and TeX source code

* Changed* July 15, 2005

TeX (pronounced "tech" with emphasis on the `ch': "teCH") is a mathematical typesetting language invented by Donald Knuth of Stanford University. It is hard to learn, but a great many mathematicians seem to have found it worth their whiles...

See Knuth's book, ISBN 0-201-13448-9,

A major advantage of TeX is that the "source" document is an ordinary text file that can be sent by e-mail, for example. This advantage has to be gained at the expense of learning the special conventions and codes for writing math in a text-only format.

I will list some of these conventions here; some are well-known, others are not. I hope this will make it possible for you to decipher what is in my Web pages until I can figure out how to put the mathematics into readable form here! Please let me know what you want to see added here -- I'm likely to forget about some stuff. There is an alphabetical list of codes at the bottom of this page, and general guides before that.

"x-sub-n" is rendered as x_n (the underscore character is "shift-minus"
on my keyboard).

"x-to-the-n" is rendered as x^n (the "carat," or "hat,"
is "shift-6" on my keyboard).

Subscripts and superscripts can be combined in either order. Thus, "x-sub-n,
squared" may be rendered as x_n^2 or as x^2_n.

When you want to write "one-over-x" as the reciprocal of x, namely
as "x-to the power: -1," the exponent becomes a two-character
superscript, so it has to be enclosed within "{" and "},"
which are TeX's special delimiters for grouping. So, you write x^{-1} to
get "reciprocal x" expressed as "x-to the power: -1."

In TeX, ordinary text is distinguished from mathematical symbols by placing
the math stuff within dollar signs - they act as some kind of non-directional
parentheses. For example, to write that

"f of x equals x-squared" you type $f(x)=x^2.$ It gets worse:
when you want the formula to be displayed on a separate line, or lines,
you enclose the math stuff within pairs of pairs of dollar signs. Thus,
$$f(x)=x^2.$$ shows up on a separate line.

I try to remove the dollar signs, so this is in here just to let you know
to ignore them if I forget, and do leave them in.

"infinity" is "\infty." It "should" be
$\infty,$ but I try to remove the dollar signs when I send math stuff by
e-mail for you to simply read, rather than take to some computer that understands
TeX.

The "integral sign" is "\int," so "the integral
from 0 to 1" is written $\int_0^1$ in TeX, or just \int_0^1 for e-mail.

The "summation sign" is "\sum," so "the sum from
n=0 to \infty" is written \sum_{n=0}^{\infty} for e-mail.

Note the braces used when there is more than one character in a subscript.

Greek letters that are not in the Roman alphabet are written using a "backslash," \, followed by the name of the Greek letter, spelled out. Thus, to write an alpha, we type \alpha, to write a capital delta, we type \Delta. If we type $\Alpha$ it just gets ignored, because a capital Alpha looks like an A (or is it really the other way around?!)

To get various arrows we write \uparrow, \downarrow, \leftarrow, \rightarrow (or simply \to); there are many other arrows! In particular, there are the "northeast" and "southeast" arrows, \nearrow, \searrow, and so on.

This may not show up much on e-mail, but it is in the Syllabus. The form is:

{numerator \over denominator}.

For example, to write the conclusion of l'Hospital's Rule, namely that the
ratio

"f(x) over g(x) converges to L," we write

"{f(x) \over g(x)} \to L."

Note the spaces after \over and \to. They have to be there to let TeX know
the "control sequence" (something that starts with \) is done...

(We can get by with writing "{f(x)\over g(x)} \to L," but not
"{f(x)\overg(x)} \to L.")

I'll try to put spaces before as well as after, but I might miss some...

First, the (TeX code for a) symbol, then a % then the explanation. The
% is the "comment-starting'' character in TeX; everything on the line
after it is ignored by TeX.

Note: TeX codes that I have "made up" are marked with an asterisk
at the end of the explanation. For example, "\ints"

means the set of all integers, denoted by a capital Z in the Blackboard
Bold font. To produce it I put this line in my TeX source file, preceding
the text:

\def\ints{\hbox{\bb Z}}

**The list:**

\ % This is a "backslash;" it starts "control sequences"
in TeX.

\{ % should not appear, but I might have missed some; it just means {.

\} % same deal; means }.

\, % same deal; means "a little space."

\dots % means "ellipsis" or ... and should not appear, but might.

\downarrow % a downward pointing arrow.

\equiv % this is the "identically equal to" symbol; it has
three horizontals

instead
of the two horizontal bars in the equals sign.

\ge % greater than or equal; also >=, but that's not TeX

\in % "belongs to" symbol -- the stylized epsilon...

\inf % stands for "infimum."

\infty % stands for the lazy-eight "infinity" symbol.

\int % means "integral sign," e.g. \int_0^1 means "integral from 0 to 1."

\ints % denotes the integers; it stands for that Z made with two slanted strokes.*

\le % less than or equal; also <=, but that's not TeX

\lim % stands for "limit," e.g. \lim_{x \to 0} means "limit as x approaches 0."

\ne % "is not equal to" symbol -- an equal sign with a slanted line thru it...

\notin % "does not belong to" symbol -- the stylized epsilon with a slanted line thru it...

\over % tells TeX to make a fraction; e.g. {xy \over x^2+y^2} means xy/(x^2+y^2).

\overline % puts a line over the following group; e.g. \overline{f(x)}
yields

what we would read as "f of ex bar," namely the complex
conjugate of f(x).

Or, we might write \overline{E} to denote the closure
of a set E in a metric space.

\pm % ""plus or minus"" symbol.

\rats % stands for that capital Q, with two left upstrokes, that we use to denote the set of rational numbers.*

\r % stands for that capital R, with two upstrokes on the left, that we use to denote the set of real numbers.*

\sqrt % puts a square root sign around or over or whatever the stuff
within the { and } that follow. Thus \sqrt{ 2 } stands for

the positive square root of 2.

\sub % stands for the set-theory symbol for "contained in or equal
to," an abbreviation for \subseteq *

\subseteq % stands for the set-theory symbol for "contained in or equal to."

\supseteq % stands for the set-theory symbol for "contains or equals."

\sum % means "summation sign" e.g. \sum_{i=0}^n means "sum as i runs from 0 thru n."

\sup % stands for "supremum."

\to % means an arrow pointing to the right.

\uparrow % - an upward pointing arrow.

This should about cover what I have used on these pages. If not, please
let me know, and I'll fix it... [* jodeit@math.umn.edu
*]

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