The Name, a Short History, and Identification

Name, Pronunciation, Short History and Present Status

The words like technology find their roots in the Greek originated term starting with the letters TEX which means art as well as technology. Hence we write the name of the program we are going to tackle with in this chapter as TeX to recall TEx. We imply that this facility, that is, TeX, is a master tool which is equipped with both technology and art.

The pronunciation of the word TeX is not what is expected. Since we recall the TEX letters the implied pronunciation does not contain the sound of the letter x. Since the letter x characterizes x it sounds like ch in the word tech or like in the German word ach. It sounds very similar to the Russian ch. Hence we pronunciate TeX same as tech.

TeX was developed by Donald E. Knuth who was a Computer Science guru at Stanford University, through a major project of the same University in 1970's. The purpose was to create a software package which removes or diminishes all difficulties in the document preparation into a very minor level. The project had been successfully completed and the product was equipped with the features which were not existing in the other available softwares. The status of TeX is still same for today as Donald E. Knuth once said like "We created a wonderful software which is the ultimate point of the document preparation. It is just a tool similar to the Stradivarius brand of the violins. It is invaluable and if you use it as virtuoso then the output is just perfection. One can create macro packages based on TeX to facilitate its utilization." during an interview (Well, these are not the exact statements of Knuth said. But the content is same within the knowledge of the author.). Indeed, after TeX was born there had been a lot of efforts to create new macro packages to facilitate its utilization.

Today, we can mention LaTeX which is an easier version enabled with a lot of simplifications and shortcuts at the expense of some quality decrement, MuTeX which was created to edit music notes in a simple way, ChemTeX which is a version to handle chemical formulae in a simple way, AMSTeX which is a collection of TeX macros to simplify mathematical formula typesetting and some others amongst most famous examples to this end.

The reason why we pronounce TeX like tech or more precisely like tecks is to remind the user TeX is essentially concerned with high-quality technical manuscripts. It emphasizes on the art and technology and reminds that it the only thing you can use if you want to produce a high fidelity document. If you merely want to obtain a document which is passably good enough, that is, something acceptable and rather readable but not really beatiful, a simpler system which is called just word processor (like Word) is sufficient. The main goal of TeX is not just to obtain a readable document but to produce finest quality.

There is one more remark about the typography of the name of TeX. As you see E is written in a displaced form. This is because of the existence of another acronym TEX which stands for the statement Text EXecutive Processor which is the name of a system which was developed by Honeywell Information Systems.

TeX is a publicly available software since it is developed by a non-profit organization. If it would be marketed in the profit-oriented media the program package would certainly gain a price about many thousands of dollars as a simple comparison of its capabilities with the profit-oriented rather simple software packages reveals. The TeX's main site in the Internet locates at the URL (Unified Resource Location) http://www.ctan.org. You can find its versions for all major operating systems there. The site has an amazing abundance of programs and utilities about TeX and related things.

The Structure and Running of TeX

TeX is in fact a programming language since it contains most of the necessary features to be so. You can use variables, constants, input and output variables, logical statements, loops, an arithmetical operation capability which suffices for the necessary calculation in printing, and macros. It has a modular structure so you can extend or build a new structure which is oriented to a more specific task, on it. These capabilities give of course a lot of superiorities to TeX over the other softwares and many computer wizards think that it is at an ultimate point where there is no more road to go. This point of TeX is emphasized by giving TeX a version number which is composed of the first few decimal digits of the transcendental number ?.

TeX works like an interpreter. When it is invoked it waits line by line input to process and creates an output file whose type is called DVI which is an acronym for the statement DeVice Independent. These files have binary contents which do not depend where they are created. This universal structure gives them the portability.

TeX needs a bundle of lines as input to produce the desired output. These lines can be composed of commands and text material to be appearing in the output. Each TeX command must start by a backslash character \ and can be composed of lower or upper case letters. The numerals can also be used in the naming but without any letters. TeX commands do generally accept or need parameters or arguments. It is possible to define new commands by combining the TeX primitive commands. The entirity of the lines which forms the input a document preparation session is defined as a TeX input file. All TeX files are assumed to have a four letter suffix .tex. The files can be created by means of any editor or editing facility existing in your system.

