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%% Thanks for your interest in A Gentle Introduction to TeX.
%% Any comments on this manual would be appreciated. These may
%% be typesetting, English, or TeX criticisms. If you decide to
%% translate this document into another language, I'd appreciate
%% receiving a copy.
%% This file is a complete TeX input file. Just run it
%% through TeX and print out the resulting "DVI" file. If you are
%% familiar with TeX, the macros at the top of the file have a few
%% switches which you may want to set. If you have problems or
%% can't run TeX at all, write to me and I'll send you a hard copy.
%% You should feel free to photocopy and/or distribute this manual.
%% My only request is that it remain in one piece and not be chopped
%% up. The only machine dependent section (#1.2) may need to be
%% rewritten for your local site, of course.
%%% Michael Doob
%% Department of Mathematics
%% The University of Manitoba
%% Winnipeg, Manitoba R3T 2N2
%% Canada
%% mdoob@uofmcc (bitnet)
%% [email protected] (internet)
%% Here is a character listing to check to be sure that no
%% unwanted translations took place within the bowels of the net.
%% Lower case letters: abcdefghijklmnopqrstuvwxyz
%% round parentheses, square brackets, curly braces: () [] {}
%% Exclaim, at, sharp, dollar, percent: ! @ # $ %
%% Caret, ampersand, star, underscore, hyphen: ^ & * _ -
%% vertical bar, backslash, tilde, backprime, plus: | \ ~ ` +
%% plus, equal, prime, quote, colon: + = ' " :
%% less than, greater than, slash, question, comma: < > / ? ,
%% period, semicolon: . ;
%% Now here come the macros used in this manual. If you are already
%% already familiar with TeX, you may want to fiddle with them.
%% In particular, the hooks are left to generate a new control
%% word index and table of contents if you change section 1.2.
%% Updated 01/04/90

%%%%%%%% Here are the fonts other than the sixteen defined in %%%%%%%%
%%%%%%%%%%%%% plain.tex that are used %%%%%%%%%%%%%%%%%%

%%%% first, choose between amr or cmr fonts %%%
\newif \ifamrfonts
\amrfontsfalse % use this line if you use the cmr fonts
%\amrfontstrue % use this line if you use the old armfonts

\ifamrfonts \font\brm=amr10 scaled \magstep1
\else \font\brm=cmr10 scaled \magstep1 \fi
\ifamrfonts \font\halfrm=amr10 scaled \magstephalf
\else \font\halfrm=cmr10 scaled \magstephalf \fi
\ifamrfonts \font\bbrm=amr10 scaled \magstep2
\else \font\bbrm=cmr10 scaled \magstep2 \fi
\ifamrfonts \font\bbbrm=amr10 scaled \magstep3
\else \font\bbbrm=cmr10 scaled \magstep3 \fi
\ifamrfonts \font\bbbbrm=amr10 scaled \magstep4
\else\font \bbbbrm=cmr10 scaled \magstep4 \fi
\ifamrfonts \font\bbbbbrm=amr10 scaled \magstep5
\else \font\bbbbbrm=cmr10 scaled \magstep5 \fi
\ifamrfonts \font\sf = amssmc10
\else \font\sf = cmss10 \fi
\ifamrfonts \font\chapfont=ambx10 scaled \magstep2
\else \font\chapfont=cmbx10 scaled \magstep2 \fi
\ifamrfonts \font\secfont=ambx10 scaled \magstep1
\else \font\secfont=cmbx10 scaled \magstep1 \fi
\ifamrfonts \font\sc= amcsc10
\else \font\sc= cmcsc10 \fi

%%%%\vbadness=10000 %%%%%% I don't want to hear about underfull vboxes
\raggedbottom %%%%% delete this line for aligned page bottoms
\widowpenalty=1000 \clubpenalty=1000 %%% I hate widows and orphans! %%%

%%%%%%%%% choosing between Canadian and American spellings %%%%%%%%%
\newif \ifcanspell
\canspelltrue %%%% Canadian spelling
\canspellfalse %%% use this line for American spelling

\def\aesthetic{\ifcanspell \ae{}sthetic\else esthetic\fi}
\def\analogue{\ifcanspell analogue\else analog\fi}
\def\cancelled{\ifcanspell cancelled\else canceled\fi}
\def\centimetre{\ifcanspell centimetre\else centimeter\fi}
\def\centre{\ifcanspell centre\else center\fi}
\def\centred{\ifcanspell centred\else centered\fi}
\def\our{\ifcanspell our\else or\fi}
\def\postcode{\ifcanspell postalcode\else zipcode\fi}
\def\province{\ifcanspell province\else state\fi}
\def\theatre{\ifcanspell theatre\else theater\fi}

%%%%%%%%%%%%%%%%%%%%%%% headline routines %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vbox {\hrule%
\line {\strut \vrule \quad \tenrm A \TeX{} intro
\ifcanspell (Canadian \else (U.S. \fi spelling)
\ifnum \secnum > 0 Section \the\secnum: \fi \sectiontitle \quad

\newif \iftitlepage \titlepagetrue
{\iftitlepage \hfil \global\titlepagefalse \else \gentleheadline \fi}

%%%%%%%%%%%%%%%%%%%% define contents and index files %%%%%%%%%%%%%%%%%%%%%%
%%% Normally the contents and index are hard coded into the input file. %%%
%% To generate new ones, use \writingcontentstrue and \writingindextrue. %%

\newif \ifwritingcontents
\newif \ifwritingindex
\newwrite\contents \newwrite\index

\writingcontentsfalse \writingindexfalse
%%% \writingcontentstrue %%% use this line to make the contents
%%% \writingindextrue %%% use this line to make the index

\ifwritingcontents \openout\contents=contents.tex \fi
\ifwritingindex \openout\index=index.tex
\def\toindex#1{\immediate\write\index{#1 \the\pageno}}
\else \def\toindex#1{} \fi

%%%%%%%%%%%%%%%%%%%% determine whether answers are printed %%%%%%%%%%%%%%%%
\newif \ifwritinganswers
\writinganswersfalse %%% use this line to suppress answer section
\writinganswerstrue %%% use this line to include answer section

%%%%%%%%%%%%%%%% footnote macro with counter %%%%%%%%%%%%
\newcount\footnotenum \footnotenum=0
\def\fnote#1{\advance \footnotenum by 1%

%%%%%%%%%%%%%%%% exercise, section, and subsection macros %%%%%%%%%%%%
\newcount\exno %%%%%% counter for exercises %%%%%%%%
\newcount\secnum \secnum=0 %%%% counter for section numbers %%%%

\vfill\eject %%%%% new section starts on a new page
%%%\ifodd\pageno \else\ \vfill\eject \fi %start on an odd page
\advance\secnum by 1 \subsecnum=0 \exno=0
\ifnum \secnum = 1 \pageno=1 \fi
\ifnum \secnum > 0
\leftline{\chapfont Section \the\secnum}
\vskip 3pt \fi
\leftline{\chapfont #1}
\vskip 1cm
\ifwritingcontents \write\contents{\string\line\string{#1
\ifnum \pageno < 0 \romannumeral-\pageno
\else \the\pageno \fi

\def\subsection#1{\advance\subsecnum by 1
\vskip 30pt
\leftline{\secfont \the\secnum .\the\subsecnum\ #1}
\string\dotfill{} \the\pageno\string}}\fi

\def\exercise{\global\advance \exno by 1
\noindent $\triangleright$ Exercise \the\secnum.\the\exno\quad

%% definitions of control sequences for characters in the typewriter font %%%
%%%% for short phrases this is easier than using a literal construction %%%%
\def\\{\char92{}} %%%%% backslash %%%%%
\def\lb{\char'173{}} %%%%% left brace %%%%%
\def\rb{\char'175{}} %%%%% right brace %%%%%
\def\sp{\char32{}} %%%%% special space symbol %%%%%

%{\obeyspaces\global\let =\ }


%%%%%%%%%%%%%% inhibit hyphenation of typewriter text %%%%%%%%%%%%%%

%%%%%%%%%%%% grouping to make input listing in typewriter type %%%%%%%
\tt \obeylines \parskip=0pt \parindent=0pt}


%%%%%%%%%%% macro to construct tables (easily) %%%%%%%%%%%%%%%%
%%% parameters: title goes between brackets, rest of the
%%% paragraph is the table
\hbox to \wd1{\bf \hss #1 \hss}
\vskip 12pt
%%%%%%%%%%% end of macro to construct tables %%%%%%%%%%%%%%%%

%%%%%%%%%%% macro to put a box around the text %%%%%%%%%%%%%%%%
\def\makebox#1#2#3% vsize, hsize, inserted text
\vbox to #1{\hrule \vss
\hbox to #2{\hss#3\hss}\vss


\def\LaTeX{{\rm L\kern-.36em\raise.3ex\hbox{\sc a}\kern-.15em

\def\AMSTeX{{$\cal A$}\kern-.1667em\lower.5ex\hbox
{$\cal M$}\kern-.125em{$\cal S$}-\TeX}

%%%%%%%%%%%% macro to put TeX references in right margin %%%%%%%%
\def \TeXref#1{%
\vadjust{\setbox0=\hbox{\sevenrm\quad\quad\TeX book: #1}%
\advance\theight by \dp0 \advance\theight by \lineskip
\kern -\theight \vbox to \theight{\rightline{\rlap{\box0}}%


%%%%%%%%%%%% macro to write the date out the date and time %%%%%%%%%%%
%%%%%%%%%%%% TeXbook p. 406 for date %%%%%%%%%%%
\def\today{\ifcase\month\or January\or February\or March\or April\or
May\or June\or July\or August\or September\or October\or November\or
December\fi \space\number\day, \number\year}

\newcount\hour \newcount\minute
\hour=\time \divide \hour by 60
\loop \ifnum \minute > 59 \advance \minute by -60 \repeat
\def\writetime{\ifnum \hour<13 \number\hour:% % supresses leading 0's
\ifnum \minute<10 0\fi% % so add it it
\ifnum \hour < 12 \ A.M. \else \ P.M. \fi
\else \advance \hour by -12 \number\hour:% % supresses leading 0's
\ifnum \minute<10 0\fi% % add it in
\number\minute \ P.M. \fi}

\rightline{\sevenrm Gentle Intro \version{} run through \TeX{}
on \today{} at \writetime}}

\topinsert \vskip 6 cm \endinsert
\centerline{\chapfont A Gentle Introduction to \TeX{}}
\vskip 15 pt
\centerline{\secfont A Manual for Self-study}
\vskip 3cm
\leftline{Michael Doob}
\leftline{Department of Mathematics}
\leftline{The University of Manitoba}
\leftline{Winnipeg, Manitoba, Canada R3T 2N2}
\leftline{[email protected]}
\leftline{[email protected]}

\secnum=-2 \pageno=-1


First the bad news: \TeX{} is a large and complicated program that
goes to extraordinary lengths to produce attractive typeset material.
This very complication can cause unexpected things to happen at times.
Now the good news: straightforward text is very easy to typeset using
\TeX\null. So it's possible to start with easier text and work up to
more complicated situations.

The purpose of this manual is to start from the very beginning and
to move towards these more complicated situations. No previous knowledge
of \TeX{} is assumed. By proceeding a section at a time, greater
varieties of text can be produced.

Here are a few suggestions: there are some exercises in each
section. Be sure and do them! The only way to learn \TeX{} is
by using it. Better yet, experiment on your own; try to make
some variations on the exercises. There is no way that you can
damage the \TeX{} program with your experiments. You can find a
complete answer to most exercises by looking at the \TeX{} source
file that was used to produce this document. You'll notice that
there are references in the right margin to {\bf The \TeX
book}\fnote{Addison Wesley, Reading,Massachusetts, 1984, ISBN
0-201-13488-9}\null. When you feel that you want more
information on a topic, that's where to look.

Incidently, there are a few fibs that appear in this manual;
they are used to hide complications (I look at this as something
like poetic license). As you become more experienced at using
\TeX{}, you'll be able to find them.

\TeX{} is a public domain program that is available for no license
fee. It was developed by Donald Knuth at Stanford University as
a major project.
In the profit-oriented market place,
the program would certainly cost many thousands of dollars. The
\TeX{} Users Group (TUG) is a nonprofit organization which distributes
copies of \TeX{}, updates software, and gives information about
new developments in both hardware and software in its publications
TUGboat and \TeX niques. Joining this users group is inexpensive;
please consider doing so. The address is:
\centerline{\TeX{} Users Group}
\centerline{P.O. Box 9506}
\centerline{Providence, RI 02940}

This manual would not have come into being without the aid of others.
In particular the proofreading and suggestions of the following
people have been invaluable:
Waleed A.~Al-Salam (University of Alberta),
Debbie L.~Alspaugh (University of California),
Nelson H.~F.~Beebe (University of Utah),
Barbara Beeton (American Mathematical Society),
Bart Childs (Texas A.~\&~M\null. University),
Mary Coventry (University of Washington),
Dimitrios Diamantaras (Temple University),
Roberto Dominimanni (Naval Underwater Systems Center),
Victor Eijkhout (University of Nijmegen),
Moshe Feder (St. Lawrence University),
Josep~M.~Font (Uviversidad Barcelona),
Jonas de Miranda Gomes
(Instituto de Matematica Pura e Aplicada - Brazil),
Rob Gross (Boston College),
Klaus Hahn (University of Marburg),
Anita Hoover (University of Delaware),
J\"urgen Koslowski (Macalester College),
Kees van der Laan (Rijksuniversiteit Groningen),
John Lee (Northrop Corporation),
Silvio Levy (Princeton University),
Robert Messer (Albion College),
Emily H.~Moore (Grinnell College),
Young Park (University of Maryland),
Craig Platt (University of Manitoba),
David Roberts (Colorado),
Kauko Saarinen (University of Jyv\"askyl\"a),
Jim Wright (Iowa State University),
Dominik Wujastyk (Wellcome Institute for the History of Medicine).

In addition several people have sent me parts or all of their local
manuals. I have received a few others on the rebound.
In particular
Elizabeth Barnhart (TV Guide),
Stephan v.~Bechtolsheim (Purdue University),
Nelson H.~F.~Beebe (University of Utah)
and Leslie Lamport (Digital Equipment Corporation),
Marie McPartland-Conn and Laurie Mann (Stratus Computer),
Robert Messer (Albion College),
Noel Peterson (Library of Congress),
Craig Platt (University of Manitoba),
Alan Spragens (Stanford Linear Accelerator Center, now of Apple Computers),
Christina Thiele (Carleton University),
and Daniel M.~Zirin (California Institute of Technology)
have written various types of lecture notes that have been most helpful.

