Sending mathematical communications electronically
     Dave Rusin   (rusin@math.niu.edu  Last updated May 15 2001)

Mathematical ideas sent via email tend to get rapid responses, and allow
for precise earmarking of passages needing comment. There is however the
problem of communicating the wide array of symbols and diagrams in a
medium which should be assumed to use only the standard keyboard symbols,
arranged in 80 (or fewer) fixed columns, with at most 25 lines viewable at
once. Here are some suggestions you can try if you need to communicate
mathematics with me. I will use these conventions when I write to
you. Most mathematicians will use techniques like these in email, on
web pages, in newsgroups and mailing lists, and in electronic journals.

For the most part, you can just type away at your message, as you see
me doing now. This has the fortunate consequence of encouraging you to
communicate in words, sentences, and paragraphs, rather than symbols. 
I urge you to take this as a blessing. But you may ask, how shall we 
handle special symbols? There are several choices commonly used:


(1) Enhanced character sets: The lowest common denominator of
electronic data exchange uses the ASCII character set: 128 characters
sufficient for typing English. There are larger character sets which
might include accented characters, other alphabets (e.g the Greek
letter mu ), or other symbols (e.g. the "plus-or-minus" symbol). These
include some fairly well-established character sets determined by commerce
(read: Microsoft diktat) or standards bodies such as ISO and Unicode.
You may find your wordprocessor or mail software offers these options.

BE RELUCTANT TO USE THESE. Different hardware and software may treat
these differently. You may think that "everybody" sees things the way you
do on your screen but I can assure you that you would be amazed to see how
many bad ways your document will appear!

The printable characters in the ASCII list are just these 94 (in ASCII order):
        !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNO
        PQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~
together with the blank space. There are two 'characters' which, alone or
together, are usually interpreted by software to mean 'carriage return';
I would encourage you to insert these (sometimes called a 'hard return')
after no more than 80 characters in a row (fewer is better) because not 
all software is smart enough to insert line breaks in a reasonable
way.  There is another character interpreted as 'tab' but what exactly
that means depends on the software which encounters it. (Likewise
there are a few other such 'control characters' included in ASCII but
their use by software is even more varied so I won't itemize these.)

Note: The 'HTML standard' assumes browsers can successfully render a
larger set of 256 characters, making most web pages capable of showing
many western European languages correctly.

In the following I will assume that you will make an electronic file
using only ASCII characters, but wish to include some other letters,
symbols, or more complex math using some other methods.


(2) "verbal method" You can always spell out what you need to say. For
example, you can write

        Let  xi  be a number which is our first approximation to a
        solution of the problem  ax+b  approx  0. Then 
        integral from  0  to  xi  of  2x  dx  is greater than  beta.

This is the method I use when I have just a few symbols I need to use. Note
that it helps legibility if you put extra spaces around the symbols you're
spelling out, and around math formulae.


(3) "ASCII art" Some people have refined this to really high levels, but
I use it just for a few special symbols which can be readily made with
symbols on the keyboard. Sometimes this takes a couple of lines of text.
WARNING: this technique usually fails if one party or the other is not
using a display with a uniform-width font. (Also remember to avoid use
of tabs -- different machines treat tabs differently.) Here are some examples:
                                         _
        If  f(x) > 0 for all  x  then  _/  f(x) dx > 0.

                           a z + b
        The mapping  z |-> -------  is an automorphism of the Riemann sphere.
                           c z + d

                                                 3    3
        We therefore conclude that  a > b  ==>  a  > b .


(4) "Programming style"  In many circumstances the easiest thing to do is
to type out commands you would use when writing a computer program in
Fortran, C, or whatever, since these require linear typewriter input anyway.
Procedures at a higher level can be expressed as you would when entering
a command to Maple, Mathematica, or some other symbolic-algebra program.
Thus one would write

        Let  Alpha  be a root of  x^3+3*x^2+5=0. Then we find that
        Integral( f(x), x, 0, Alpha) happens to equal Sqrt(Sin(x)).

There are a lot of variations here of course -- some programmers write
x**3 for  x^3; Mathematica users can type 3x^2 instead of  3*x^2.
Since you are addressing a human, not a computer, consistency is less
important.

Note however that an expression with lots of parentheses and so on
will be very hard for a human to read even if it is correct syntax for
a program. I suppose I would use ASCII art to display such a mess.


(5) "TeX input" Far and away the most common format for preserving
important mathematical expression these days is as an input file for
Knuth's TeX typesetting program. It is an invaluable skill for serious
math students to learn the TeX commands anyway, so you might want to use
this in your informal communications as well. TeX input is for the most
part straight text, but anything which is not ordinary Latin-alphabet
text is rendered with specific mark-ups which the TeX program will
accurately display in a final copy. You would for example say

        Let  $f$  be a mapping  $A  \times   B \longrightarrow C$ 
        which has the property that  $x \ge y$ and $z \ge w$
        imply $f(x,z) \ge f(y,w)$. Then $\integral_0^\alpha f dx=0$.

If there is ever any confusion about just what is meant, one can
always use some precise TeX syntax to specify exactly what should appear
on a printed page.