The creation of the DVI file from a TeX input file can be realized in two ways, either interactive or batch mode. In the interactive mode each line of the input is given separately but consecutively. To invoke an interactive mode under SuSE Linux you can give the command tex at the commandline interpreter prompt by facing with the following display.

demiralp@jaguar:~/bil101 > tex
This is TeX, Version 3.14159 (Web2C 7.3.1)

** ** is the global TeX prompt and, to proceed, you must give either a TeX command or a filename for the TeX input file to be processed. If you choose to give a filename which does not exist then TeX complains by a warning message and prompts you to give another filename. If so then you can try to give the correct filename or to stop the session by giving a Control D, that is, end-of-file character. If you stop the session of TeX then you see a message announcing No pages of output. If you give the correct filename for the TeX input file, say sample.tex then the filename of the file between the parentheses is returned in the next line and a single star prompt appears in the following line like the following display

demiralp@jaguar:~/bil101 > tex
This is TeX, Version 3.14159 (Web2C 7.3.1)
**sample.tex
(sample.tex)
*
This happens when the file sample.tex does not contain any terminating command. So what we are going to do for the finalization of the session is to give either end or maybe more securely bye command at this prompt. If the file sample.tex would contain one of the commands end or bye then the above display would replace with the following one.

demiralp@jaguar:~/bil101 > tex
This is TeX, Version 3.14159 (Web2C 7.3.1)
**sample.tex
(sample.tex [1] )
Output written on sample.dvi (1 page, 220 bytes).
Transcript written on sample.log.

If we do not want to give a filename but to give the content of a TeX input file line by line then we need to switch to line insertion mode where everything typed at the prompt * will be assumed the content of a single line without regarding how long it is until we type an end-of-line character which is generally invoked by the Enter or Return key. This switching can be accomplished by giving the TeX command \relax at the prompt **. If done, the prompt changes from the double star prompt to the single star one. When the line insertion mode is activated as told above the session can be terminated by giving one of the TeX commands end or bye or entering an end-of-file character by pressing the Ctrl and d keys together. In the case of termination via end or bye commands an output file created under the name texput.dvi and the text format display of the session is collected into a file whose name ise texput.log. However, in the case where the session is terminated by entering Ctrl d no dvi file is created although all the text format display of the session is saved into the file texput.log.

The batch mode is quite straight forward. We write all necessary TeX commands and the text material to create what we want in the output display into a file whose name contains a four letter suffix .tex. This file is the TeX input file. We run TeX by specifying the name of the input file after tex command followed by at least one space character at the commandline interpreter of the system. If there is no syntax error in the input file then this command runs successfully and a dvi file for the display and a log file for the log of the session is created and this is announced on the screen. For example, in our SuSE system we can give the display of an attempt to run a TeX file sample.tex and its result as follows.

demiralp@jaguar:~/bil101 > tex sample
This is TeX, Version 3.14159 (Web2C 7.3.1)
(sample.tex [1] )
Output written on sample.dvi (1 page, 216 bytes).
Transcript written on sample.log.

The interactive mode is more pedagogical because of the instant reaction of TeX after entering each line. Therefore one can learn more rapidly by using this mode. On the other hand the batch mode can also be in the same way but this time the errors can be seen and removed after the run of the commands in the given input file as a single entity. If the user is a little bit experienced about the preparation of a TeX input file then it can use the batch mode. For advanced user it is a matter of preference to use bacth or interactive mode.

The steps we have mentioned here are called Editing of the TeX Input File and Running of the TeX Input File or simply Editing and Running steps. The editing stage covers the creation of the input file to be processed by TeX. One can use any editors which can insert the TeX commands and the text material into a file or can revise the content of an existing TeX input file. There is no restriction on the format of the input file but its name is assumed to have a four letter suffix .tex. The running stage involves the processing of the input file by TeX. This can be done via interactive or batch mode as we told above. A successful running stage always creates a DVI and LOG files unless a special structure is given to the input file not to produce any output page. DVI and LOG files are created under the names which are composed of a common basename same as the .tex suffixed input filename and the suffixes .dvi and .log

Displaying the DVI Files

After the successful editing and running stages the next step is the displaying of the output included in the DVI file in binary form. For this task we need a GUI (Graphical User Interface) specific software which is called Driver. Most common driver for X window system is called xdvi. It is publicly available without any license fee. It is used by giving the basename of the DVI file after the command xdvi by leaving at least one space. This can be done only in a X Window session through a terminal window or console of X Window. If done, a new window with some buttons at the right margin of the window, is created. The frame which does not contain buttons is used for the display of the output file (DVI file). The buttons at the right margin are for some specific purposes like magnification or switching among the pages. When the pointer of the mouse is located on one of these buttons and is clicked then the corresponding action which is called event in the graphics programming terminology is realized. The actions taken via mouse clicking can also be realized by pressing some keys on the keyboard. We do not intend to give every information about xdvi although the student will be kept responsible to use it at the level where all mouse clicking and key pressing capabilities are considered. This information can be found in the documentation for xdvi like manual pages.