\vfill \eject

\line{Introduction \dotfill{} i}
\line{Contents \dotfill{} iii}
\line{1.~Getting Started \dotfill{} 1}
\line{\qquad{}1.1 What \TeX{} is and what \TeX{} isn't \dotfill{} 1}
\line{\qquad{}1.2 From \TeX{} file to readable output,
the big set up \dotfill{} 2}
\line{\qquad{}1.3 Let's do it! \dotfill{} 4}
\line{\qquad{}1.4 \TeX{} has everything under control \dotfill{} 7}
\line{\qquad{}1.5 What \TeX{} won't do \dotfill{} 8}
\line{2.~All characters great and small \dotfill{} 9}
\line{\qquad{}2.1 Some characters are more special than others \dotfill{} 9}
\line{\qquad{}2.2 Typesetting with an accent \dotfill{} 10}
\line{\qquad{}2.3 Dots, dashes, quotes,
$\textfont0=\tenbf \mathinner {\ldotp \ldotp \ldotp }$ \dotfill{} 13}
\line{\qquad{}2.4 Different fonts \dotfill{} 15}
\line{3.~The shape of things to come \dotfill{} 19}
\line{\qquad{}3.1 Units, units, units \dotfill{} 19}
\line{\qquad{}3.2 Page shape \dotfill{} 20}
\line{\qquad{}3.3 Paragraph shape \dotfill{} 22}
\line{\qquad{}3.4 Line shape \dotfill{} 26}
\line{\qquad{}3.5 Footnotes \dotfill{} 27}
\line{\qquad{}3.6 Headlines and footlines \dotfill{} 28}
\line{\qquad{}3.7 Overfull and underfull boxes \dotfill{} 29}
\line{4.~$\Bigl\{$Groups, $\bigl\{$Groups,
$\{$and More Groups$\}\bigr\}\Bigr\}$ \dotfill{} 31}
\line{5.~No math anxiety here! \dotfill{} 33}
\line{\qquad{}5.1 Lots of new symbols \dotfill{} 33}
\line{\qquad{}5.2 Fractions \dotfill{} 37}
\line{\qquad{}5.3 Subscripts and superscripts \dotfill{} 38}
\line{\qquad{}5.4 Roots, square and otherwise \dotfill{} 39}
\line{\qquad{}5.5 Lines, above and below \dotfill{} 39}
\line{\qquad{}5.6 Delimiters large and small \dotfill{} 40}
\line{\qquad{}5.7 Those special functions \dotfill{} 41}
\line{\qquad{}5.8 Hear ye, hear ye! \dotfill{} 42}
\line{\qquad{}5.9 Matrices \dotfill{} 43}
\line{\qquad{}5.10 Displayed equations \dotfill{} 45}
\line{6.~All in a row \dotfill{} 48}
\line{\qquad{}6.1 Picking up the tab \dotfill{} 48}
\line{\qquad{}6.2 Horizontal alignment with more sophisticated patterns
\dotfill{} 51}
\line{7.~Rolling your own \dotfill{} 55}
\line{\qquad{}7.1 The long and short of it \dotfill{} 55}
\line{\qquad{}7.2 Filling in with parameters \dotfill{} 57}
\line{\qquad{}7.3 By any other name \dotfill{} 60}
\line{8.~To err is human \dotfill{} 62}
\line{\qquad{}8.1 The forgotten bye \dotfill{} 62}
\line{\qquad{}8.2 The misspelled or unknown control sequence \dotfill{} 62}
\line{\qquad{}8.3 The misnamed font \dotfill{} 64}
\line{\qquad{}8.4 Mismatched mathematics \dotfill{} 64}
\line{\qquad{}8.5 Mismatched braces \dotfill{} 65}
\line{9.~Digging a little deeper \dotfill{} 68}
\line{\qquad{}9.1 Big files, little files \dotfill{} 68}
\line{\qquad{}9.2 Larger macro packages \dotfill{} 69}
\line{\qquad{}9.3 Horizontal and vertical lines \dotfill{} 70}
\line{\qquad{}9.4 Boxes within boxes \dotfill{} 72}
\line{10.~Control word list \dotfill{} 77}
\line{11.~I get by with a little help\dotfill{} 80}

\section{Getting Started}

\subsection{What \TeX{} is and what \TeX{} isn't}

First of all, let's see what steps are necessary to produce a
document using \TeX\null. The first step is to type the file
that \TeX{} reads. This is usually called the \TeX{} file or the
input file, and it can be created using a simple text editor (in
fact, if you're using a fancy word processor, you have to be sure
that your file is saved in ASCII or nondocument mode without any
special control characters). The \TeX{} program then reads your
input file and produces what is called a DVI file (DVI stands for
DeVice Independent). This file is not readable, at least not by
humans. The DVI file is then read by another program (called a
device driver) that produces the output that is readable by
humans\TeXref{23}. Why the extra file? The same DVI file can be
read by different device drivers to produce output on a dot
matrix printer, a laser printer, a screen viewer, or a
phototypesetter. Once you have produced a DVI file that gives
the right output on, say, a screen viewer, you can be assured
that you will get exactly the same output on a laser printer
without running the \TeX{} program again.

The process may be thought of as proceeding in the following way:
{\hbox{edit text\quad} \atop \longrightarrow}
\lower .6cm \makebox{1.5cm}{1.8cm}%
{\hbox{\quad \TeX{} program\quad} \atop \longrightarrow}
\lower .6cm \makebox{1.5cm}{1.8cm}%
{\hbox{\quad device driver\quad} \atop \longrightarrow}
\lower .6cm \makebox{1.5cm}{1.8cm}%

This means that we don't see our output in its final form when it
is being typed at the terminal. But in this case a little
patience is amply rewarded, for a large number of symbols not
available in most word processing programs become available. In
addition, the typesetting is done with more precision, and the
input files are easily sent between different computers by
electronic mail or on a magnetic medium.

Our focus will be on the first step, that is, creating the \TeX{}
input file and then running the \TeX{} program to produce
appropriate results. There are two ways of running the \TeX{}
program; it can be run in batch mode or interactively. In
batch mode you submit your \TeX{} input file to your computer;
it then runs the \TeX{} program without further intervention and
gives you the result when it is finished. In interactive mode
the program can stop and get further input from the user, that
is, the user can interact with the program. Using \TeX{}
interactively allows some errors to be corrected by the user,
while the \TeX{} program makes the corrections in batch mode as
best it can. Interactive is the preferred mode, of course. All
personal computer and many mainframe implementations are
interactive. On some mainframes, however, the only practical
method of running \TeX{} is in batch mode.

\subsection{From \TeX{} file to readable output, the big set up}

{\parskip = 0pt \noindent
[Note from MD:
{\sl This is the only system dependent section in the manual and
may be replaced by a local guide. No reference is made to it
outside of the section itself. The following local information
should be included:
\item{$\bullet$} What initial steps, if any, should be taken by the reader
to permit the running of \TeX{} and your local device driver(s).
\item{$\bullet$} How to run \TeX\null.
\item{$\bullet$} How to read the log file.
\item{$\bullet$} How to preview and/or print the dvi file.

The following sample is applicable here at the University of
Manitoba. We use a locally written editor (MANTES) on an
Amdahl running MVS; I'm fairly certain that it's the worst possible case.}]

In this section we'll see how to run \TeX{} at the University of
Manitoba. It is assumed that the reader is familiar with MANTES
and can create text files using it.

First, there are several things that must be done {\bf one time
only}. To start you must do the following (you type in the
material that looks like typewriter type):
\item{(1)} allocate the files that \TeX{} will use by typing the
following lines (while in MANTES):
\itemitem{} C: {\tt alloc da=source.tex format=vb,256,6144}
\itemitem{} C: {\tt alloc da=dvi format=fb,1024,6144}

\item{(2)} Create a file called RUNTEX in your MANTES aggregate
containing the following JCL:


The name {\tt } is replaced by your own user id, of
course. The use of upper case and the spaces must be followed

\item{(3)} Create a file called PRINTTEX in your MANTES aggregate
containing the following JCL:


Once you have completed these three steps, you are ready to run a
\TeX{} job. The files you have created will allow you to produce
about ten pages of ordinary text.

Here are the steps you use {\bf each time you run a job}.

\item{(1)} create a MANTES file containing your \TeX{} input.
\item{(2)} save and submit your file using the commands
\itemitem{} C: {\tt save f/l to da=source.tex noseq}
\itemitem{} C: {\tt submit runtex}
\item{(3)} when you get a message saying that your job is finished,
enter the command
\itemitem{} C: {\tt out ; list ttyout}

In this command, {\tt } is replaced by your user id with
a dollar sign appended. This file listing will tell you of any
errors that might have occurred. It is an abbreviated version of
what is called the ``log file''; we will use the term ``log
file'' to refer to the ttyout file produced by \TeX\null.

If you want, you can check on the status of your job while it is
executing by using the command
\itemitem{} C: {\tt q }

When you are finished looking at the log file, the command {\tt
end scratch} will throw away the log file while the command {\tt
end release} will send the log file to the printer, and it can
then be picked up with your \TeX{} output.

\item{(4)} when your output from RUNTEX program is error free, \TeX{}
will have created a legal DVI file. To print it, use the
\itemitem{} C: {\tt submit printtex nohold}

As in (3), you can check on the status of your job while it is
\item{(5)} Pick up your output at the I/O window, sixth floor
Engineering building. It usually takes about twenty minutes for the
output to be ready. Ask for it by ``{\tt }''.

The files created are large enough for running \TeX{} jobs of about
10 pages. A job of that size will take about one second of CPU time
to run through \TeX\null. It will take about 15 seconds of CPU time to
print 10 pages on the Xerox 8600 using the current device driver.

You can print your own copy of this manual using the command
{\tt \%docu tex}. You can also find lots of other useful
online information about \TeX{} by using {\tt \%texinfo}.

\subsection{Let's do it!}

So, from our viewpoint, the name of the game is to create the \TeX{}
file that produces the right readable output. What does
a \TeX{} file look like? It consists of characters from an
ordinary terminal, that is, upper and lower case letters,
numbers, and the usual punctuation and accent characters (these
are the usual ASCII characters). Text, for the most part, is just
typed normally. Special instructions usually start with one of a
few special symbols such as {\tt \#} or {\tt \&} (these will be
described in more detail later). Here is an example of a \TeX{}
input file:

Here is my first \\TeX\\ sentence.
\toindex{TeX{} } \toindex{ }

First note that the characters in this example look like
typewriter type. We use these characters with all examples that
are meant to be typed from the computer terminal. Second, note
that the backslash appears three times in the text. We'll soon
see that this is one of the special symbols mentioned
previously, and it is used very frequently when making \TeX{}
documents. Make a file containing this example. Use the \TeX{}
program to make a DVI file and a device driver to see what you
have produced. If all goes well, you'll have a single page with
the following single sentence:

Here is my first \TeX\ sentence.

There will also be a page number at the bottom of the page. If
you've gotten this far, congratulations! Once you can produce one
\TeX{} document, it's just a matter of time before you can do
more complicated ones. Now let's compare what we typed in with
what we got out. The straightforward words were just typed in
normally, and \TeX{} set them in ordinary type. But the word ``\TeX'',
which can't be typed in on a terminal because the letters
aren't on the same line, is entered by using a word starting with
a backslash, and \TeX{} made the proper interpretation. Most
symbols that are not ordinary letters, numbers, or punctuation
are typeset by entering a word starting with a backslash. If we
look a little closer, we'll note that the word ``first'' is also
The first two letters have been joined together and there isn't
a separate dot over the letter ``i''. This is standard
typesetting practice: certain letter combinations are joined up
to form what are called {\sl ligatures}. There is actually a good
\aesthetic{} reason for this. Compare the first two letters of
``first'' and ``{f}irst'' to see the difference. We note that
{\tt \\bye} appears in the input file with no corresponding word
in the final copy.
This a typesetting instruction that tells \TeX{} that the input
is finished. We'll learn about lots of different typesetting
instructions as we go along.

Let's look at the log file that was created when we ran \TeX
\null. It may vary slightly at your site, but should look
something like this:
2. This is TeX, MVS Version 2.9 (no format preloaded)
5. ** File SRC.TEX <-- DD=SRC
6. (src.tex [1]
7. Output written on DVI (1 page, 256 bytes).
8. Transcript written on TEXLOG.

This is the file that will contain any error messages. On line 6,
{\tt (src.tex\ } indicated that \TeX{} has started reading that
file. The appearance of {\tt [1]} indicates that page 1 has been
processed. If there were errors on page 1, they would be listed
at that point.

\vskip .5cm
\exercise Add a second sentence to your original \TeX{} file to get:
Here is my first \\TeX\\ sentence.
I was the one who typeset it!

Use \TeX{} and look at your output. Is the second sentence on a
new line?

\exercise Now add this line to the beginning of your file:
Guess what will happen when you run the new file through
\TeX\null. Now try it and see what happens.

\exercise Add three or four more sentences to your file.
Use letters, numbers, periods, commas, question marks,
and exclamation points, but don't use any other symbols.

\exercise Leave a blank line and add some more sentences.
You can now get new paragraphs.

We have now seen a major principle concerning the preparation of
\TeX{} input files. The placement of the text on your computer
terminal does not necessarily correspond to the placement of the
text on your output. You can not, for example, add space between
words in your output by adding spaces in your input file. Several
consecutive spaces and one space will produce exactly the same
output. As would be expected, a word at the end of one line will
be separated from the first word of the following line. In fact,
sometimes when working on a file that will be heavily edited, it
is convenient to start each sentence on a separate line. Spaces
at the beginning of a line, however, are always ignored.

\exercise Add the following sentence as a new paragraph, and then
typeset it:
Congratulations! You received a grade of 100\% on your latest
The {\tt\%} sign is used for comments in your input file.
Everything on a line following this symbol is ignored. Notice
that even the space that normally separates the last word on one
line from the first word on the next line is lost. Now put a
backslash in front of the {\tt\%} sign to correct the
sentence. \toindex{\$} \toindex{\%}

\exercise Add the following sentence as a new paragraph:
You owe me \$10.00 and it's about time you sent it to me!
This will produce an error in your log file (if your
implementation of \TeX{} is interactive, that is, sends you
messages and waits for answers, just hit the carriage return or
enter key when you get the error message). You will get output,
but not what you might expect. Look at the log file and see
where the error messages are listed. Don't worry about the
actual messages. We'll have a lot more to say about errors
(including this one) later. Now fix the error by putting a
backslash in front of the {\tt \$} sign, and typeset the result
(there are a small number of characters like the per cent and
dollar signs that \TeX{} uses for its own purposes. A table of
these characters will be provided shortly).

