Wednesday 30 July 2008

Math Symbols Are NOT All Created Equal

I have a friend named Dave Refro who writes and edits questions for one of those high stakes tests that is used for admission into certain graduate programs and uses his job as an excuse for his fascination with archiving old math journal articles. Some folks garden, Dave archives. He spends hours pouring through journals and abstracts and fits together articles with a common theme. If you read almost any math discussion on line, you will probably have come across one of Dave's responses to a question with numerous links to how the question was addressed, discussed, and argued over through history.

Fortunatly for me, Dave sometimes finds an article that he thinks might be of interest to me, and when he gets a stack of them, I get a big present in the mail and my wife knows I will be taking my meals in the den for a few days. In a stack of journal articles he sent recently, (THANKS Dave!) there was one particularly interesting article by Florian Cajori from 1923. In the article Cajori points out two interesting things about the equal sign that every one uses; and that is one of the interesting things he points out, is that EVERYONE uses it. Even in 1923, it was one of the most ubiquitous math symbols in the world and today there are still only about four math symbols that you could write and they would not only be understood, but written exactly the same way whether you found yourself in darkest Africa, the Far East, or downtown Los Angeles. It seems like the perfect symbol, and as Robert Recorde said when he created the symbol in 1557 in his "Whetstone of Witte", the use of "a pair of paralleles, or Gemowe(twin, from the same root as Gemini) lines of one length ... bicause noe 2 thynges can be moare equalle." In fact, Recorde's equal sign had much longer lines than is common today, sort of like == but longer .

Indeed, one wonders why it hadn't been thought of years before, and assume that it immediatly became the most common of mathematical notations....ahhh, but not so. The other thing Cajori commented on that I think would surprise young students is that it took over a hundred years for the symbol to become accepted. So what symbol did mathematicians use before the good old == signs? Well, many of them used nothing. The early development of algebra occurred with a very rhetorical approach. When people wanted to write 7x+5 = 26, they would say," the product of seven and some quantity when added to five will equal twenty-six." Ok, they probably said it in Latin, and sometimes they did write numbers in place of the words for numbers, but for equals, they often wrote out the Latin aequales or some variation of it. Frequently they used abbreviations instead of full words and so "p" would stand for plus and "m" for minus...and they would shorten aequales as "aeq" or just "ae". By the time that Recorde had his inspirational stroke, lots of other people had decided THEY had a really good symbol. A pair of vertical lines, ||, was used by Xylander (Wilhelm Holzman) in his translation of Diophantus, Arithmetica only a few years later, and Regiomontous had used a single horizontal line for equality almost a century earlier. Descartes used a which was probably drawn from the "ae" abbreviation for aequalis. Descartes symbol became a popular competitor on the continent, finding favor with Huygens and the Bernoulli's, while many of the he English mathematicians, Wallis, Barrow, and Newton, followed Recorde's lead. Others used the "gemowe" lines of Recorde for other meanings, Descartes used it to mean +/- in his Geometrie, and Johann Caramuel used them where we would use a decimal point, so Pi would be 3==1415 etc.

So what brought the divided world into a common accord? It took a brand new idea, a reveolutionary idea, the calculus. As if by divine providence, the two great minds that created the calculus, almost in unison, tended to publish their versions with a common symbol for equality, Recorde's "gemowe" lines. They disagreed on almost every other symbol they used, but in the last half of the 1600's and the early 1700's the = sign rose to world dominance. In Cajori's words, "The fact that both Newton and Leibniz used Recorde's symbol led to its general adoption."

If I can get my students to understand how long and difficult it is to get mathematicians to accept a symbol, perhaps they will not be too surprised if their College Prof goes into a rant when they use the symbol "ln" for the natural log... and if they accept that the symbol exists (honest, they don't all accept it's use), I can't begin to imagine how they will react if you pronounce it differently than they would. Wait for them to say it first!

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