it cannot interpret in the wrong mode.
+On the other hand, the dollar signs are easier to type, and easier to
+see in your input file.
+
+In Maths mode most symbols are typeset as if they represent single-letter
+variables. A string of three letters will be set as if those three variables
+should be multiplied together. Fancy features like kerns and ligatures, which
+are used in normal text to help the reader interpret letter-strings as words,
+are turned off. Letters are set in the special font known as {\em Maths
+italic\/} which is usually used for variables.
+
+Almost all spaces that you type are ignored. \AllTeX\ thinks that it
+knows better than you do how Mathematics should be spaced, and it is
+probably right to think so.
+
+Don't stay in Maths mode for too long just because you are too lazy to type a
+few |$|~signs. Everything between the~|$|s should be Maths. A common
+mistake by beginners is to forget that a punctuation sign, like a comma, may
+have a different meaning in Maths from its meaning in text. In
+\begin{quote}
+the scalars $a$, $b$ and $c$
+\end{quote}
+we have a textual list containing three mathematical objects, so the input file
+contains
+\begin{verbatim}
+ the scalars $a$, $b$ and $c$
+\end{verbatim}
+That comma is a textual one. The lazy typist types
+\begin{verbatim}
+ the scalars $a, b$ and $c$
+\end{verbatim}
+and obtains
+\begin{quote}
+ the scalars $a, b$ and $c$
+\end{quote}
+On the other hand, in
+\begin{quote}
+the vector $(a,b,c)$
+\end{quote}
+there is a single Mathematical object, so it is correct to type
+\begin{verbatim}
+ the vector $(a,b,c)$
+\end{verbatim}
+or, equally well,
+\begin{verbatim}
+ the vector $(a, b, c)$
+\end{verbatim}
+These commas are part of the Mathematical notation.
+
+\subsection{Basic symbols}
+The basic symbols are the numerals {\tt 1}, {\tt 2}, \ldots, the Latin
+letters {\tt a}, {\tt b}, \ldots, {\tt z}, {\tt A}, \ldots, {\tt Z}, and the
+Greek letters |\alpha|, |\beta|, |\gamma|, \ldots,
+|\omega|, |A|, |B|, |\Gamma|, \ldots,~|\Omega|.
+If you don't know the standard English spellings of Greek letters, look on
+page~43 of \shortlamp. Upper-case Greek letters which are conventionally the
+same as their Latin equivalents do not have special commands. Some Greek letters
+have variants: |\varepsilon|, for example.
+
+The obvious symbols for operators are the keyboard symbols |+|
+and~{\tt -}. If you forget to go into Maths mode (a common temptation
+when typing a table of data), the symbol~{\tt -} will not look like a
+minus sign. Outside Maths mode the~|+| will look like a plus
+sign, but the spacing will be wrong. In Maths mode \AllTeX\ knows what
+is the proper spacing to put around binary operators like~\verb|+|
+and~{\tt -}; it also knows the proper spacing to surround binary
+relations like~{\tt =}. Try typing the following both inside Maths
+mode and outside it, and compare the results.
+\begin{verbatim}
+ 1 +2 = 3 4-1 = 3
+ 1 -4 = -3 -2+7 =+5
+\end{verbatim}
+Also try~{\tt >} outside Maths mode: you may be surprised.
+\label{symsec}
+
+\subsection{Sub- and superscripts}
+Subscripts are introduced with~|_|: for example, |x_n| gives~$x_n$.
+If there is more than one thing in the subscript you have to use braces, as in
+\verb!x_{n+1}! for~$x_{n+1}$. You can type |x_{n}| for $x_{n}$ if you
+want, but it makes your input file less readable.
+
+Superscripts are done similarly, using~|^|: thus |y^3| for~$y^3$ and
+|y^{-1}| for~$y^{-1}$.
+
+A sub- and superscript can be put on the same symbol in either order:
+|x_n^2| and |x^2_n| both produce~$x_n^2$. Double subscripts or
+superscripts are obtained by using braces in the obvious way: |x_{n_2}|
+and |n^{m^2}|.
+
+To put a sub- or superscript {\em before\/} a symbol, precede it with
+|{}|. Otherwise the sub- or superscript attaches itself to the previous
+thing, which may well be something like $+$ or~$=$.
+
+In an expression such as ${(X+Y)^2}$, strictly speaking \TeX\ thinks it is
+putting the superscript on the right parenthesis if
+you type \verb!(X+Y)^2!, and
+it positions the superscript in accordance with that thought. If this really
+offends you, you can force \TeX\ to share your logic by typing
+\verb!{(X+Y)}^2!, but you may not always prefer the result.