\subsection{\TeX{} has everything under control}

We have seen that the backslash has a special role. Any word
starting with a backslash will be given a special interpretation
when \TeX{} reads it from your input file. Such a word is called
a {\sl control sequence}. There are, in fact, two types of
control sequences: a {\sl control word\/} is a backslash followed
by letters only (for example, {\tt \\TeX}) and a {\sl control
symbol\/} is a backslash followed by a single nonletter (for
example, {\tt \\\$})\null. Since a space is a nonletter, a
backslash followed by a space is a legitimate control symbol.
When we want to emphasize that a space is present, we will use a
special symbol {\tt\sp} to indicate the space, as in the control
symbol {\tt\\\sp}. This convention is used in {\bf The \TeX
book} as well as in this manual.

When \TeX{} is reading your input file and comes to a backslash
followed by a letter, it knows that a control word is being
read. So it continues reading the name of the control word until
a nonletter is read. So if your file contains

\displaytext{\tt I like \\TeX!}

the control word {\tt \\TeX} is terminated by the exclamation
point. But this presents a problem if you want to have a space
after a control word. If you have, for example, the sentence

\displaytext{\tt I like \\TeX and use it all the time.}

in your input file, the control word {\tt\\TeX} is terminated by
the space (which is, of course, a nonletter). But then you won't
have a space between the words ``\TeX{}'' and ``and''; inserting
more spaces won't help, since \TeX{} doesn't distinguish between
one space and several consecutive spaces. But if you put the
control symbol {\tt\\\sp} after a control word, you will both
terminate the control word and insert a space. It's really easy
to forget to put in something like {\tt\\\sp} after a control
word. I promise you that you will do it at least once while
you're learning to use \TeX\null.

\exercise Make an input file that will produce the following

I like \TeX! Once you get the hang of it, \TeX{} is really easy
to use. You just have to master the \TeX nical aspects.

Most control words are named so as to give a hint of their uses.
You can use {\tt \\par} to make a new paragraph, for example,
instead of putting in a blank line.

\subsection{What \TeX{} won't do}

\TeX{} is very good at setting type, but there are things that \TeX{}
can't do well. One is making figures. Space can be left
to insert figures, but there are no graphic procedures built into
the language (at present). Some implementations allow graphic
instructions to be inserted using the {\tt \\special} control
word but these are exceptional and definitely site dependent.

\TeX{} sets type in horizontal straight lines and not in
straight lines at other angles. In particular, it is generally
not possible to make insertions in ``landscape mode'', that is,
with the text rotated by so that the baseline is parallel with
the long edge of the paper, or to include text that has a curve for
a baseline. Perspective type (gradually increasing or
decreasing in height) is not handled well by \TeX\null.

We have seen that there is an ``edit, \TeX{}, driver'' cycle that
is necessary for each different copy of output. This is true
even when the output device is a terminal. In particular, it's
not possible to type the input file and see the results on the
screen immediately without going through the full cycle. Some
implementations have both text and graphics displays with
reasonably quick turnaround (a few seconds for a single page); as
hardware becomes less expensive and processors become faster, we
may see improvement.

\section{All characters great and small}

\subsection{Some characters are more special than others}

We saw in the last section that most text is entered at the
terminal as sentences of ordinary words just as when typing with
a typewriter. But we also saw that, in particular, the backslash
could be used for at least two different purposes. It can be
used for symbols (or combinations of symbols) that don't appear
on the keyboard such as typing {\tt \\TeX} to get \TeX\null. It
can also be used to give \TeX{} special instructions such as
typing {\tt \\bye} to indicate the end of the input file. In
general, a word starting with a backslash will be interpreted by
\TeX{} as one requiring special attention. There are several
hundred words that \TeX{} knows, and you can define more
yourself, and so the backslash is very important. We'll spend a
lot of time as we proceed learning some of these words;
fortunately we'll only need to use a small number of them most of
the time.

There are ten characters which, like the backslash, are used
by \TeX{} for special purposes, and we now want to give the
complete list.
\TeXref{37--38}What if we want to use a sentence with one of these
special characters in it? With this in mind we'll ask the following
\item{(1)} What are the different special characters?
\item{(2)} How do we use a special character if we really want to
typeset it in our text?

Here is a table of each special character, its purpose, and the
method of typesetting the special character itself if you need

\maketable [Characters requiring special input]
\strut \hfil#\hfil & \quad # \hfil & \quad\tt # \hfil\cr
\bf Character & \bf Purpose & \bf Input for literal output \cr
\noalign{\hrule} \noalign{\smallskip}
$\backslash$ & Special symbols and instructions & \$\\backslash\$ \cr
$\{$ & Open group & \$\\\lb\$ \cr
$\}$ & Close group & \$\\\rb\$ \cr
\% & Comments & \\\% \cr
\& & Tabs and table alignments & \\\& \cr
\~{} & Unbreakable space & \\\~{}\lb\rb \cr
\$ & Starting or ending math text & \\\$ \cr
\^{} & Math superscripts & \\\^{}\lb\rb \cr
\_{} & Math subscripts & \\\_{}\lb\rb \cr
\# & Defining replacement symbols & \\\# \cr


\subsection{Typesetting with an accent}

Now we're going to start using some of \TeX{}'s goodies! So far
we've just been using \TeX{} to make our output look attractive,
but now we'll start to do things that are difficult or impossible
on the typewriter. In particular, we're going to look at accents
now. How do you produce an accent when the symbol doesn't appear on
the keyboard? Just as with the symbol \TeX{}, it is necessary to
enter a word starting with a backslash. For the word ``premi\`ere'',
as a first example, you need to type in {\tt premi\\`ere}
(you may have to hunt around to find the ``back prime''
or ``grave'' symbol {\tt `} on your keyboard, but it's there
somewhere\fnote{If you have a very old or obscure
keyboard and the back prime is {\it really\/} not there, you can
use {\tt \\lq\lb\rb} instead. Similarly {\tt \\rq\lb\rb} can be
used for the symbol {\tt '}. You can think of the symbols as
being abbreviations for ``left quote'' and ``right quote.'' In
addition, {\tt \\lq\\lq\lb\rb} and {\tt \\rq\\rq\lb\rb} give the
usual quotation marks. But this won't work as a method to produce
an accent over the following letter, so you're really better off
with a proper keyboard.}). In general, to put an accent
on a letter, the appropriate control sequence precedes the letter.

Here are some examples:

\strut \hfil\tt # & \quad # \hfil\cr
\bf \TeX{} input & \bf \TeX{} output \cr
\noalign{\hrule} \noalign{\smallskip}
\\`a la mode & \`a la mode \cr
r\\'esum\\'e & r\'esum\'e \cr
soup\\c{\sp}con & soup\c con \cr
No\\"el & No\"el \cr
na\\"\\i{\sp}ve & na\"\i ve \cr

We see several principles illustrated by these examples. Most
accents are produced by using a control symbol with a similar
shape. A few of them are produced by control words containing a
single letter. Some care must be used in this case, for a space
must be used to terminate the control word. If you have {\tt
soup\\ccon} in your file, for example, \TeX{} will look for the
control word {\tt\\ccon}\fnote{We'll see that there is another
method when we look at the grouping concept in Section~4.}.

Notice that there is a control word {\tt\\i} also. This produces
the letter ``i'' without the dot over it; this allows an accent
to be put over the lower part of the letter. There is an
analogous control word {\tt\\j} that produces a dotless ``j''
for accenting purposes.

\maketable [Accents that may be immediately followed by a letter]
\strut \hfil # & \quad \hfil \tt # \hfil & \hfil \quad # \hfil\cr
\bf Name & \bf \TeX{} input & \bf \TeX{} output \cr
\noalign{\hrule} \noalign{\smallskip}
grave & \\`o & \`o \cr
acute & \\'o & \'o \cr
circumflex & \\\^{}o & \^o \cr
umlaut/dieresis/tr\'emat & \\"{}o & \"o \cr
tilde & \\\~{}o & \~o \cr
macron & \\={}o & \=o \cr
dot & \\\char'056{}o & \.o \cr


\maketable [Accents requiring an intervening space]
\strut \hfil # \hfil & \quad \hfil \tt # \hfil & \hfil \quad # \hfil\cr
\bf Name & \bf \TeX{} input & \bf \TeX{} output \cr
\noalign{\hrule} \noalign{\smallskip}
cedilla & \\c o & \c{o} \cr
underdot & \\d o & \d{o} \cr
underbar & \\b o & \b{o} \cr
h\'a\v{c}ek & \\v o & \v{o} \cr
breve & \\u o & \u{o} \cr
tie & \\t oo & \t oo \cr
Hungarian umlaut& \\H o & \H{o} \cr


\TeX{} also allows some letters from languages other than English to
be typeset.

\maketable [A\kern-1pt vailable foreign language symbols]
\strut \hfil # \hfil & \quad \hfil \tt # \hfil & \hfil \quad # \hfil\cr
\bf Example& \bf \TeX{} input & \bf \TeX{} output \cr
\noalign{\hrule} \noalign{\smallskip}
\AE gean, \ae sthetics & \\AE, \\ae & \AE, \ae \cr
\OE uvres, hors d'\oe uvre & \\OE \\oe & \OE, \oe \cr
\AA ngstrom & \\AA, \\aa & \AA, \aa \cr
\O re, K\o benhavn & \\O, \\o & \O, \o \cr
\L odz, \l\'odka & \\L, \\l & \L, \l \cr
Nu\ss & \\ss & \ss \cr
?`Si? & {?}{`} & ?` \cr
!`Si! & {!}{`} & !` \cr
Se\~nor & \\\~{} & \~{} \cr
& \lb\\it\\\$\rb & {\it\$} \cr


Typeset the sentence in each of the following exercises:

\exercise Does \AE schylus understand \OE dipus?

\exercise The smallest internal unit of \TeX{} is about 53.63\AA.

\exercise They took some honey and plenty of money wrapped up in a
{\it \$}5 note.

\exercise \'El\`eves, refusez vos le\c cons! Jetez vos cha\^\i nes!

\exercise Za\v sto tako polako pijete \v caj?

\exercise Mein Tee ist hei\ss.

\exercise Peut-\^etre qu'il pr\'ef\`ere le caf\'e glac\'e.

\exercise ?`Por qu\'e no bebes vino blanco? !`Porque est\'a avinagrado!

\exercise M\'\i\'\j n idee\"en worden niet be\"\i nvloed.

\exercise Can you take a ferry from \"Oland to \AA land?

\exercise T\"urk\c ce konu\c san ye\u genler nasillar?

\def\bdots{$\textfont0=\tenbf \ldots$} %%% boldface version of \ldots

\subsection{Dots, dashes, quotes, \bdots}

Typing has always been a compromise: the small number of keys
(compared to the number of typeset symbols available) has forced
some changes on the typist. When preparing material using \TeX{},
there is no need to be so restricted. In this section we'll look
at some differences between typing and using \TeX\null.

There are four types of dashes that are used. The hyphen is used
for combining words into one unit as with mother-in-law.
The en-dash is used to indicate a sequence of page numbers, years
or such things. The em-dash is a grammatical symbol. The minus
sign is used for negative numbers. Here they are with their uses:

\maketable [Different types of dashes]
\strut \hfil # \hfil & \quad \hfil \tt # \hfil & \hfil \quad # \hfil
& \hfil # \hfil\cr
\bf Name& \bf \TeX{} input & \bf \TeX{} output & \bf Example\cr
\noalign{\hrule} \noalign{\smallskip}
hyphen & - & - & The space is 3-dimensional. \cr
en-dash & {-}{-} & -- & Read pages 3--4. \cr
em-dash & {-}{-}{-} & --- & I saw them---there were 3 men alive. \cr
minus sign & \$-\$ & $-$ & The temperature dropped to $-$3 degrees. \cr

\exercise I entered the room and---horrors---I saw both my
father-in-law and my mother-in-law.

\exercise The winter of 1484--1485 was one of discontent.

Another difference between typing and using \TeX{} is the use of
quotation marks. Opening and closing quotation marks are the
same on a typewriter. They are produced in \TeX{} by using the
apostrophe or prime key {\tt '} and backprime key {\tt `}\null.
An open quote is produced by {\tt ``} and the
close quote by {\tt ''}\null.\TeXref{3} Similarly the
opening single quote is produced by {\tt `} and the closing
single quote by {\tt '}\null. Notice that there is no need
to use the usual typing quotation mark (it normally gives a close
quote, but don't count on it).

\exercise His ``thoughtfulness'' was impressive.

\exercise Frank wondered, ``Is this a girl that can't say `No!'?''

Sometimes ellipses (three dots) are used to indicate a passage
of time or missing material. Typing three periods will give
three dots very close together. The dots with proper spacing will
be produced by having {\tt \\dots} in your input file.

\exercise He thought, ``\dots and this goes on forever, perhaps to the
last recorded syllable.''

Another problem with dots is that the period after an
abbreviation normally has less space after it than does the
period at the end of a sentence. There are two ways to solve this
problem: the dot can be followed by either {\tt\\\sp} or {\tt\~{}}
to change the spacing.
The second alternative will give an unbreakable space; when such
a space is between two words, these words must typeset on the
same line. The input {\tt Prof.\~{}Knuth} would cause those two
words to be typeset on one line. This is desirable for names
like Vancouver, B.~C. and Mr.~Jones so that ``Mr.'' and ``Jones''
do not end up on separate lines. Notice that no backslash is used
with the unbreakable space.

\exercise Have you seen Ms.~Jones?

\exercise Prof.~Smith and Dr.~Gold flew from
Halifax N.~S. to Montr\'eal, P.~Q. via Moncton, N.~B.
L.~A. to Washington, D.~C., via Fargo, N.~D.

\subsection{Different fonts}

The most obvious difference between typewritten text and \TeX{}
output is undoubtedly the different fonts or types of symbols
used. When \TeX{} starts up it has sixteen fonts available. Some
of these are used for technical purposes, but even so there are
several different typefaces available as can be seen in the
following table. A complete list of the sixteen fonts is given in
Appendix F of {\bf The \TeX{}book}.
Most of them are used automatically; a mathematical subscript,
for example, is put in smaller type by \TeX{} with no special
user intervention.