There are some recently developed desktop environments like KDE or Gnome where some other drivers are under development. For example, SuSE Linux KDE contains a driver for DVI files under the name kdvi. It is at the level of version number 0.4 at the moment when these pages are in preparation. The capabilities of this newly developed driver is not so much comparable with xdvi hence we are not going to give a special emphasis on this software.

The Microsoft Windows series or NT use some other type of drivers like dvi2scr although there are some efforts to develop the version xdvi for these systems. The other platforms also have similar drivers and they can be downloaded with a satisfactory documentation from the main site of TeX without paying anything. We suffice the information given here without attempting for further detailing.

Document Preparation With TeX

Plain Text, Text Alignment and Blocks, Glues, Fonts

Plain Text

Now we can start to learn how a TeX input file can be prepared in the editing step by opening a file whose name is sample1.tex. You can use any editor for this purpose and it is quite natural to use your favorite editor. Assume that we write the following lines into this file by maintaining their format.

The interactibe mode is more
pedagogical because of the instant reaction
of TeX after entering each line. Therefore one can
learn more rapidly by using this mode. On the other hand the
batch mode
can also be
in the same way bur this time the errors
can be seen and removed after the run of
the commands in the
given input file as a single entity.
If the user is a little
bit
experienced about the preparation of a TeX input file
then it can use the batch mode.
For advanced users it is a
matter of preference
to use batch or interactive mode.


where the content of the file is exactly same as one of the paragraphs located above. After saving this file and running TeX for this file as follows

demiralp@jaguar:~/bil101 > tex sample1
This is TeX, Version 3.14159 (Web2C 7.3.1)
(sample1.tex [1] )
Output written on sample1.dvi (1 page, 872 bytes).
Transcript written on sample1.log.

we obtain the following output on the display if the file sample1.dvi is driven by xdvi.

The interactive mode is more pedagogical because of the instant reaction of TeX after entering each line. Therefore one can learn more rapidly by using this mode. On the other hand the batch mode can also be in the same way but this time the errors can be seen and removed after the run of the commands in the given input file as a single entity. If the user is a little bit experienced about the preparation of a TeX input file then it can use the batch mode. For advanced user it is a matter of preference to use batch or interactive mode.

Text Alignment and Blocks

If we now compare the output display with the content of the input file then we see that the appearences are different. The content of the input file is left adjusted and there is no alignment at the right margin of the file and the number and content of each line in the output is very different than the input file. This is because of the alignment capability of TeX. It is better to explain the procedure TeX uses for creating lines now. TeX reads all lines of the input file and makes a single strip by changing each character with its roman counterpart and deleting all newline characters without caring how long the final strip is. TeX can use many different fonts and the default font type is called Roman. The default size of the font is ten points where 72.22 points are equivalent to one inch.

Each character has a display in a rectangular frame which has three dimensions: width, height and depth. The frame is divided into two vertical segments separated by a horizontal line which is called baseline. The distance between the top edge and the baseline is height whereas the distance from the baseline to the bottom edge is called depth. The depth of a character may be zero depending on its shape. The width is the length of the baseline. TeX concatenates two characters such that their baselines match on the same line. If there is one or more spaces or tabs between two consecutive characters then TeX inserts just a single space. Since the heights may vary from character to character TeX inserts extra vertical space at the top and the bottom of the each character (if necessary) to get the same height and depth for the entire strip obtained after the concatenation of all existing characters in the file. If the strip is long than the default size of the display then TeX tries to divide the strip into consecutive substrips. There may be three options to be encountered at this cutting stage. There remains a space at the cut edge in the first possible case. Then TeX tries to increase the spaces between the words in the strip such that the cut operation leaves no space at the end of each substrips. The second case presents no spaces at the end of the substrips and needs nothing to do although the probability of this occurrence is quite low. In the third case the rightmost words of some substrips may necessitate the partitioning of the word into two segments. This is called hyphenation and done in according to the hyphenation rules of the English primarily and German secondarily. Although the defaults are English and German hyphenations one can add some other hyphenation schemes globally. On the other hand it is possible to mark the locations where the word can be divided. All you have to do for this purpose is to insert \- characters into the hyphenation positions. If the word is divided then its first segment is concatenated by the dash character - and spacing is readjusted.