Sending TeX input has the advantage that the other party, should
they need to do so, can actually run TeX on your mail to them and display
the final product on their display or on paper. This results in the
prettiest and least-ambiguous transmission. It's also the most work for
the reader. I would tend to shy away from this approach unless I needed
to express something really complex.

Many people use a pseudo-TeX approach for mildly difficult passages:
keep TeX syntax but without the  $'s and  \'s  (and sometimes the {'s and }'s)
thus you can easily speak of  alpha_3^2  without getting all worked up
about TeX syntax. In particular, note that superscripts are marked with "^"
and subscripts with "_", both in TeX and in informal math communication.

TeX formatters and viewers are available, for free, for very many platforms.
Visit the Tex Users Group (tug.org) or the CTAN archives. TeX is 
extremely stable: I personally wrote documents in 1983 which look better
than anything MS Word can make now, and the .tex file I would write today
would be identical to the one I wrote then. Can any other word-processing
package boast more than two decades' longevity?


(6) "Wordprocessor input" One virtue of TeX input is that it uses only
ASCII characters and so is displayable. Except for fancy formatting 
instructions, it is also human-readable. 

Other word-processing programs may or may not share these
features. If you create a message using WordPerfect or Micro$oft Word,
for example, you should look for an option like "Save as Flat Text".
This should create a document which, while it may lose some formatting
information, is human-readable. Old WordStar documents mark formatting
with very minor control codes as TEX does, and so could be exchanged
just as readily. 

One the other hand I would DISCOURAGE the use of 'attachments', which allow 
you to send documents with email. You may not know this but the simple act
of attaching an HTML file or DOC file to your email creates an extremely
long ASCII file which encodes it, sometimes in multiple formats, swamping
the recipient's mailbox and requiring sometimes complicated decoding,
possibly to no good end if the other person does not use the same
word-processing package. 

With a little thought you can express yourself without all the
word-processor options. Do you really need to use italics if you want
to emphasize something? You could _underline_ it or *emphasize* it or 
SHOUT it or  s-p-e-l-l  it out. Be creative, but focus on your thoughts, 
not on the presentation.

Besides, word-processors make smilies look funny :-)


(7) "Graphics format" If you already know just how the message is
supposed to look -- especially if what you want to send is a real
picture (nicer than some ASCII-art imitation) -- you can send a binary file
in some accepted format. What consitutes 'accepted' varies with the
recipient. I'm pretty flexible; not everyone is. Among your options:

    A .pdf file (which can be read by free Adobe Acrobat readers on
     many computer platforms); this can include text and/or images
    A .dvi file (this is the output of the TeX program, and since most
     mathematicians use TeX all the time, they have dvi-viewers handy).
    A .ps (postscript) file: postscript is a language developed for
     transmission of files to a wide variety of graphics printers. 
     Nowadays most people can view postscript files on their screens.
Personal-computer owners tend to store graphic images in other
formats, such as .gif or .jpg files. These were intended for transmitting
photographs, but if you have some math picture you want to send, you can
write it on paper, scan it in a scanner, and then have your scanning PC
store the image in one of these formats (or .bmp or ...). I have seen
this approach used by people who want an illustration to, say, a complicated
Euclidean construction such as one studies in secondary-school geometry.

I believe most people who surf the world-wide-web use browsers which
will automatically display images presented in any of these images.

Note that internet standards for transmitting email do not allow binary
files to be sent directly. Many mailers allow the attachment of binary
files to email; if the mail software uses what's known as MIME encoding,
the resulting ASCII file can be unpacked on the recipient's end by most
modern mail-readers. (I may be the world's last hold-out, using old software
which forces me actually to look at the encoded file.)

There are other ways to exchange the binary file, by the way.
An old standard (UUENCODE / UUDECODE) can encode binary files with ASCII
characters. Or, the two parties can agree on an FTP transfer: for example, 
the sender can place the file at some publicly readable FTP site and 
simply send the address (URL) to the receiver. 

As you can see, exchanging binaries requires more work, so it should
be limited to occasions on which nothing else will do. (And that says nothing
about the work needed to create the binary file in the first place!)

Actually, current technology exceeds current use. It is
possible to send a movie or a 3D object or a spoken message or even a
full-length multicolor video multimedia presentation with stereoptic
3D display. All these can be captured electronically with the right
combination of hardware and software, transmitted as a binary file,
and then displayed on the other end with, again, appropriate hardware
and software. However I have to question whether this amount of
presentation is warranted by a simple question-and-answer exchange of
a short amount of mathematics.


SUMMARY

Really you should have no trouble expressing most of your ideas as straight
text. A reader with a fair amount of experience in mathematical communication
should be able to follow along with you if you choose to mix several of
the embellishments mentioned above. I find the simplest methods sufficient
for all but the most formal communications, but I welcome any comment on
your own experiences. Feel free to communicate to me in any conceivable way,
and we'll iron out the details as we go.

Just for the record: the human language most commonly expected in a
mathematics discussion is English. Certainly French and German are also
quite commonly used in print. A well-trained mathematician should probably
be able to handle Russian, Italian, and/or a language specific to a
subdiscipline; however, your chances of being understood decrease greatly
if you use something besides the three most common languages.

dave