To change from the usual (roman) typeface to italic, the control
word {\tt \\it} is used. To change back to the usual roman
typeface, use {\tt \\rm}. For example, you could have the
following sentence: {\tt I started with roman type, \\it switched
to italic type, \\rm and returned to roman type}. This would give
the following: I started with roman type, \it switched to italic
type,\fnote{Notice that the comma and this footnote are in
italic type, and this looks a little funny. We'll see that there
is another method for changing fonts when we talk about grouping
in Section 4.}
\rm and returned to roman type.

Typefaces other than italic are also available. Those given below
are the ones most commonly used. They are available immediately
when \TeX{} starts. A little later we'll see how to use other
typefaces that aren't available when \TeX{} starts.

\global\advance\footnotenum by 1
\maketable [Font Samples]
\strut \hfil # \hfil & \quad \hfil \tt # \hfil & \hfil \quad # \hfil\cr
\bf Font name & \bf \TeX{} switch sequence & \bf Sample of typeface \cr
\noalign{\hrule} \noalign{\smallskip}
Roman & \\rm & \rm This is roman type. \cr
Boldface & \\bf & \bf This is boldface type. \cr
Italic & \\it & \it This is italic type. \cr
Slanted & \\sl & \sl This is slanted type. \cr
Typewriter & \\tt & \tt This is typewriter type. \cr
Math symbol${}^\the\footnotenum$ & \\cal & $\cal SOME\hbox{\quad}SCRIPT
\hbox{\quad}LETTERS$\rm. \cr

This example is cheating since you need to know a little about
mathematics input to use it, but I wanted to include it anyway.
You'll be able to use these letters after you study the use of
mathematical symbols.}

The slanted and italic fonts seem quite similar at first. It is
easy to see the difference between the letter ``a'' in each
sample. When changing from a slanted or italic font to a roman
font, the last letter in the first font will lean towards the
first letter of the roman font; this looks cramped, and to
compensate there is a little extra space that can be added called
the {\sl italic correction}. This is done using the control
symbol {\tt \\/}.
In the following sentence, there is no italic correction after
the first sequence of italic letters but there is after the
second sequence; see the difference: {\it If} the italic
correction is not used the letters are too close together, but
{\it if\/} the correction is used, the spacing is better. There
is no need for the italic correction when the italic characters
are followed by a comma or a period, but your text will
definitely look better if you use it before quotation marks or

It is possible to use fonts other than those initially defined in
\TeX{} when desired (assuming that they are available on
your computer system, of course). Different sizes can be used
with the aid of the control word {\tt \\magstep}. To define your
new font you will have to know its name on your computer system.
For example, the roman typeface is called ``cmr10'' on most
systems. If you use {\tt \\font\\bigrm = cmr10 scaled \\magstep
1}, you can then use {\tt\\bigrm} in exactly the same way as you
use {\tt\\it} or {\tt\\rm}.

Switching via {\tt\\bigrm} will give roman type that is larger
than the usual by 20\%\null. {\tt \\font\\bigbigrm = cmr10
scaled \\magstep 2} will define a font that is about 44\% larger
than the usual roman typeface. The sizes {\tt \\magstep~0} to
{\tt \\magstep~5} are available. On most computers {\tt \\magstephalf}
is also available; this represents an enlargement of
about 9.5\%\null. Here are some samples at various sizes:

\maketable [Different Font Magnifications]
\strut \tt # \hfil & \quad # \hfil \cr
\bf Magnification & \bf Sample \cr
\noalign{\hrule} \noalign{\smallskip}
\\magstep 0 & \rm Sample text at magstep 0. \cr
\\magstephalf & \halfrm Sample text at magstephalf. \cr
\\magstep 1 & \brm Sample text at magstep 1. \cr
\\magstep 2 & \bbrm Sample text at magstep 2. \cr
\\magstep 3 & \bbbrm Sample text at magstep 3. \cr
\\magstep 4 & \bbbbrm Sample text at magstep 4. \cr
\\magstep 5 & \bbbbbrm Sample text at magstep 5. \cr

It's also possible to use completely new typefaces. These
are dependent on system availability, of course. Many systems
have a file called CMSS10 which is a font that contains sans
serif letters. Using {\tt \\font\\sf = cmss10} will allow the
control word {\tt\\sf} to be used in the same manner as {\tt\\bf}.
Having made this definition, the input
\displaytext{\tt \\sf Here is a sample of our new Sans Serif font.}
will produce
\displaytext{\sf Here is a sample of our new Sans Serif font.}

\exercise What problem might have arisen if we had used {\tt \\ss}
instead of {\tt \\sf} to turn on the Sans Serif font? Hint: if
the answer doesn't occur to you at first, think about the German
alphabet and it will finally come to you.

\exercise Typeset a paragraph of magnified sans serif text.

The extra fonts available may vary from site to site. The ones
in the following table are available at most places.

\maketable [External Fonts Names]
\strut \tt # \hfil & \quad \tt # \hfil & \quad \tt # \hfil
& \quad \tt # \hfil & \quad \tt # \hfil
& \quad \tt # \hfil \cr
\noalign{\hrule} \noalign{\smallskip}
CMMI8 &CMMI9 &CMR10 &CMR12 &CMR17 &CMR5 \cr
CMR6 &CMR7 &CMR8 &CMR9 &CMSLTT10 &CMSL10 \cr
CMSS10 &CMSS12 &CMSS17 &CMSS8 &CMSS9 &CMSY10 \cr
CMU10 &CMVTT10 \cr

Here's a little background about these names. The first two
letters {\tt CM} stands for Computer Modern, the name given to
this font family by the designer. The number at the end is the
point size: 10 point type is the normal size, 7 point is the
normal size for subscripts, and 5 point is the normal size for
subscripts of subscripts, 12 point is 20\% larger than 10 point,
etc. If the letters {\tt CM} are followed by {\tt B}, it is a
boldface type. Similarly {\tt R} indicates roman, {\tt I} is for
italics, {\tt CSC} is for small caps, {\tt SL} is for slanted,
{\tt SS} is for sans serif, {\tt SY} is for symbols,
and {\tt TT} is for typewriter type.

\exercise Find the fonts available on your system, and print out
all the letters and numbers in several of them.

\exercise The font {\tt CMR12} is 20\% larger than {\tt CMR10}.
Also {\tt \\magstep~1} magnifies the type by 20\%. Take some
text and print it using {\tt CMR12} then using {\tt CMR10} scaled
by {\tt \\magstep~1}. The results are quite different!

\section{The shape of things to come}

In this section we want to see how to make text have different
shapes or sizes. One may use \TeX{} with the default sizes, of
course, just as we have been doing so far. We'll now become more
creative with our output. In discussing the size of various
parts of a page of text, there are several units of measure we
can use.

\subsection{Units, units, units}

\TeX{} can measure length using many types of units. The most
common are the inch, the \centimeter, the point, the pica. The
abbreviations for these units are {\tt in}, {\tt cm}, {\tt pt},
and {\tt pc} respectively. The point is defined by the equation 1
inch = 72.27 points, and the pica by 1 pica = 12 points. Thus a
point is rather small and is about the size of a decimal point.
Here is a sample to give an idea of the comparative sizes:
\def\pip{\vrule height 4 true pt }

$$ \hbox{1 inch: } \hbox to 1 true in{\pip\hrulefill\pip}$$
$$ \hbox{1 \centimeter: }\hbox to 1 true cm{\pip\hrulefill\pip}$$
$$ \hbox{20 points: }\hbox to 20 true pt{\pip\hrulefill\pip}$$
$$ \hbox{1 pica: } \hbox to 1 true pc{\pip\hrulefill\pip}$$

Thus points are used to make fine changes; a pica is about the
distance between the baselines of two consecutive lines of
(unmagnified) normal text. \TeX{} is quite exact about
dimensions; internally its smallest unit is less than one
four-millionth of an inch. Hence it is the resolution of the
output device that will determine the accuracy of the output.

There are two other units that are sometimes useful that have
different sizes in different fonts. The {\tt ex} is about the
height of a small ``x'' and the {\tt em} is a little smaller than
the width of a capital ``M''\null.

The shape of the output is generally determined by control
words. There are many such words; these allow very fine control
of the resulting text. But most of the time only a small number
of them are necessary.

\subsection{Page shape}

The text on a page consists of three basic parts. There is the
headline above the main text; this often contains a chapter
title, section title, or a page number, and may differ on odd and
even pages. Below this is the main text, which includes any
footnotes. And finally there is the footline that might contain a
page number.

In the examples we have seen so far, the headline has been blank.
The footline has contained either a \centred{} number or, if {\tt
\\nopagenumbers} had been used, has also been blank. We'll have
more to say about headlines and footlines later in this section.
For the moment let's concentrate on the main text.

To finish one page and start a new one, {\tt \\vfill \\eject} can
be used. The control word {\tt \\eject} forces the completion of
the current page, while {\tt \\vfill} causes any left over
vertical space to be gathered at the bottom of the page (you might
try leaving out the {\tt \\vfill} as an experiment to see how the
vertical \analog{} of justified horizontal lines of text works).

The current horizontal width of the text on the page is described
by the control word {\tt \\hsize}. It can be changed, say to 4
inches, by {\tt \\hsize = 4 in\ } at any desired time using
methods to be described in the next few sections. The value of
{\tt \\hsize} in effect when the paragraph is completed
determines the width of the paragraph. As can be seen in this
paragraph, the text width can be changed (in this case to 4
inches) for a single paragraph. Because it represents the
current value of the horizontal width, expressions such as {\tt
\\hsize = 0.75\\hsize} can make a new value that is relative
to the old one. \par}

The vertical \analog{} of {\tt \\hsize} is {\tt \\vsize}, which
is the current height of the main text. It can be reset, just
like {\tt \\hsize}. Thus, {\tt \\vsize = 8 in\ } can be used to
change the vertical height of the main text. Note that {\tt \\vsize}
is the size of the text exclusive of any headline or
footline material.

Text can also be shifted across the page. The upper left corner
of the text is one inch down and one inch in from the upper left
corner of the page. The control words {\tt \\hoffset} and {\tt \\voffset}
are used to shift the text horizontally and
vertically. Thus {\tt \\hoffset = .75 in} and {\tt \\voffset =
-.5 in} would shift the text an additional .75 inches to the
right and .5 inches up. Most of the time you'll need to set {\tt
\\hoffset}, {\tt \\voffset}, and {\tt \\vsize} at the beginning
of your document only.

\global\advance\footnotenum by 1
\maketable [Control words for page sizes]
\strut \hfil # & \quad \hfil \tt # \hfil & \hfil \quad # \hfil\cr
\bf Name & \bf \TeX{} Control Word & \bf \TeX{} default value (inches)\cr
\noalign{\hrule} \noalign{\smallskip}
horizontal width & \\hsize & 6.5 \cr
vertical width & \\vsize & 8.9 \cr
horizontal offset${}^\the\footnotenum$ & \\hoffset & 0 \cr
vertical offset${}^\the\footnotenum$ & \\voffset & 0 \cr

\footnote{}{\llap{$^\the\footnotenum$}The text is initially 1 inch
down and 1 inch in from upper left corner of the page.}

\exercise Enter a paragraph of text that is a few lines long.
Take several copies of your paragraph and put {\tt \\hsize = 5
in\ } before one and {\tt \\hsize = 10 cm\ } in front of a second
one. Try a few other values for {\tt \\hsize}.

\exercise Put {\tt \\hoffset = .5 in\ } and {\tt \\voffset = 1
in\ } before the first paragraph of your previous exercise.

\exercise Take your previous exercise and put {\tt \\vsize = 2
in\ } before the first paragraph.

In the previous section we saw that it is possible to use fonts
of larger size by using the {\tt \\magstep} control word. It's
also possible to magnify the entire document at once. If {\tt
\\magnification = \\magstep 1} appears at the top of your document,
then all of your output is magnified by 20\%\null. The other
values of {\tt \\magstep} can be used, too. {\bf It should be
emphasized that {\tt \\magnification} can only be used before
even a single character is typeset.} This magnification creates
a problem in units; if the document is to be magnified by 20\%
and {\tt \\hsize = 5 in} appears in the \TeX{} input file, should
the final output have an {\tt \\hsize} of 5 inches or should it
be magnified by 20\% to 6 inches? Unless told otherwise, all the
dimensions will be magnified so that, in this case, {\tt \\hsize}
would be 6 inches in the final output. This means that all the
dimensions will increase uniformly when {\tt \\magnification} is
used. But for some special situations it might not be desirable
for this to happen; for example, you might want to leave a space
of exactly 3 inches to insert a figure. In this case any unit
can be modified by {\tt true} so that {\tt \\hsize = 5 true in}
will always set the line width to 5 inches, regardless of
magnification. \TeXref{59--60} \toindex{magnification}

\exercise Put {\tt \\magnification = \\magstep 1} as the first
line in one of your input files and see how the output changes.

\subsection{Paragraph shape}

When \TeX{} is reading the text to be typeset from your input
file, it reads in a paragraph at a time and then typesets it.
This means that there is a lot of control over paragraph shapes,
but also that some care must be taken. We've already seen that
{\tt \\hsize} can be used to control the width of a single
paragraph. But suppose the following appeared as part of your
input file:

\\hsize = 5 in
Four score and seven years $\ldots$
$\ldots$ from this earth.
\\hsize = 6.5 in


What is the width of the paragraph? The {\tt \\hsize} was
changed at the start of the paragraph and again at the end.
Since the paragraph was not completed (by putting in a blank line
or {\tt \\par}) until after the second change, the paragraph
would be typeset with a width of 6.5 inches. However, if a blank
line were inserted before {\tt \\hsize = 6.5 in}, then the
paragraph would be typeset with a width of 5 inches. So, in
general, when a paragraph is set, the values of the parameters
that are in effect when the paragraph is completed are the ones
that are used.

Here is a table with some paragraph parameters:

\maketable [Some paragraph shape parameters]
\strut \hfil # & \quad \hfil \tt # \hfil & \hfil \quad # \hfil\cr
\bf Function & \bf \TeX{} Control Word & \bf \TeX{} default\cr
\noalign{\hrule} \noalign{\smallskip}
width & \\hsize & 6.5 inches \cr
indentation on first line & \\parindent & 20 points\cr
distance between lines & \\baselineskip & 12 points\cr
distance between paragraphs & \\parskip & 0 points \cr


The control word {\tt \\noindent} may be used at the beginning of
a paragraph to avoid the automatic indentation on the first line.
This control word will only affect the paragraph being set when
it is invoked (of course {\tt \\parindent = 0 pt} will cause no
indentation for all paragraphs).