TeX assumes that the display will be locate on a standard sheet of paper which is either letter (11x8.5 inches and used mostly in USA) or A4 which is a standard mostly used in Europe. The default size of the display can be changed by using the TeX commands \hsize, \vsize, \hoffset, \voffset. The created substrings are vertically juxtapositioned by adding vertical spaces between the substrings to create a new vertical strip. This may not fit to the vertical size of the default page of the display. If so then TeX tries to divide this vertical strip into pages by cutting it at the vertical size of the default display. Each of the created pages is located into a virtual sheet of paper such that the location of the left upper corner of the page has the coordinates defined by \hoffset, \hoffset. The coordinates of any location on the sheet are measured from left upper corner of the sheet to the right and down. The default values for horizontal and vertical offsets are 1 inch.

A single line can be left or right justified or centered in the output display. This can be accomplished by using the TeX commands \leftline, \rightline, and \centerline. All these commands needs a single argument which may involve any space or tab characters. The text material to be left or right justified or centered must be given in block structure. A block in TeX is a group of commands and text material which are encompassed by the left and right braces {, }.

One more facility about the alignment is the deliberately breaking of the lines within a predefined tolerance. We can mention two TeX commands to this end: \break, and \goodbreak. \break breaks the line at the point it is given and increases the intermediate spaces such that the new formed line is terminated by the last word before the command. This may not be so beatiful appearence unless there are sufficiently many words are existing in the new formed line to leave spaces which do not irritate the observer of the display. \goodbreak breaks the line at the point it is issued such that some reasonable amount and size of spaces are used and then the remaning portion of the line (if any) is filled by the sufficient spaces. The appaerance is like the output of the \leftline command. Of course, there is no strict rules how and when these commands should be used. It is a matter of the preference of the designer of the TeX document although, in the case official documentation, there may be some flexible restrictions to be obeyed. An underfull hbox warning message is announced when \break command is used.

Now we can see how these commands work from the processing of the file sample2.tex whose content is given below

\leftline{This line is left justified.}
\rightline{This line is right justified.}
\centerline{This line is centered.}
\bye

where the output is as follows

This time is left justified.

This time is right justified.

This time is centered.

       This                        line                        is                        broken.

This line is goodbroken.

Glues

You can create spaces between the words or lines. The spaces can be divided into two categories: skips and glues. skip is a vertical or horizintal space such that it can be divided into pieces if it exceeds the borders of line or page. Whereas glue is an indivisible space. skipneeds a reference material to be activated. However gluedoes not behave so. For example if you want to leave a blank space at the beginning of the page you can not use the command \vskip 2cm because if you do so it is ineffective whereas the command \vglue 2cm affects the display as you want.

To create a vertical space either \vskip or \vglue commands are used. These command must be followed some lengths. The units which can be used for these lengths are pt, mm, cm, in where pt (point) stands for the unit employed for typographical purposes and 72.22 pt is equivalent to 1 inch.

The horizontal space can be created by using \hskipor \hglue commands. These must be followed by some lengths also.

There are some special purpose skips or glues like \baselineskip, or \parskip \topskip. In TeX each line has a baseline where the bases of the characters are located. Each character of TeX has three dimensions: width, height, and depth. If we assume that each character is located into a rectangle then the length of the horizontal edge is width while the height of the rectangle is the sum of the height and the depth of the character. This hypothetical rectangle is divided into two vertical pieces by a hypothetical horizontal line. When two separate characters are joined by matching their rectangles horizontally such that these hypothetical horizontal lines match on the same horizontal direction. This line is called baseline. When TeX builds the lines it takes the characters and combines their rectangular frames such that all baselines are matched on the same horizontal line. Therefore if one wants to leave spaces between the lines than it must specify a baselinskip value otherwise TeX uses its default value. The command for this purpose is \baselineskip and needs a length value. If the last line of the page does not fit the last portion of the page then TeX throws it to the next page and an extra space is left blank in the page. When the file is interpreted TeX complains about the underfull vertical box which is the page under consideration. This circumstance can be avoided by giving a flexibility to the command. This can be done by giving parameters to the command for plus or minus flexibilities. For example, the command \baselineskip 20pt plus 2pt minus 2ptcreates 20 point deep vertical spaces between the consecutive lines and TeX is enabled to use the skip value not exactly 20 points but some other value between 18 points and 22 points inclusive. This removes the possibility of mismatching the last line with the end of the page. TeX automatically adjust the situation. You have to be careful about the order of the parameters plus and minus. One or both of them can be skipped but if both of them are given then plus must precede the minus. The numbers in the lengths may not need prepending spaces although it is better to use the spaces for clarity.