+
+\subsection{Modifying symbols}
+To turn~$x$ into~$x'$ type~|x'|. You do not need to think of the prime as
+a superscript.
+
+Some common modifiers are exemplified in
+\[
+\begin{tabular}{cc@{\qquad}cc}
+|\bar{x}| & $\bar{x}$ & |\tilde{x}| & $\tilde{x}$\\
+|\hat{x}| & $\hat{x}$ & |\vec{x}| & $\vec{x}$
+\end{tabular}
+\]
+A few more such decorations are shown on page~51 of \shortlamp. If any
+of them is used over an~$i$ or a~$j$ then the dotless versions of
+those letters should be used: |\imath| and |\jmath|.
+
+There are wide versions of |\hat| and |\tilde|:
+\[
+\begin{tabular}{c@{\qquad}c}
+\verb!\widehat{a+b}! & $\widehat{a+b}$\\
+|\widetilde{1-\theta}| & $\widetilde{1-\theta}$
+\end{tabular}
+\]
+There are also wide versions of |\bar| and |\vec| but with less
+obvious names: I'll cover these in a later tutorial.
+
+Logically, a decoration such as |\hat| may modify the whole of a
+subscripted expression such as~$x_2$; you usually mean `the estimate of~$x_2$'
+rather than `the second part of~$\hat{x}$'. However, both $\hat{x_2}$ and
+$\widehat{x_2}$ simply look wrong, so you have to let aesthetics triumph over
+logic and type |\hat{x}_2|.
+
+\subsection{Dots}
+To get a line of dots to show that some items have been missed out, use
+|\ldots| if the missing items are normally aligned on their baselines,
+such as letters, or |\cdots| if the missing items are normally aligned on
+the centreline, such as binary operators. If the missing items are part of a
+textual list, don't forget to come out of Maths mode and to put a comma at the
+end of the dots.
+\[
+\begin{tabular}{c}
+|for $i=1$, $2$, \ldots, $10$|\\
+|the vector $(x_1, x_2, \ldots, x_n)$|\\
+|$a_1 + a_2 + \cdots + a_n$|\\
+|$y_1 = y_2 = \cdots = y_7$|
+\end{tabular}
+\]
+If you think that the dollar signs round the numerals in the first example are
+unnecessary, try embedding that phrase in a piece of italic text.
+
+\subsection{Square roots}
+Type |\sqrt{2}| to obtain $\sqrt{2}$. The same technique works for more
+complicated expressions than~$2$: you don't have to do anything to make the
+root sign the right size. For example,
+\begin{quote}
+\verb!\sqrt{n^2+6}! \qquad $\sqrt{n^2+6}$
+\end{quote}
+Other roots, such as cube roots, are obtained by putting in an optional
+argument:
+\begin{quote}
+|\sqrt[3]{8} = 2| \qquad $\sqrt[3]{8} = 2$
+\end{quote}
+The simple symbol for a square root is |\surd|.
+
+Don't abuse \TeX's wizardry by using |\sqrt| for a large expression in
+text or in a complicated display. The mess obscures the message.
+
+\subsection{Displayed Maths}
+To get a single line of displayed Maths, type the contents between |\[|
+and~|\]|. You should not start a paragraph with displayed Maths, but may
+end one. If the displayed Maths is in the middle of a paragraph, remember not
+to leave blank lines around it in your input file.
+
+Displayed Maths may also be typed between |$$| and~|$$|, but the
+effect is not quite the same. For example, the document option
+\latexword{fleqn} aligns displayed Maths on the left if you use |\[| and
+|\]|, but not if you use |$$|.
+
+To put a short piece of text in displayed Maths, insert it in |\mbox|,
+remembering to include any necessary spaces that would be ignored in Maths
+mode.
+\begin{verbatim}
+ \[ a=b \mbox{ if } c=d \]
+\end{verbatim}
+Don't try to use |\mbox| in a similar way to put short text
+between pieces of Maths in text: it inhibits line-breaks.