A more flexible way to control the width of a paragraph is to use
{\tt \\rightskip} and {\tt \\leftskip}. Setting {\tt \\leftskip
= 20 pt} causes the left margin of the paragraph to be moved in
an extra twenty points. Negative values may be assigned to {\tt
\\leftskip} to move the left margin out. The control word {\tt
\\rightskip} does the same to the right side of the paragraph.
The single control word {\tt \\narrower} is the equivalent of
setting both {\tt \\leftskip} and {\tt \\rightskip} equal to {\tt
\\parindent}. This is quite useful for long quotations, and
this paragraph is an example of its use. As with {\tt \\hsize},
the value of {\tt \\leftskip} and {\tt \\rightskip} in effect
when the paragraph is completed is the one which will apply to
the whole paragraph.


\exercise Make two paragraphs with the following specifications:
the left margin of both paragraphs is indented by 1.5 inches,
the right margin of the first paragraph is indented by 0.75
inches, and the right margin of the second paragraph is indented
by 1.75 inches.

\def\hangparagraph{ Lines can be made with different lengths
within one paragraph by using {\tt \\hangindent} and {\tt
\\hangafter}. The amount of the indentation is determined by value
of {\tt \\hangindent}. If {\tt \\hangindent} is positive, the
indentation is made from the left, and if it is negative it is
made from the right. The lines on which the indentation occurs is
controlled by {\tt \\hangafter}. If {\tt \\hangafter} is
positive then it determines the number of lines at full width
before the indentation starts. Thus if {\tt \\hangindent = 1.75
in} and {\tt \\hangafter = 6}, then the first six lines will be
at full width and the rest will be indented by 1.75 inches from
the left. On the other hand if {\tt \\hangindent = \hbox{-1.75}
in} and {\tt \\hangafter = -6}, then the first six lines will be
indented by 1.75 inches from the right and the rest will be at
full width. \TeX{} resets to the default values {\tt \\hangindent
= 0 pt} and {\tt \\hangafter = 1} after each paragraph. These
control words are useful for paragraphs with ``hanging indents''
and for flowing a paragraph around space reserved for a figure.
\TeXref{355}The control word {\tt \\hang} at the beginning of the
paragraph will cause the first line to be of full width ({\tt
\\hangafter=1}) and the rest of the paragraph to be indented by
the current value of {\tt \\parindent}. But you do have to use
{\tt \\noindent} if you want the first line to extend all the way
to the left margin.

\hangafter=6 \hangindent=1.75in

Here is the previous paragraph repeated with {\tt \\hangafter = -6}
and {\tt \\hangindent = -1.75 in}.

\hangafter=-6 \hangindent=-1.75in \hangparagraph

The control word {\tt \\parshape} can be used to make paragraphs with
a greater variety of shapes.

Another useful control word for setting paragraphs is {\tt \\item}.
It can be used to make various types of itemized lists,
and is invoked using the pattern {\tt \\item\lb$\ldots$\rb}\null.
This causes the next paragraph to be formed with every line
indented by {\tt \\parindent} and, in addition, the first line
labeled on the left by whatever is between the braces. It is
usually used with {\tt \\parskip = 0 pt}, since that control word
determines the vertical space between the different items. The control
word {\tt \\itemitem} is the same as {\tt \\item} except that the
indentation is twice as far, that is, twice the value of
{\tt \\parindent}.
\TeXref{102}Here is an example:

\parskip = 0pt \parindent = 30 pt
Answer all the following questions:
\item{(1)} What is question 1?
\item{(2)} What is question 2?
\item{(3)} What is question 3?
\itemitem{(3a)} What is question 3a?
\itemitem{(3b)} What is question 3b?

will produce

{\parskip = 0pt \parindent = 30 pt
Answer all the following questions:
\item{(1)} What is question 1?
\item{(2)} What is question 2?
\item{(3)} What is question 3?
\itemitem{(3a)} What is question 3a?
\itemitem{(3b)} What is question 3b?


\exercise Make a paragraph several lines long and use it with
the {\tt \\item} control word to see the ``hanging indent.'' Now
take the same paragraph and use it with different values of {\tt
\\parindent} and {\tt \\hsize}.

Now let's see how to put space between paragraphs. The control
word {\tt \\parskip} is used to determine how much space is
normally left between paragraphs. So if you put {\tt \\parskip =
12 pt} at the beginning of your \TeX{} source file, there will be
12 points between paragraphs unless other instructions are
given. The control word {\tt \\vskip} can be used to insert extra
vertical space between paragraphs. If {\tt \\vskip 1 in} or {\tt
\\vskip 20 pt} appears between two paragraphs, then the extra
space is inserted.

There are a couple of peculiarities of {\tt \\vskip} that seem
quite strange at first. If you have {\tt \\vskip 3 in} and the
skip starts two inches from the bottom of the page, the rest of
the page is skipped, but the extra one inch is {\it not\/}
skipped at the top of the next page. In other words, {\tt \\vskip}
{\bf will not insert space across page boundaries}. In
fact, {\tt \\vskip~1~in} will have no effect at all if it happens
to appear at the top of a page! For many types of vertical
spacing this is quite appropriate. The space before a section
heading, for example, should not continue across page boundaries.

A similar phenomenon occurs at the beginning of your document.
If, for example, you want a title page with the title about half
way down the page, you can not insert the space at the top of the
page using {\tt \\vskip}.

But what if you really want some blank space at the top of a
page? You could start the page with {\tt\\\sp} but this in effect
typesets a one line paragraph containing a blank. So while
nothing is typeset, the extra space due to the values of {\tt
\\baselineskip} and {\tt \\parskip} will add extra space. An easier
method is to use {\tt \\vglue} instead of {\tt \\vskip} to get
the desired result. Thus {\tt \\vglue 1 in\ }will leave one inch
of blank space at the top of the page.\TeXref{352}

We can note in passing that there is another more general method
to put material at the top of a page using the control words {\tt
\\topinsert} and {\tt \\endinsert}. If {\tt \\topinsert $\ldots$
\\endinsert} is used within a page, the material between
{\tt \\topinsert} and {\tt \\endinsert} will appear at the top
of the page, if possible. For example:

\\vskip 1 in
\\centerline\lb Figure 1\rb
is useful for inserting figures of a prescribed size.

There are also some special control words for making small
vertical skips. They are {\tt \\smallskip}, {\tt \\medskip}, and
{\tt \\bigskip}. Here is the size of each one:

\def\hrl{\hrule width .5 in}

\centerline{{\tt \\smallskip}: \vbox{\hrl \smallskip \hrl} \quad
{\tt \\medskip}: \vbox{\hrl \medskip \hrl} \quad
{\tt \\bigskip}: \vbox{\hrl \bigskip \hrl}

\subsection{Line shape}

For most material \TeX{} does a good job of breaking up a
paragraph into lines. But sometimes it's necessary to add
further instructions. It's possible to force a new line by
inserting {\tt \\hfill \\break} in your input file. It's also
possible to put some text on a line by itself using the control
word {\tt \\line\lb $\ldots$\rb}; then the material between the
braces will be restricted to one line (although the result will
be spread out over the whole line and may be horrible). The
control words {\tt \\leftline\lb $\ldots$\rb}, {\tt \\rightline\lb
$\ldots$\rb}, and {\tt \\centerline\lb $\ldots$\rb} will,
respectively, set the material between the braces on a single
line on the left margin, on the right margin, or \centred. Thus

\leftline{I'm over on the left.}
\centerline{I'm in the @centre.}
\rightline{I'm on the right.}
\line{I just seem to be spread out all over the place.}

produces the four lines:

\hbadness = 10000
\leftline{I'm over on the left.}
\centerline{I'm in the \center.}
\rightline{I'm on the right.}
\line{I just seem to be spread out all over the place.}

Other types of spacing can be obtained by using the control word {\tt
\\hfil}. This causes all the extra space in the line to be
accumulated at the position where {\tt \\hfil} appears.
Thus if we alter our last example to
{\tt \\line\lb I just seem to be spread out \\hfil all over the place.\rb}
we will get \hfil\break
\line{I just seem to be spread out \hfil all over the place.}
If we have more than one {\tt \\hfil}, they will divide any extra
space among themselves equally. Hence
{\tt \\line\lb left text \\hfil \centre{} text\\hfil right text.\rb}
will produce
\line{left text \hfil \centre{} text \hfil right text.}

\exercise Typeset the following line:

\line{left end \hfil left tackle \hfil left guard \hfil \centre{} \hfil
right guard \hfil right tackle \hfil right end}

\exercise Typeset the following line with twice as much space between
``left'' and ``right-\centre{}'' as between ``right-\centre{}'' and

\line{left \hfil \hfil right-\centre{} \hfil right}

It's possible to move horizontally using {\tt \\hskip} in a
manner analogous with {\tt \\vskip}.

\exercise What happens with the following input:\hfil\break
{\tt \\line\lb\\hskip 1 in ONE \\hfil TWO \\hfil THREE\rb}

The right justification can be canceled by using the
control word {\tt \\raggedright}.


The general pattern to make footnotes using \TeX{} is
{\tt \\footnote\lb$\ldots$\rb\lb$\ldots$\rb}\null.

The footnote mark goes in between the first set of braces. Some
available marks are {\tt \\dag (\dag)}, {\tt \\ddag (\ddag)}, {\tt \\S
(\S)}, and {\tt \\P (\P)}\null. The text of the footnote
goes between the second set of braces. The use of numbers as
marks is a little less straightforward. The footnote%
\footnote{${}^{21}$}{This is the footnote at the bottom of the page.}
at the bottom of the page was produced by using {\tt \\footnote\lb\$\lb
\rb\^{}\lb{21}\rb\$\rb\lb This is the footnote at the
bottom of the page.\rb} after the word ``footnote'' in the
text. This construction is somewhat complicated; we'll see why
it's this way when we know a little more about typesetting
mathematics. But for the moment we can look at it as a way of
getting the job done. You may want to use {\tt \\rm} within the
footnote to ensure that the right font appears. Usually you will
not want a space between the text and the control word {\tt

\exercise Make up a page with a relatively long footnote spanning
several lines.

\exercise Make up a page with two different footnotes on it.

\subsection{Headlines and Footlines}

\headline={\hfil \tenrm Page \the\pageno} %example for this section

The lines for title and page numbers that go above and below the
main text are produced by using {\tt \\headline=\lb$\dots$\rb}
and {\tt \\footline=\lb$\dots$\rb}\null.

The principle is the same as using the control word {\tt \\line\lb
$\dots$\rb} within the usual text on the page. A helpful
control word is {\tt \\pageno} which represents the current page
number. Thus {\tt \\headline=\lb\\hfil \\tenrm Page \\the\\pageno\rb}
would cause the page number to appear in the upper
right corner preceded by the word ``Page'' (now look at the upper
right corner of this page). It is safer to explicitly name the
font that you want to use (in this case {\tt \\tenrm} to use the
10 point roman font), since you are never guaranteed that a
particular font will be in use when the headline or footline is
set. The control word {\tt \\the} takes the internal value of the
next word if it is an appropriate control word and prints it as
text. You can also use the control word {\tt \\folio} instead of
{\tt \\the \\pageno}. The difference is that {\tt \\folio} will
give roman numerals when {\tt \\pageno} is negative.

You can also assign values to {\tt \\pageno} if you want your
document to use a different sequence of page numbers from the
usual. Roman numerals can be produced by using negative numbers;
{\tt \\pageno=-1} at the beginning of a document will cause page
numbers to be in roman numerals.

Different headlines can be produced for even and odd pages in
the following manner:

\\headline=\lb\\ifodd \\pageno \lb$\dots$\rb \\else \lb$\dots$\rb

where the material between the first set of braces is for the
right-hand pages and the material between the second set of
braces is for the left-hand pages.

\exercise Change the footline so that the page number is \centred{}
with a en-dash on either side.

\subsection{Overfull and underfull boxes}

One of the most frustrating experiences for the new \TeX{} user
is the occurrence of overfull and underfull boxes. They are duly
noted in the log file and are also written to the screen when \TeX{}
is being run interactively. Overfull boxes are also marked
on the output by a slug (a large filled-in black rectangle that
looks like this: \vrule width \overfullrule) in the right
margin. Various obscene terms are applied to this slug. It
appears even though there is nothing wrong with the \TeX{} input.
So why is the slug there and what can be done about it?

A good way to visualize the way \TeX{} organizes a page is to
think of the printed material as being put into boxes. There are
two types of boxes: {\sl hboxes\/} and {\sl vboxes}. Most of the
time these correspond to the organization of horizontal text
into lines and vertical paragraphs into pages. In particular, it
is the spacing of the words in a hbox corresponding to a line of
text that causes the slug to appear.

Recall that \TeX{} reads in the complete paragraph before
deciding how to break it up into lines. This is better than
working a line at a time, since a slight improvement in one line
might cause catastrophic changes farther down in the paragraph.
When words are gathered together to form a line, space is added
between the words to justify the right margin. Very large spaces
between words is obviously undesirable; the badness of the line
is a measure of how badly the words are spaced. An underfull
hbox means that there is too much space between words. More
specifically, the badness of any line is a number between 0
(perfect) and 10000 (horrible). There is a parameter called {\tt
\\hbadness} whose default value is 1000\null. Any line whose
badness is greater than {\tt \\hbadness} is reported as an
underfull hbox. If the value of {\tt \\hbadness} is increased,
then fewer underfull hboxes will be reported. In fact {\tt
\\hbadness = 10000} will suppress the reporting of underfull
hboxes altogether. Similarly, if the words must be squeezed into
a line so that the spaces are very small or even so the words
extend a little into the right margin, then an overfull hbox
results. \toindex{hbadness}

%%%%%%%%%%%%%% restore original headline %%%%%%%%%%%%%%%%%%%%%%%%
\headline={\iftitlepage \hfil \global\titlepagefalse
\else \gentleheadline \fi}

In a similar way, sometimes \TeX{} allows a line to be slightly
longer than {\tt \\hsize} in order to achieve a more balanced
appearance. The control word {\tt \\tolerance} determines when
this happens. If the badness of a line is greater than {\tt
\\tolerance}, \TeX{} will make the line longer by adding a new word
at the end of the line, even though it might exceed the value of
{\tt \\hsize}. A line which is only slightly longer is set
without being reported. The control word {\tt \\hfuzz} determines
how much excess is allowed; the default is {\tt \\hfuzz
=~0.1~pt}. A line that is more than slightly longer than {\tt \\hsize}
is obviously a serious problem; \TeX{} puts a slug in
the margin to make its point forcefully. It's possible to avoid
overfull hboxes altogether by increasing the value of {\tt \\tolerance}.
With {\tt \\tolerance = 10000} there will never be
overfull boxes or slugs. The default is {\tt \\tolerance = 200}\null.

