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