Reading my email math -- written in a mixture of e-mail typing
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,

The TeXbook, by Donald E. Knuth, Addison Wesley, 1989.

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.

Subscripts and superscripts

"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."

Dollar signs are math "delimiters" in TeX

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.

Special math symbols: sum, integral, etc.

"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

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?!)

More Special math symbols: arrows

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...


A list of the special TeX-like 'symbols' you may find here

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... [ ]

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