The width of the slug is determined by the control word {\tt
\\overfullrule}. Including {\tt \\overfullrule = 0 pt} in your
file will delete any further printing of the slugs. The
overfull boxes will still be there, of course, and they will
probably be harder to spot.

So we see why overfull boxes and underfull boxes are reported; we
can also change the reporting by changing the values of {\tt
\\badness}, {\tt \\hfuzz}, and {\tt \\tolerance}. In addition, a
small value of {\tt \\hsize} obviously makes it more difficult to
set lines and causes more overfull and underfull hboxes to be
reported. These are warnings that you ignore at your own peril!

Adding new possibilities of hyphenation will sometimes eliminate
an overfull box. \TeX{} has automatic hyphenation and usually
finds good places to hyphenate a word; however, it's possible to
add hyphenations to let lines break at new places. For example,
the automatic hyphenation will never put a hyphen in the word
``database''. If you type in {\tt data\\-base}, it will allow a
hyphen to be inserted after the second letter ``a'' in the word.
More generally, if you put {\tt \\hyphenation\lb data-base\rb} in
the beginning of your input file, then all occurrences of the
word ``database'' will allow hyphenation after the letter ``a''.
\TeXref{28} The log file will show the possible hyphenations on
the line containing the overfull or underfull box. Sometimes the
best solution to an overfull or underfull hbox is a little
judicious editing of the original document.

Our discussion has involved the setting of type into lines, that is,
the horizontal page structure. There are several vertical \analog{}s.
Overfull and underfull hboxes indicate how well words are gathered
into lines. Similarly, overfull and underfull vboxes are reported
when paragraphs are gathered to form pages. A large table that
can't be broken in the middle, for example, can produce an underfull
vbox that is reported in the log file when the page being typeset is
completed. The control word {\tt \\vbadness} works for the vertical
placement of text in the same way as {\tt \\hbadness} works for
horizontal text.

\exercise Take a few paragraphs and print them using various (small)
values of {\tt \\hsize} to see what kind of overfull boxes result.
Repeat with various values of {\tt \\hbadness}, {\tt \\hfuzz}, and
{\tt \\tolerance}.

\section{$\Bigl\{$Groups, $\bigl\{$Groups,
$\{$and More Groups$\}\bigr\}\Bigr\}$}

The concept of gathering text into groups allows \TeX{} input
files to be greatly simplified. A new group is started by the
character {\tt\lb} and terminated by the character {\tt \rb}\null.
Changes made within a group will lose their effect when the
group terminates. So, for example if {\tt \lb \\bf three
boldface words\rb} appears in your text, the opening brace starts
the group, the {\tt \\bf} control word changes to a boldface
font, and the closing brace finishes up the group. Upon
completion of the group the font being used is the one in effect
before the group started. This is the (easier) way of having a
few words in a different font. It's also possible to have groups
nested within groups.

\def\sectiontitle{$\Bigl\{$Groups, $\bigl\{$Groups, $\dots \bigr\}\Bigr\}$}

As another example, size changes can be made in the text that are
only temporary. For example
\\hsize = 4 in
\\parindent = 0 pt
\\leftskip = 1 in
will produce a paragraph that is four
(this is an easy mistake to make).

{\hsize = 4 in
\parindent = 0 pt
\leftskip = 1 in
will produce a paragraph that is four inches wide with the text
offset into the paragraph by one inch regardless of the settings
in effect before the start of the group. This paragraph is set
with those values. After the end of the group, the old settings
are in effect again. Note that it is necessary to include
{\tt \\par} or to use a blank line before the closing brace to end
the paragraph, since otherwise the group will end and \TeX{} will
go back to the old parameters before the paragraph is actually
typeset (this is an easy mistake to make). \par}

When a control word (like {\tt \\centerline}) acts on text
following it in braces, that text is implicitly in a group. Thus
{\tt \\centerline\lb\\bf A bold title\rb} will produce a \centred{}
boldface line, and the text following that line will be in
whatever font was in effect before the {\tt \\centerline} was

The empty group {\tt\lb\rb} is useful. One use allows accents to be
typeset with no accompanying letter. For example, {\tt \\\~{}\lb\rb}
will print a tilde with no letter under it. It can also be used to
stop \TeX{} from eating up consecutive spaces. Hence {\tt I use
\\TeX\lb\rb{} all the time} will leave a space after ``\TeX'' in the
output. This is an alternative to using {\tt \\\sp} as we did in

Grouping can also be used to avoid spaces in the middle of a word
when including accents. Either {\tt soup\\c\sp con} or {\tt
soup\\c\lb c\rb on} will produce the word soup\c{c}on.

\exercise Change the dimensions of one paragraph on a page using the
grouping idea.

\exercise Mathematicians sometime use the word ``i{f}f'' as an
abbreviation for ``if and only if''\null. In this case it looks
better if the first and second letter ``f'' are {\sl not\/}
joined as a ligature. How do you do this (there are several

It's really easy to forget to match braces properly. The effect
can be dramatic; if you get output that suddenly changes to an
italic font for the rest of the document, a mismatched brace is
probably the cause. If you have an extra {\tt \lb} \TeX{} will
give a message in the log file: {\tt (\\end occurred inside a
group at level 1)}. An extra {\tt\rb} will result in the message
\hbox{\tt! Too many \rb's.}

Here's a little hint to help you keep track of the braces in more
complicated groups: put the opening brace on a line by itself and
do the same for the closing brace. If there are braces nested
within the original ones, put them on separate lines also, but
indent them a few spaces. The text within the nested braces can
also be indented since \TeX{} ignores all spaces at the beginning
of a line. The matching braces will then stand out when you look
at your \TeX{} source file. In fact, if your editor is smart
enough, you can create the two lines with the braces first and
then insert the appropriate material within them with automatic

\exercise In section 2 we changed fonts the following method:
{\tt I started with roman type, \\it switched to italic type,
\\rm and returned to roman type}. Get the same result using the
idea of grouping.

\section{No math anxiety here!}

\TeX{} is at its best when typesetting mathematics. The
conventions for doing this are many and complex, and the ability
of \TeX{} to take them into account makes the production of high
quality, attractive mathematical output possible. If you plan to
produce papers with mathematical symbols in them, this section
will give you all the basics necessary for creating beautiful
output in almost all circumstances; \TeX{} may be used without
any mathematics, of course, and if this is your goal, then the
following two subsections are probably sufficient for your needs.

\subsection{Lots of new symbols}

Mathematical text is inserted into normal text in two possible
ways: it can be {\sl in-line\/}, that is, as part of the usual
lines of text. It can also be {\sl displayed}, that is, \centred{}
in a gap between the usual text. The results in the spacing and
placement of symbols can be quite different in each case. The
in-line equation $\sum_{k=1}^{\infty} {1\over k^2} = {\pi^2\over6}$
is easily seen to be different from the same equation when
$$\sum_{k=1}^{\infty} {1\over k^2} = {\pi^2\over6}.$$

Since the spacing and the fonts used for mathematics are quite
different from those of ordinary text, \TeX{} needs to be told
when mathematics rather than normal text is being typeset. This
is done using the {\tt\$} symbol. More specifically, mathematics
is typeset in-line by surrounding the symbols to be entered by
single dollar signs: {\tt \$$\ldots$\$}, and is typeset displayed
by surrounding the symbols to be entered by double dollar signs:
{\tt \$\$$\ldots$\$\$}\null. Hence {\tt \$x = y+1\$} will give
$x=y+1$ in-line while {\tt \$\$x = y+1.\$\$} will display

The spacing for both in-line and displayed mathematics is
completely controlled by \TeX\null. Adding spaces to your input
has no effect at all. What if you need a space or some text in
the middle of some mathematics? You can insert text by inserting
it into an hbox (don't worry about the definition of an hbox for
now): {\tt \\hbox\lb$\ldots$\rb}\null. This is particularly
useful for displayed mathematics. Hence ``$x=y+1 \hbox{ whenever
} y=x-1$'' can be typeset using {\tt \$x=y+1 \\hbox\lb\ whenever
\rb y=x-1\$}. Note the spaces on either side of the word within
the braces. Usually it isn't necessary to insert spaces into
mathematical text, but here are the control sequences that do the

\maketable [Adding space to mathematical text]
\strut \hfil # & \quad \hfil\tt# \hfil \quad
& \hbox to 2cm{\hrulefill\vrule height 8pt#\vrule height 8pt\hrulefill} \cr
Name & \rm Control Sequence & \hfil{}$\gets$Size$\to$\cr
\noalign{\hrule} \noalign{\smallskip}
Double quad & \\qquad &\qquad \cr
Quad & \\quad &\quad \cr
Space & \\\sp\ &\ \cr
Thick space & \\; &$\;$\cr
Medium space & \\> &$\>$\cr
Thin space & \\, &$\,$\cr
Negative thin space & \ \\!\ &$\!$\cr


If you look closely at the negative thin space, you'll notice
that, unlike the other entries, the two arms overlap. This is
because the negative space is one in the opposite direction, that
is, while the other control sequences increase the amount of
space between two symbols being typeset, the negative thin space
decreases the space between them, even if it causes them to

\exercise Typeset the following: $C(n,r) = n!/(r!\,(n-r)!)$\null. Note the
spacing in the denominator.

You shouldn't have any blank lines between the dollar signs
delimiting the mathematical text. \TeX{} assumes that all the
mathematical text being typeset is in one paragraph, and a blank
line starts a new paragraph; consequently, this will generate an
error message. This turns out to be useful, for one of the
easiest errors to make is to forget to put in the trailing dollar
sign(s) after the mathematical input (I promise that you'll do it
at least once while learning \TeX{}); if \TeX{} allowed more than
one paragraph to be between the dollars signs, then one omitted
trailing dollar sign might cause the rest of the document to be
typeset as mathematics.

Most mathematical text is entered in exactly the same way for
in-line typesetting as for displayed typesetting (except for the
surrounding dollar signs, of course). The exceptions, such as
aligning multiline displays and placing equation numbers at the
left or right margin will be discussed in the last part of this

Many new symbols can appear when typesetting mathematics. Most
of the ones that actually appear on the keyboard can be used
directly. The symbols {\tt + - / * = ' | < > (} and {\tt )} are
all entered directly. Here they are as mathematics: $+ - / * =\>
'\> | <\> >\> ( \> )$.

\exercise Typeset the equation $a+b=c-d=xy=w/z$ as in-line and
displayed mathematical text.

\exercise Typeset the equation $(fg)' = f'g + fg'$ as in-line and
displayed mathematical text.

Many other symbols, as you would expect, are predefined control
words. All Greek letters are available. Here is a table of them:

\maketable [Greek letters]
\strut \hfil$#$ & \quad \tt# \hfil \qquad &\hfil$#$ & \quad \tt# \hfil \qquad
&\hfil$#$ & \quad \tt# \hfil \qquad &\hfil$#$ & \quad \tt# \hfil \cr
\noalign{\hrule} \noalign{\smallskip}
\alpha & \\alpha &\beta & \\beta &\gamma & \\gamma &\delta & \\delta \cr
\epsilon & \\epsilon & \varepsilon & \\varepsilon & \zeta & \\zeta &
\eta & \\eta \cr
\theta & \\theta & \vartheta & \\vartheta & \iota & \\iota & \kappa &
\\kappa \cr
\lambda & \\lambda & \mu & \\mu & \nu & \\nu & \xi & \\xi \cr
o & o & \pi & \\pi & \rho & \\rho & \varrho & \\varrho \cr
\sigma & \\sigma & \varsigma & \\varsigma & \tau & \\tau & \upsilon &
\\upsilon \cr
\phi & \\phi & \varphi & \\varphi & \chi & \\chi & \psi & \\psi \cr
\omega & \\omega & \Gamma & \\Gamma & \Delta & \\Delta & \Theta &
\\Theta \cr
\Lambda & \\Lambda & \Xi & \\Xi & \Pi & \\Pi & \Sigma & \\Sigma \cr
\Upsilon & \\Upsilon & \Phi & \\Phi & \Psi & \\Psi & \Omega & \\Omega \cr


\exercise Typeset $\alpha\beta=\gamma+\delta$ as in-line and displayed
mathematical text.

\exercise Typeset $\Gamma(n) = (n-1)!$ as in-line and displayed
mathematical text.

Sometimes accents are put above or below symbols. The control
words used for accents in mathematics are different from those
used for normal text. The normal text control words may not be
used for mathematics and vice-versa.

\maketable [Mathematical accents]
\strut \hfil$#$ & \quad \tt# \hfil \qquad &\hfil$#$ &
\quad \tt# \hfil \qquad &\hfil$#$ & \quad \tt# \hfil \qquad \cr
\noalign{\hrule} \noalign{\smallskip}
\hat o & \\hat o & \check o & \\check o & \tilde o & \\tilde o \cr
\acute o & \\acute o & \grave o & \\grave o & \dot o & \\dot o \cr
\ddot o & \\ddot o &\breve o & \\breve o & \bar o & \\bar o \cr
\vec o & \\vec o & \widehat {abc} & \\widehat \lb abc\rb
& \widetilde {abc} & \\widetilde \lb abc\rb\cr


Binary operators combine two mathematical objects to get another
object. Ordinary addition and multiplication, for example,
combine two numbers to get another number, and so they are binary
operators. When a binary operator such as $+$ or $\times$ is
typeset, a little extra space is put around it. Here is a list of
some of the available binary operators:

\maketable [Binary operators]
\strut \hfil$#$ & \quad \tt# \hfil \qquad &\hfil$#$ & \quad \tt# \hfil \qquad
&\hfil$#$ & \quad \tt# \hfil \qquad &\hfil$#$ & \quad \tt# \hfil \cr
\noalign{\hrule} \noalign{\smallskip}
\cdot & \\cdot &\times & \\times &\ast & \\ast &\star & \\star \cr
\circ & \\circ & \bullet & \\bullet & \div & \\div & \diamond & \\diamond \cr
\cap & \\cap & \cup & \\cup & \vee & \\vee & \wedge & \\wedge \cr
\oplus & \\oplus &\ominus & \\ominus & \otimes &\\otimes &\odot &\\odot \cr


Ellipses are commonly used with binary operators. The control
word {\tt \\cdots} will raise the dots so that they are level
with the binary operator. Thus {\tt \$a + \\cdots + z\$} will
produce $a + \cdots + z$. The control word {\tt \\ldots} will
put the dots on the baseline, and so {\tt \$1\\ldots n\$}
produces $1\ldots n$.