+
+\subsection{Words as labels}
+Sometimes you want to attach natural-language words to Mathematical symbols to
+label them. For example, you might have analogous quantities associated with
+the rows and columns of a rectangular array, and wish to indicate this by using
+the same symbol, say~$Q$, with different subscripts. It simply will not do to
+type |Q_{rows}|, because this gives $Q_{rows}$, where the subscript looks
+like the product of $r$~by $o$ by \ldots. And it is no good putting {\tt rows}
+in an |\mbox|, because it will come out too big. Once something has been
+put in a box, it doesn't change size. You have to type
+|Q_{\rm rows}| to get $Q_\mathrm{rows}$.
+(Did you remember the caveat about fonts?)
+
+If this seems too much trouble, you might decide to abbreviate to $Q_r$
+and~$Q_c$. But this will not do either,
+because the subscripts look like variables
+into which numbers, say, could be substituted. If you don't want to mislead
+your readers, you should type |Q_{\rm r}|.
+
+\subsection{Fractions}
+A built-up fraction is made with |\frac|:
+\begin{quote}
+|\frac{n}{m}| \qquad $\displaystyle\frac{n}{m}$
+\end{quote}
+This comes out larger in displayed Maths than in text. It is better to use the
+solidus, as in $n/m$, for most fractions in text, with the exception of a few
+simple common fractions like~$\frac{1}{2}$.
+
+Of course, fractions can be put inside other fractions with no bother:
+\[
+\begin{tabular}{c}
+\verb!\frac{a(b+c)}{5 + \frac{1}{xy}}! \\[2\jot]
+$\displaystyle \frac{a(b+c)}{5 + \frac{1}{xy}}$
+\end{tabular}
+\]
+
+\subsection{Binary operators}
+In the golden olden days of golf-ball typewriters, it was a luxury to
+a Mathematician to have the symbol for direct sum, or for union.
+\AllTeX\ not only has the symbols; it knows that they are operators,
+and gives them the correct spacing for infix operators, and has
+reasonably good ideas about where to break lines near them. A few of
+the common ones are:
+\[
+\begin{tabular}{cc@{\qquad}cc@{\qquad}cc}
+\verb!+! & $+$ & \verb!-! & $-$ & \verb!\pm! & $\pm$\\
+\verb!\times! & $\times$ & \verb!\div! & $\div$ & \verb!\oplus! & $\oplus$\\
+\verb!\cup! & $\cup$ & \verb!\cap! & $\cap$ & \verb!\wedge! & $\wedge$.
+\end{tabular}
+\]
+There are many more on page~44.
+
+In fact, \AllTeX\ is even cleverer than this. If a binary operator
+doesn't find itself between two things it can operate on then it
+becomes a simple symbol, and spaces and line-breaks adjust
+accordingly. You should have noticed this if you did the exercise
+suggested above.%in Section~\ref{symsec}.
+
+\subsection{Binary relations}
+\AllTeX\ also knows about infix relations, such as
+\[
+\begin{tabular}{cc@{\qquad}cc@{\qquad}cc}
+\verb!=! & $=$ & \verb!\in! & $\in$ & \verb!\subset! & $\subset$\\
+\verb!<! & $<$ & \verb!\leq! & $\leq$ & \verb!\perp! & $\perp$.
+\end{tabular}
+\]
+More are shown on page~44. Don't confuse~$\in$ with either of the epsilons.
+
+Compare \verb!\mid! with~\verb!|!. The former is a relation, while the latter
+is just a symbol. So which should you use for `divides'?
+
+Relations can be negated by preceding them with |\not|:
+%\begin{quote}
+\[
+\verb!Z_2 \times Z_2 \not\cong Z_4! \qquad
+Z_2 \times Z_2 \not\cong Z_4
+\]
+%\end{quote}
+This doesn't work quite right for~$\in$, so there is the special command
+|\notin|. Also, |\ne| is a useful shorthand for |\not=|.
+
+\subsection{Fonts in Maths}
+(Did you remember the caveat about fonts?)
+
+For something like script letters use |\cal|, as in |${\cal F}(x)$|
+for $\mathcal{F}(x)$. The braces give the scope of |\cal|: for a
+single Mathematical letter such as~$\mathcal H$ you can get away with
+|$\cal H$|. Only upper-case Latin letters may be modified by |\cal|.
+
+In some branches of Mathematics, constants are shown in Roman type. So the base
+of natural logarithms is |{\rm e}|.
+
+For bold letters, you can use |\bf| to modify
+Latin letters and upper-case Greek ones:
+\[
+\verb!{\bf Mv} = a{\bf w}! \qquad \mathbf{Mv} = a\mathbf{w}
+\]
+For lower-case Greek letters, and for non-letters, you have to use a
+cumbersome construction:
+\begin{quote}
+\verb!\mbox{\boldmath $\lambda$}! \qquad \mbox{\boldmath $\lambda$}
+\end{quote}
+Because of the box, this does not change size properly in sub- and
+superscripts.