The command \parskip is used to make spaces between the paragraphs. If you leave a blank line or use the command \par in the TeX file then TeX inserts a default space between two consecutive lines. Specifying a parskip changes the default value for paragraph skipping. The command must be followed by the desired skip value.

The command \topskip is used to make spaces between the headline and the first line of the page. If not given then TeX uses the default value. The command must be followed by the desired skip value.

Fonts

As all typographical utilities do TeX uses fonts. Font is a collection of characters. For TeX the numbers of the characters in a font is limited by 256. Each font can be considered as matrix box which is composed of 256 (16x16) cells. Each cell is related to a single different character. In a font some cells may be empty depending on the type of the font. TeX uses 10 pts roman characters for plain text by default. However this can be changed. In fact, TeX can use 65536 different fonts at the same time. There are two major ways for calling a font inside a TeX document. One and easiest way is to give a calling-by-acronym-command. For example, \rm, is used to call roman font of the current size in the document. Similarly, \it, \sl, \bf, \tt commands are used to call italic, slanted, extended boldface, typewriter fonts respectively. Therefore, one of these commands can be given at any place of the document source file. From that place to the next place where another font calling command is given the effect of thecommand continues. In other words, these commands change the default value for the font when they are given. It is also possible to give these commands without changing the default value of the font. For this purpose, the command must be given in a block which is defined by a TeX material encompassed by the characters { and }.

The other way for font calling is to use the command \font. This command must be followed by a name which is at our disposal for the font, an equal sign, and the real name of the font in the TeX package. For example \font myfont=cmbx10
where myfont is the name to be given to the font through the document by us. cmbx10 is the name of the font in the TeX package. In this name first two characters forms an acronym for the Computer Modernstatement, bx stands for Extended Boldface and 10 denotes trhe size of the font, that is, ten points. It is also possible to scale the fonts. This can be done by giving extra parameters to the above command. The parameter is scaled which must be followed by a value for the scaling. This value equals thousand times of the scaling ratio. That is, 1000 means a scaling ratio 1 which does not change the measures whereas 2500 means that all sizes will be multiplied by a factor of 2.5. In other words a scaling of 2500 value will change 10 pts to 25 pts. Scaling ratio can be given as the parameters \magstep1, \magstep2, \magstep3, \magstep4, \magstep5. Each parameter here scales the sizes by a corresponding power of 1.2. For example \magstep1 and \magstep2scales with the ratios 1.2 and 1.44 respectively.

Mathematical and Display Modes, Formulae Typesetting

Mathematical Mode

You can write mathematical formulae by using TeX. There are two available modes for this purpose. The Mathematical Mode can be created by encompassing the TeX commands and related material by $ signs. Anything given between these signs are considered mathematical objects to be inserted into lines by TeX. So this mode enables us to combine text material and mathematical formulae at the same line.

Display Mode

The Display Mode can be created by encompassing the TeX commands and related material by $$ signs. Anything given between these signs are considered mathematical objects to be inserted between lines by TeX. So this mode enables us to typeset mathematical formulae separately.

Formulae Typesetting

This task can be done by combining the convenient TeX commands by obeying some rules. The coming examples will give sufficient information about this job.

Examples for Typesetting via TeX

The following examples are taken from the document file gentle.tex. They are self-explanatory so the students are recommended to investigate them and get more information and experience about TeX.

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.

Does \AE schylus understand \OE dipus?

Does Æ schylus understand dipus?

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

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

They took some honey and plenty of money wrapped up in a £5 note.

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

É&egraveves, refusez vos leçons! Jetez vos chaî nes!

Za\v sto tako polako pijete \v caj?

Zato tako polako pijete c?