\exercise Typeset: $x\wedge (y\vee z) = (x\wedge y) \vee (x\wedge z)$.

\exercise Typeset: $2+4+6+\cdots +2n = n(n+1)$.

A relation indicates a property of two mathematical objects. We
already know how to show two objects equal, or how to show one
number less than or greater than another number (since these are
symbols on most terminal keyboards). To negate a relation, the
control word {\tt \\not} is put in front of the relation. Here
are some relations:

\maketable [Relations ]
\strut \hfil$#$ & \quad \tt# \hfil \qquad &\hfil$#$ & \quad \tt# \hfil \qquad
&\hfil$#$ & \quad \tt# \hfil \qquad &\hfil$#$ & \quad \tt# \hfil \cr
\noalign{\hrule} \noalign{\smallskip}
\leq & \\leq &\not\leq & \\not \\leq
& \geq & \\geq & \not\geq & \\not \\geq \cr
\equiv & \\equiv & \not\equiv & \\not \\equiv
& \sim & \\sim & \not\sim & \\not \\sim \cr
\simeq & \\simeq & \not\simeq & \\not \\simeq
& \approx & \\approx & \not\approx & \\not \\approx \cr
\subset & \\subset & \subseteq & \\subseteq
& \supset & \\supset & \supseteq & \\supseteq \cr
\in & \\in & \ni & \\ni & \parallel & \\parallel & \perp & \\perp \cr


Typeset: $\vec x\cdot \vec y = 0$ if and only if $\vec x \perp \vec y$.

Typeset: $\vec x\cdot \vec y \not= 0$ if and only if $\vec x \not\perp \vec y$.

Here are some other available mathematical symbols:\TeXref{435--438}

\maketable [Miscellaneous symbols ]
\strut \hfil$#$ & \quad \tt# \hfil \qquad &\hfil$#$ & \quad \tt# \hfil \qquad
&\hfil$#$ & \quad \tt# \hfil \qquad &\hfil$#$ & \quad \tt# \hfil \cr
\noalign{\hrule} \noalign{\smallskip}
\aleph & \\aleph & \ell & \\ell & \Re & \\Re & \Im & \\Im \cr
\partial & \\partial & \infty & \\infty & \| & \\| & \angle & \\angle \cr
\nabla & \\nabla & \backslash &\\backslash & \forall & \\forall
& \exists & \\exists \cr
\neg & \\neg & \flat & \\flat & \sharp & \\sharp & \natural & \\natural \cr


\exercise Typeset: $(\forall x\in \Re)(\exists y\in\Re)$ $y>x$.


There are two methods of typesetting a fraction: it can be
typeset either in the form $1/2$ or in the form ${1\over2}$\null.
The first case is just entered with no special control sequences,
that is, {\tt \$1/2\$}\null. The second case uses the control
word {\tt \\over} and the following pattern: {\tt \lb
\\over \rb}\null. Hence {\tt \$\$\lb a+b
\\over c+d\rb.\$\$} gives $${a+b\over c+d}.$$

\exercise Typeset the following: ${a+b\over c}\quad {a\over b+c}
\quad {1\over a+b+c} \not= {1\over a}+{1\over b}+{1\over c}$.

\exercise Typeset: What are the points where
${\partial \over \partial x} f(x,y) = {\partial \over \partial y}
f(x,y) = 0$?

\subsection{Subscripts and superscripts}

Subscripts and superscripts are particularly easy to enter using
\TeX\null. The characters {\tt \_{}} and {\tt \^{}} are used to
indicate that the next character is a subscript or a superscript.
Thus {\tt \$x\^{}2\$} gives $x^2$ and {\tt \$x\_{}2\$} gives
$x_2$\null. To get several characters as a subscript or
superscript, they are grouped together within braces. Hence we
can use {\tt \$x\^{}\lb 21\rb\$} to get $x^{21}$ and {\tt \$x\_{}\lb
21\rb\$} to get $x_{21}$\null. Notice that the
superscripts and subscripts are automatically typeset in a
smaller type size. The situation is only slightly more
complicated for a second layer of subscripts or superscripts.
You can {\sl not\/} use {\tt \$x\_{}2\_{}3\$} since this could
have two possible interpretations, namely, {\tt \$x\_{}\lb 2\_{}3\rb\$}
or {\tt \$\lb x\_{}2\rb\_{}3\$}; this gives two
different results: $x_{2_3}$ and ${x_2}_3$, the first of which is
the usual mathematical subscript notation. Thus you must put in
the complete braces to describe multiple layers of subscripts and
superscripts. They may be done to any level.

To use both subscripts and superscripts on one symbol, you use
both the {\tt \_{}} and {\tt \^{}} in either order. Thus either
{\tt \$x\_{}2\^{}1\$} or {\tt\$x\^{}1\_{}2\$} will give $x_2^1$.

Typeset each of the following: $e^x \quad e^{-x} \quad
e^{i\pi}+1=0 \quad x_0 \quad x_0^2 \quad {x_0}^2 \quad 2^{x^x}$.

\exercise Typeset:
$\nabla^2 f(x,y) = {\partial^2 f \over\partial x^2}
+ {\partial^2 f \over\partial y^2}$.

A similar method is used for summations and integrals. The input
of {\tt \$\\sum\_{}\lb k=1\rb\^{}n k\^{}2\$} will give $\sum_{k=1}^n
k^2$, and {\tt \$\\int\_{}0\^{}x f(t) dt\$} will
give $\int_0^x f(t) dt$.

Another use of this type of input is for expressions
involving limits. You can use
{\tt \$\\lim\_\lb n\\to \\infty\rb (\lb n+1 \\over n\rb)\^{}n = e\$}
to get $\lim_{n\to \infty} ({n+1\over n})^n = e$.

\exercise Typeset the following expression: $\lim_{x\to 0}
(1+x)^{1\over x}=e$.

\exercise Typeset: The cardinality of $(-\infty, \infty)$ is $\aleph_1$.

\exercise Typeset: $\lim_{x\to {0^+}} x^x = 1$.

Here's a hint to make integrals look a little nicer: look at the
difference between $\int_0^x f(t) dt$ and $\int_0^x f(t)\, dt$\null.
In the second case there is a little extra space after
$f(t)$, and it looks nicer; {\tt \\,} was used to add the
additional space.

\exercise Typeset the following integral: $\int_0^1 3x^2\,dx = 1$.

\subsection{Roots, square and otherwise}

To typeset a square root it is only necessary to use the construction
{\tt \\sqrt\lb$\ldots$\rb}\null. Hence {\tt \$\\sqrt\lb x\^{}2+y\^{}2\rb\$}
will give $\sqrt{x^2+y^2}$\null. Notice that \TeX{} takes care of the
placement of symbols and the height and length of the radical.
To make cube or other roots, the control words {\tt \\root} and {\tt
\\of} are used. You get $\root n \of {1+x^n}$ from the
input\TeXref{130--131} {\tt \$\\root n \\of \lb1+x\^{}n\rb\$}.

A possible alternative is to use the control word {\tt \\surd}; the
input {\tt \$\\surd 2\$} will produce $\surd 2$.

\exercise Typeset the following: $\sqrt2 \quad \sqrt {x+y\over x-y}
\quad \root 3 \of {10}$ \quad $e^{\sqrt x}$.

\exercise Typeset: $\|x\| = \sqrt{x\cdot x}$.

\exercise Typeset:
$\phi(t) = {1 \over \sqrt{2\pi}} \int_0^t e^{-x^2/2}\,dx$.

\subsection{Lines, above and below}

Use the constructions {\tt \\overline\lb$\ldots$\rb} and
{\tt \\underline\lb$\ldots$\rb} to put lines above or below mathematical
expressions. Hence {\tt \$\\overline\lb x+y\rb=\\overline x + \\overline
y\$} gives $\overline{x+y}=\overline x + \overline y$\null.
But notice that the lines over the letters are at
different heights, and so some care is necessary. The use of {\tt
\\overline\lb\\strut x\rb} will raise the height of the line
over\TeXref{130--131} $x$\null.

To underline non-mathematical text, use {\tt \\underbar\lb$\dots$\rb}.

\exercise Typeset the following: $\underline x \quad \overline y
\quad \underline{\overline{x+y}}$.

\subsection{Delimiters large and small}

The most commonly used mathematical delimiters are brackets,
braces, and parentheses. As we have seen, they may be produced
by using {\tt [ ] \\\lb\ \\\rb\ ( )} to get $[\>]\>\{\>\}\>(\>)\>$.
Sometimes larger delimiters increase the clarity of
mathematical expressions, as in $$\bigl(a\times(b+c)\bigr) \bigl((a\times
b)+c\bigr).$$ To make larger left delimiters the
control words {\tt \\bigl}, {\tt \\Bigl}, {\tt \\biggl}, and {\tt
\\Biggl} are used in front of the delimiter; similarly, {\tt \\bigr},
{\tt \\Bigr}, {\tt \\biggr}, and {\tt \\Biggr} are used
for the right delimiters. Hence {\tt \$\\Bigl[\$} and {\tt \$\\Bigr]\$}
will produce $\Bigl[$ and $\Bigr]$.


Here is a table to compare the size of some of the delimiters.

%% \everycr can add 4 points between lines in the following table %%
\everycr={\noalign{\vskip 4 pt}}
\maketable [Delimiters of various sizes]
\strut \hfil$#$ & \quad \tt# \hfil \quad\qquad
&\hfil$#$ & \quad \tt# \hfil \quad\qquad
&\hfil$#$ & \quad \tt# \hfil \quad\qquad
&\hfil$#$ & \quad \tt# \hfil \cr
\noalign{\hrule} \noalign{\smallskip}
\{ & \\\lb & \} & \\\rb & ( & ( & ) & )\cr
\bigl\{ & \\bigl\\\lb & \bigr\} & \\bigr\\\rb & \bigl( & \\bigl( & \bigr) &
\Bigl\{ & \\Bigl\\\lb & \Bigr\} & \\Bigr\\\rb & \Bigl( & \\Bigl( & \Bigr) &
\biggl\{ & \\biggl\\\lb & \biggr\} & \\biggr\\\rb & \biggl(
& \\biggl( & \biggr) & \\biggr) \cr
\Biggl\{ & \\Biggl\\\lb & \Biggr\} & \\Biggr\\\rb & \Biggl(
& \\Biggl( & \Biggr) & \\Biggr)\cr


If you want, you can let \TeX{} choose the size of delimiter by
using the control words {\tt \\left} and {\tt \\right} before
your delimiters.
\TeXref{148}Thus {\tt \\left[$\ldots$\\right]} will cause the material
to be enclosed by brackets that are appropriately big. {\bf Note
well:} each use of a {\tt \\left} delimiter must have a matching
{\tt \\right} delimiter (although the delimiters themselves may be
different). Hence {\tt \$\$\\left|\lb a+b \\over
c+d\rb\\right|.\$\$} gives $$\left|{a+b \over c+d}\right|.$$

\maketable [Mathematical delimiters]
\strut \hfill$#$ & \quad \tt # \qquad\qquad &
\hfill$#$ & \quad \tt # \qquad\qquad &
\hfill$#$ & \quad \tt # \cr
\noalign{\hrule} \noalign{\smallskip}
( & ( & ) & ) & [ & [ \cr
] & ] &\{ & \\\lb & \} & \\\rb \cr
\lfloor & \\lfloor &\rfloor &\\rfloor & \lceil & \\lceil \cr
\rceil & \\rceil &\langle & \\langle & \rangle & \\rangle \cr
/ & / & \backslash & \\backslash &| & | \cr
\| & \\| &\uparrow & \\uparrow & \Uparrow & \\Uparrow \cr
\downarrow & \\downarrow & \Downarrow & \\Downarrow
& \updownarrow & \\updownarrow \cr
\Updownarrow & \\Updownarrow \cr


\exercise Typeset $\bigl \lceil \lfloor x \rfloor \bigr \rceil
\leq \bigl \lfloor \lceil x \rceil \bigr \rfloor$.

\subsection{Those special functions}

There are several types of functions that appear frequently in
mathematical text. In an equation like ``$\sin^2x + \cos^2x =
1$'' the trigonometric functions ``sin'' and ``cos'' are in roman
rather than italic type. This is the usual mathematical
convention to indicate that it is a function being described and
not the product of three variables. The control words {\tt
\\sin} and {\tt \\cos} will use the right typeface automatically.
Here is a table of these and some other special functions:

\maketable [Special mathematical functions]
\strut \tt {\\}#\hfil && \quad \tt {\\}#\hfil \cr
\noalign{\hrule} \noalign{\smallskip}
sin & cos & tan & cot & sec & csc & arcsin & arccos \cr
arctan & sinh & cosh & tanh & coth & lim & sup & inf \cr
limsup & liminf & log & ln & lg & exp & det & deg \cr
dim & hom & ker & max & min & arg & gcd & Pr \cr


\exercise Typeset: $\sin(2\theta) = 2\sin\theta\cos\theta \quad
\cos(2\theta) = 2\cos^2\theta - 1 $.

\exercise Typeset: $$\int \csc^2x\, dx = -\cot x+ C
\qquad \lim_{\alpha\to 0} {\sin\alpha \over \alpha} = 1
\qquad \lim_{\alpha\to \infty} {\sin\alpha \over \alpha} = 0.$$

\exercise Typeset: $$\tan(2\theta) = {2\tan\theta \over

\subsection{Hear ye, hear ye!}

There is a particular macro that is used in almost every
mathematical paper, and is different enough to require a special
explanation. This is the {\tt \\proclaim} macro. It is used when
stating theorems, corollaries, propositions, and the like. The
paragraph following {\tt \\proclaim} is broken into two parts:
the first part goes up to and including the first period that is
followed by a space, and the second part is the rest of the
paragraph. \TeXref{202--203}The idea is that the first part
should be something like ``Theorem 1.'' or ``Corollary B.'' The
second part is the statement of the theorem or corollary. Here
is an example:

\proclaim Theorem 1 (H.~G.~Wells). In the country of the blind,
the one-eyed man is king.

\noindent gives

\proclaim Theorem 1 (H.~G.~Wells). In the country of the blind,
the one-eyed man is king.

The statement of the theorem may contain mathematical expressions, of

\exercise Typeset:
\proclaim Theorem (Euclid). There exist an infinite number of

\exercise Typeset:
\proclaim Proposition 1.
$\root n \of {\prod_{i=1}^n X_i} \leq {1 \over n} \sum_{i=1}^n X_i$
with equality if and only if $X_1=\cdots=X_n$.