+
+\subsection{Writing Maths}
+The ability to produce beautiful Mathematical formulae is no licence to produce
+poor Mathematical writing. Remember that relations are verbs. It is impossible
+to parse the sentence
+\begin{quote}
+Therefore $n=56$ is the sample size.
+\end{quote}
+but
+\begin{quote}
+The equation $x^2+9=0$ has no real roots.
+\end{quote}
+is fine.
+
+Don't start a sentence with notation: the reader doesn't get the right visual
+clue. If possible, avoid putting notation immediately after {\em any\/}
+punctuation, unless it is part of a list. This saves the reader from having to
+work out if the punctuation is Mathematical or textual. Similarly, avoid
+abbreviations like `iid' and `e.g.'\ which might be mistaken for notation at a
+first glance.
+
+\section{Exercises}
+\begin{qn}
+The zeros of the quadratic $ax^2 + bx + c$ are
+\[
+\frac{-b\pm \sqrt{b^2 - 4ac}}{2a} .
+\]
+\end{qn}
+
+\begin{qn}
+The upper 5\% point of the $\chi^2_6$ distribution is 12.592.
+\end{qn}
+
+\begin{qn}
+If $\nu= n_1 + n_2 -2 $ and
+\[
+s^2 = \frac{(n_1 -1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 -2}
+\]
+then
+\[
+\frac{\bar{X}_1 - \bar{X}_2}{s\surd(\frac{1}{n_1} + \frac{1}{n_2})}
+\]
+is distributed as $t_\nu$.
+\label{tdist}
+\end{qn}
+
+\begin{qn}
+By choosing bases, it follows that the subspaces $Z_1$, \ldots, $Z_r$ span~$V$;
+hence it follows that $V$ is the direct sum $V=Z_1 \oplus \cdots \oplus Z_r$,
+as asserted.
+\end{qn}
+
+\begin{qn}
+If $\mathcal M$ and $\mathcal N$ are subspaces of a finite-dimensional inner product
+space $\mathcal V$ then
+\[
+(\mathcal{M} + \mathcal{N})^\perp = \mathcal{M}^\perp \cap \mathcal{N}^\perp
+\]
+and
+\[
+(\mathcal{M} \cap \mathcal{N})^\perp = \mathcal{M}^\perp + \mathcal{N}^\perp.
+\]
+Moreover, $\mathcal{M}^\perp \cong \mathcal{V}/\mathcal{N}$.
+\end{qn}
+
+\begin{qn}
+The sum of squares for the linear model $V_\mathrm{protein}
++ V_\mathrm{fishmeal}$
+is $1559378$.
+\end{qn}
+
+\begin{qn}
+\newcommand{\bbeta}{\mbox{\boldmath $\beta$}}
+\newcommand{\bep}{\mbox{\boldmath $\varepsilon$}}
+The usual regression equation is
+$\mathbf{Y} = \mathbf{X}\bbeta + \bep$, where
+$\mathbf{Y}$ is an $n\times 1$ vector, $\mathbf{X}$
+is an $n\times p $ matrix, $\bbeta$
+is the $p\times1$ vector of unknown parameters, and $\bep$ is the $n\times1$
+vector of random errors. The least-squares estimate $\widehat{\bbeta}$ of the
+parameters is given by
+\[
+\widehat{\bbeta} = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{Y}.
+\]
+\end{qn}
+
+\begin{qn}
+The $T$-orders are $p(x)^{e_1}$, $p(x)^{e_2}$ and $p(x)^{e_3}$, where $e_1 >e_2
+\geq e_3$. This implies that $p(x)^{e_1}\mid\eta(x)^{e_1-d}$ and hence that
+$\eta(x) = \psi(x)p(x)^d$ for some polynomial $\psi(x)$.
+\end{qn}
+
+
+\begin{qn}
+We have $t\in A\setminus B$ if and only if \[t \in A \mbox{ and } t\notin B.\]
+\end{qn}
+
+
+\begin{qn}
+Pascal's triangle is based on the identity
+\[
+{}^{n-1}\mathrm{C}_{k} + {}^{n-1}\mathrm{C}_{k-1} = {}^{n}\mathrm{C}_{k}.
+\]
+\end{qn}
+
+\end{Article}