Mein Tee ist hei\ss.

Mein Tee ist heiß.

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

Peut-être qu'il préf&egravere le caf? glacé.

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

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

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

Can you take a ferry from &Oumlland to Å land?

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

T&uumlrkçe konusan yegenler nasillar?

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

I entered the room andhorrorsI saw both my father-in-law and my mother-in-law.

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

The winter of 14841485 was one of discontent.

His ``thoughtfulness'' was impressive.

His "thoughtfulness" was impressive.

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

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

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

He thought, "...and this goes on forever, perhaps to the last recorded syllable."

Have you seen Ms.~Jones?

Have you seen Ms. Jones?

Prof.~Smith and Dr.~Gold flew from Halifax N.~S. to Montr\'eal, P.~Q. via Moncton, N.~B.

Prof. Smith and Dr. Gold flew from Halifax N. S. to Montréal, P.Q. via Moncton, N. B.

\line{\hskip 1 in ONE \hfil TWO \hfil THREE}

          ONE                                 TWO                                 THREE

i{f}f if{}f if{f}

I started with roman type {\it switched to italic type}, and returned to roman type.

I started with roman type switched to italic type, and returned to roman type.

$C(n,r) = n!/(r!\,(n-r)!)$

C(n,r) = n!/(r! (n-r)!)

$a+b=c-d=xy=w/z$
$$a+b=c-d=xy=w/z$$

a+b=c-d=xy=w/z
a+b=c-d=xy=w/z

$(fg)' = f'g + fg'$
$$(fg)' = f'g + fg'$$

(fg)' = f'g + fg'
(fg)' = f'g + fg'

$\alpha\beta=\gamma+\delta$
$$\alpha\beta=\gamma+\delta$$

?ß=?+?
?ß=?+?

$\Gamma(n) = (n-1)!$
$$\Gamma(n) = (n-1)!$$

?(n) = (n-1)!
?(n) = (n-1)!

$x\wedge (y\vee z) = (x\wedge y) \vee (x\wedge z)$

$2+4+6+\cdots +2n = n(n+1)$

2+4+6+...+2n = n(n+1)

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

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

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

${a+b\over c}\quad {a\over b+c}\quad {1\over a+b+c} \not= {1\over a}+ {1\over b}+{1\over c}$.

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

$e^x \quad e^{-x} \quad e^{i\pi}+1=0 \quad x_0 \quad x_0^2 \quad {x_0}^2 \quad 2^{x^x}$.

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

$\lim_{x\to 0} (1+x)^{1\over x}=e$.

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

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

$\int_0^1 3x^2\,dx = 1$.

$\sqrt2 \quad \sqrt {x+y\over x-y} \quad \root 3 \of {10}$ \quad $e^{\sqrt x}$.

$\|x\| = \sqrt{x\cdot x}$.

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

$\underline x \quad \overline y \quad \underline{\overline{x+y}}$.

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

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

sin(2?) = 2sin?? cos(2?) = 2cos²? - 1.

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

$$\tan(2\theta) = {2\tan\theta \over 1-\tan^2\theta}.$$

\proclaim Theorem (Euclid). There exist an infinite number of primes.

\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 $X_1=\cdots=X_n$.

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

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

	Getting Started.......................................................1

	All Characters Great and Small................................9

\settabs \+ \hskip 1cm&\hskip 1 cm&\hskip 1 cm& \cr
\moveright 2 in
\vbox{
\hrule width 3 cm
\+ \vrule height 1 cm & \vrule height 1 cm & \vrule height 1 cm
& \vrule height 1 cm \cr
\hrule width 3 cm
\+ \vrule height 1 cm & \vrule height 1 cm & \vrule height 1 cm
& \vrule height 1 cm \cr
\hrule width 3 cm
\+ \vrule height 1 cm & \vrule height 1 cm & \vrule height 1 cm
& \vrule height 1 cm \cr
\hrule width 3 cm
}

\def\boxtext#1{%
\vbox{%
\hrule
\hbox{\strut \vrule{} #1 \vrule}%
\hrule
}%
}
\moveright 2 in \vbox{\offinterlineskip
\hbox{\boxtext{6}\boxtext{1}\boxtext {8}}
\hbox{\boxtext{7}\boxtext{5}\boxtext{3}}
\hbox{\boxtext{2}\boxtext{9}\boxtext{4}}
}