Matrices are typeset using combinations of the alignment
character {\tt \&} and the control word {\tt \\cr} to indicate
the end of the line. Start with {\tt \$\$\\pmatrix\lb$\dots$\rb\$\$}.
Into the space between the braces go the rows of the matrix, each
one ended by {\tt \\cr}. The entries are separated by the
{\tt \&}\null. For example the input

a & b & c & d \cr
b & a & c+d & c-d \cr
0 & 0 & a+b & a-b \cr
0 & 0 & ab & cd \cr

gives as printed output
a & b & c & d \cr
b & a & c+d & c-d \cr
0 & 0 & a+b & a-b \cr
0 & 0 & ab & cd \cr

The matrix entries in our examples have all been \centred{}
within their columns with a little space on each side. They can
be made flush right or flush left by inserting {\tt \\hfill}
before or after the entry. Notice the differences between the
following example and the previous one.

a & b & c \hfill & \hfill d \cr
b & a & c+d & c-d \cr
0 & 0 & a+b & a-b \cr
0 & 0 & ab \hfill & \hfill cd \cr

gives as printed output

a & b & c \hfill & \hfill d \cr
b & a & c+d & c-d \cr
0 & 0 & a+b & a-b \cr
0 & 0 & ab \hfill & \hfill cd \cr

\exercise Typeset
$$ I_4 = \pmatrix{ 1 &0 &0 &0 \cr
0 &1 &0 &0 \cr
0 &0 &1 &0 \cr
0 &0 &0 &1 \cr}$$

It's possible to have matrices that use other delimiters. Using
{\tt \\matrix} instead of {\tt \\pmatrix} will leave off the
parentheses, so the delimiters must be explicitly included using
{\tt \\left} and {\tt \\right}. Here is how we can change the
matrix of our first example.

$$ \left |
a & b & c & d \cr
b & a & c+d & c-d \cr
0 & 0 & a+b & a-b \cr
0 & 0 & ab & cd \cr
\right | $$

gives as printed output
$$ \left |
a & b & c & d \cr
b & a & c+d & c-d \cr
0 & 0 & a+b & a-b \cr
0 & 0 & ab & cd \cr
\right | $$

It's even possible to use {\tt \\left.} and {\tt \\right.} to
indicate that the opening or closing delimiter is deleted (note
the use of the period).

\exercise Use a matrix construction to typeset

$$ |x| = \left\{ \matrix{ x & x \ge 0 \cr
-x & x \le 0 \cr} \right.$$

This exercise and more general constructions of this type
may also be typeset using the {\tt \\cases} macro.

Sometimes ellipses are used within matrices. The control words
{\tt \\cdots}, {\tt \\vdots}, and {\tt \\ddots} can be used to
insert horizontal, vertical, and diagonal dots.

Thus we can use
$$ \left [
aa & \cdots & az \cr
\vdots & \ddots & \vdots \cr
za & \cdots & zz \cr
\right ] $$

to get as printed output
$$ \left [
aa & \cdots & az \cr
\vdots & \ddots & \vdots \cr
za & \cdots & zz \cr
\right ] $$

Matrices may also be typeset in-line, but they are pretty ugly
unless they have a small number of rows.

\subsection{Displayed equations}

All of the mathematics covered so far has identical input whether
it is to be typeset in-line or displayed. At this point we'll
look at some situations that apply to displayed equations only.

The first is that of aligning multiline displays. This is done with
the alignment character {\tt \&} and the control words {\tt \\cr} and
{\tt \\eqalign}. Starting with {\tt \$\$\\eqalign\lb$\dots$\rb\$\$},
the equations to be aligned are entered with each one terminated by
{\tt \\cr}. In each equation there should be one alignment symbol
{\tt \&} to indicate where the alignment should take place. This is
usually done at the equal signs, although it is not necessary to do
so. For example

a+b &= c+d \cr
x &= w + y + z \cr
m + n + o + p &= q \cr

a+b &= c+d \cr
x &= w + y + z \cr
m + n + o + p &= q \cr

Displayed equations can be numbered at either the right or left
margin. When the control word {\tt \\eqno} appears in a displayed
equation, everything after the control word is put at the right
margin. Hence {\tt \$\$ x+y=z. \\eqno (1)\$\$} yields
$$ x+y=z. \eqno (1)$$
To number an equation at the left margin, use {\tt \\leqno}
in place of {\tt \\eqno}.

It's possible to number aligned equations by using the control
word {\tt \\eqalignno}. The alignment character {\tt \&} is
used to separate the equation from the equation number.
a+b &= c+d & (1) \cr
x &= w + y + z \cr
m + n + o + p &= q & * \cr

a+b &= c+d & (1) \cr
x &= w + y + z \cr
m + n + o + p &= q & * \cr

Use {\tt \\leqalignno} to put the equation numbers on the left.

Finally, suppose some text needs to appear in the middle of a
displayed equation. This can be done by putting it in an hbox.
We will describe hboxes in more detail in the next section.
For now we want to use them to temporarily resume using ordinary
roman type and to also allow the insertion of space between
words (remember that all spaces are ignored when typesetting
mathematics). Hence
{\tt \$\$X=Y \\hbox\lb{} if and only if \rb x=y.\$\$} will give
$$X=Y \hbox{ if and only if } x=y.$$
Note carefully the spaces in the hbox.

\exercise Do some of the challenge problems on pages 180--181 of The
\TeX book.

\section{All in a row}

It's not uncommon to want to put a table in the middle of some
text. Fortunately \TeX{} makes it easy to do this. In fact there
are two separate methods of aligning text. The first is by using
the tabbing environment. This is similar to setting the tab
stops on a typewriter. Each line is handled individually,
according to set tab columns, but with greater flexibility than
that provided by a typewriter. The second is the horizontal
alignment environment which typesets the whole table at once
using a prescribed pattern.

\subsection{Picking up the tab}

To align material using the tabbing environment, you must first
set the tab positions using the {\tt \\settabs} control word.
Having done this, a line to use these tabs starts with the
control symbol {\tt \\+ } and ends with {\tt \\cr } (remember
that the actual spacing on lines in the input file in
unimportant). \toindex{settabs}

The easiest way to use the {\tt \\settabs} control word is to put
the text into equal columns.
Using {\tt \\settabs 4 \\columns} will set the tabs that will
produce four equal columns. The tabbing is then done by using
the alignment character {\tt \&} to move to the next tab stop.
So, for example, \toindex{columns}

\settabs 4 \columns
\+ British Columbia & Alberta & Saskatchewan & Manitoba \cr
\+ Ontario & Quebec & New Brunswick & Nova Scotia \cr
\+ & Prince Edward Island & Newfoundland \cr

will produce the table

\settabs 4 \columns
\+ British Columbia & Alberta & Saskatchewan & Manitoba \cr
\+ Ontario & Quebec & New Brunswick & Nova Scotia \cr
\+ & Prince Edward Island & Newfoundland \cr

Notice that it is possible to skip over some tab positions, and
it is not necessary to use all of the tabs in a given line. To
make the same table using five columns, it is only necessary to
use {\tt \\settabs 5 \\columns} to reset the tab stops; then the
same three lines from the last example will produce:

\settabs 5 \columns
\+ British Columbia & Alberta & Saskatchewan & Manitoba \cr
\+ Ontario & Quebec & New Brunswick & Nova Scotia \cr
\+ & Prince Edward Island & Newfoundland \cr

In this example, the columns are smaller, of course. In fact
there are two overlapping entries in the last row. This is
because \TeX{} will tab to the next tab position even if (unlike
a typewriter) it means going backward on the page.

There is an interesting relationship between grouping and
tabbing. The {\tt \\settabs} values are only applicable to the
group in which it is defined, as would be expected. Thus it is
possible to temporarily change the tab settings by grouping
within braces. In addition, each table entry is in a group of
its own. Hence we may make a single entry boldface, for example,
by using {\tt \\bf} without braces. In addition, for any column
but the last one it is possible to \centre{} the entry or to
align it either on the left or on the right, or to fill a column
with a line or dots. Each entry has an implicit {\tt \\hfil} at
the end so that it will be at the left of the column by default.
Adding {\tt \\hfil} at the beginning of the entry will then cause
it to be \centred{}, just as with the {\tt \\line} control word.
Adding {\tt \\hfill} to the beginning will cause the entries to
be pushed to the right ({\tt \\hfill} acts just like {\tt \\hfil}
in that it absorbs excess space; when both {\tt \\hfil} and {\tt
\\hfill} appear, the {\tt \\hfill} takes precedence).

\settabs 4 \columns
\+ \hfil British Columbia & \hfill Alberta \qquad & \bf Saskatchewan
@hskip 2.5in & Manitoba \cr
\+ \hfil Ontario & \hfill Quebec \qquad & \bf New Brunswick
@hskip 2.5in & Nova Scotia \cr
\+ \hfil --- & \hfill * \qquad & \bf Newfoundland
@hskip 2.5in & Prince Edward Island \cr
\+ \dotfill && \hrulefill & \cr

will produce a table with the first column \centred{}, the second
column flush right with a {\tt \\qquad} of padding, and the third
column boldface. The control words {\tt \\dotfill} and {\tt \\hrulefill}
give alternative column entries.

\settabs 4 \columns
\+ \hfil British Columbia & \hfill Alberta \qquad & \bf Saskatchewan
& Manitoba \cr
\+ \hfil Ontario & \hfill Quebec \qquad & \bf New Brunswick
& Nova Scotia \cr
\+ \hfil --- & \hfill * \qquad & \bf Newfoundland &
Prince Edward Island \cr
\+ \dotfill && \hrulefill & \cr

\exercise Take the table of Canadian provinces above and \centre{} each
entry within its column.

The tab positions can be set with much more flexibility than just
in equal columns. The general pattern is to use a sample line of
the form {\tt \\settabs \\+ $\ldots$ \& $\ldots$ \& $\ldots$ \\cr}.
The spacing between the alignment characters {\tt \&}
determines the position of the tabs. For example, {\tt \\settabs
\\+ \\hskip 1 in \& \\hskip 2 in \& \\hskip 1.5 in \& \\cr} would
set the first tab one inch from the left margin, the next another
two inches further in, and the third 1.5 inches more. It's also
possible to use text to determine the distance between tabs. So,
for example, another possible sample line is
{\tt \\settabs \\+ \\quad Province \\quad \& \\quad Population \\quad
\& \\quad Area \\quad \& \\cr}.
The tab column would then be just wide enough to accept the headings
with a quad of space on each side. Here's a more complete example:

\beginuser \obeyspaces
\\settabs \\+ \\quad Year \\quad \& \\quad Price \\quad
\& \\quad Dividend \& \\cr
\\+ \\hfill Year \\quad \& \\quad Price \\quad \& \\quad Dividend \\cr
\\+ \\hfill 1971 \\quad \& \\quad 41--54 \\quad \& \\qquad \$2.60 \\cr
\\+ \\hfill 2 \\quad \& \\quad 41--54 \\quad \& \\qquad \$2.70 \\cr
\\+ \\hfill 3 \\quad \& \\quad 46--55 \\quad \& \\qquad \$2.87 \\cr
\\+ \\hfill 4 \\quad \& \\quad 40--53 \\quad \& \\qquad \$3.24 \\cr
\\+ \\hfill 5 \\quad \& \\quad 45--52 \\quad \& \\qquad \$3.40 \\cr

gives \TeXref{247}

\settabs \+ \quad Year \quad & \quad Price \quad & \quad
Dividend \quad & \cr
\+ \hfill Year \quad & \quad Price \quad & \quad Dividend \cr
\+ \hfill 1971 \quad & \quad 41--54 \quad & \qquad \$2.60 \cr
\+ \hfill 2 \quad & \quad 41--54 \quad & \qquad \$2.70 \cr
\+ \hfill 3 \quad & \quad 46--55 \quad & \qquad \$2.87 \cr
\+ \hfill 4 \quad & \quad 40--53 \quad & \qquad \$3.24 \cr
\+ \hfill 5 \quad & \quad 45--52 \quad & \qquad \$3.40 \cr

\exercise Take the table given above and move it closer to the
\centre{} of the page.

\exercise One way to \centre{} a block of text, possibly several
lines long, is to use: {\tt \$\$\\vbox\lb$\ldots$\rb\$\$}. Use
this to \centre{} the table given above. Does the
{\tt \\settabs} line need to be included in the {\tt \\vbox}?

\exercise Improve your last result by putting a line under the
column heads. The control word {\tt \\hrule} will insert a
horizontal line if introduced between two rows of a table. Now
repeat with the control word {\tt \\strut} after the {\tt \\+ }
of the line containing the column heads. (A {\tt \\strut}
effectively makes the spacing between lines a little greater. The
size can be altered from the default.)\TeXref{82} Note the extra
space that results.

\exercise Make the following table with decimal alignment, that
is, with the decimal points above each other (think of the dollar
figure as being right aligned and the cents figure as being left
aligned against the decimal point):
\settabs \+ \hskip 2 in & \hskip .75in & \hskip 1cm& \cr
\+ &Plums &\hfill\$1&.22 \cr
\+ &Coffee &\hfill1&.78 \cr
\+ &Granola &\hfill1&.98 \cr
\+ &Mushrooms & &.63 \cr
\+ &{Kiwi fruit} & &.39 \cr
\+ &{Orange juice} &\hfill1&.09 \cr
\+ &Tuna &\hfill1&.29 \cr
\+ &Zucchini & &.64 \cr
\+ &Grapes &\hfill1&.69 \cr
\+ &{Smoked beef} & &.75 \cr
\+ &Broccoli &\hfill\underbar{\ \ 1}&\underbar{.09} \cr
\+ &Total &\hfill \$12&.55 \cr

\exercise Devise a method to make a rough table of contents by using
{\tt settabs} and having entries looking something like:\hfil\break
\leftline{\tt Getting Started \\dotfill \& \\hfill 1}
\leftline{\tt All Characters Great and Small \\dotfill \& \\hfill 9.}

\subsection{Horizontal alignment with more sophisticated patterns}

The {\tt \\settabs} environment is not difficult to use, and once
the pattern is set, it can be used repeatedly in different
portions of the text that follows. It does have some drawbacks,
however. For one, the column size must be set before the entries
are known. Also, even though in one case we wanted the third
column to be boldface, it had to be specified in each line.
These problems can be handled more easily by using the {\tt
\\halign} environment.

The general pattern in the {\tt \\halign} is as follows: