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stoutfellow) wrote2006-12-12 07:58 pm
stoutfellow) wrote2006-12-12 07:58 pm
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Ramble, Part 8: What's Gray and Has Integer Solutions?
Diophantus of Alexandria was the last important mathematician of the classical era. At least, he may have been; there's enough uncertainty about when he lived to make that a question. The best guess I know of puts him somewhere in the second century AD. His primary interest was number theory; specifically, he was interested in rational-number solutions of equations in two or more variables. He is also notable for taking the first tentative steps towards modern algebraic notation. More under the cut.
Diophantus' most important work was Arithmetica, a compendium of numerical problems. Here's a typical example, together with his method of solution. (My interpolations are in brackets.)
Modern mathematicians use the phrase "Diophantine equations" to describe problems of this type, with the slightly stronger requirement that integer - whole-number - solutions are sought. It turns out that, given an equation or set of equations of the type Diophantus considered, it's possible to set up a (possibly different) set of equations with the property that rational solutions of the first set correspond to integer solutions of the second. I'll discuss this a bit more when we get to Fermat.
Before I can described Diophantus' innovations in notation, I have to say a few words about Greek numerals. The most common way of representing numbers in Greek, at this time, was as follows. The Greek alphabet had 24 letters; three archaic letters, which had been discarded from everyday use, were added, for a total of 27. The first nine of these were used to represent the numbers 1 through 9. The next nine represented 10 through 90; the last nine, 100 through 900. By way of comparison, think of adding the character * at the end of the English alphabet (to bring the total to 27). Then the number 243 would be represented by the string of letters "tmc" - "t" for 200, "m" for 40, "c" for 3. The largest number representable by this system would be 999, or "*ri". (The inflexibility of this system prompted Archimedes, in The Sand-Reckoner, to devise a method for representing very large numbers, coming very close to inventing logarithms.)
Prior to the time of Diophantus, numerical problems of this type were described purely in words, what historians of mathematics call "rhetorical algebra". Diophantus took the step forward to "syncopated algebra", in which certain commonly occurring words are replaced by abbreviations. In particular, he introduced a symbol for an unknown - like, but not the same in effect, as modern algebra's "x". He couldn't use a single Greek letter, since all of them already had specific numerical values; instead, he devised a symbol, resembling our lower-case "s", apparently derived from the first two letters of the Greek word "arithmos". To refer to the square of the unknown, he used the first two letters of "dynamis" ("power"); for the cube, the first letters of "kybos" ("cube"); and, for higher powers, up to about the sixth, various combinations of these. Addition was represented by concatenation, as if we were to write "2cb5sq3vr2" instead of 2x3+5x2+3x+2. For subtraction, he used a symbol derived from "leipsis" ("lacking"); all terms with negative coefficients were grouped together after this symbol.
Why is this significant? Because well-chosen notation, by downplaying some factors and emphasizing others, can make reasoning more or less efficient. I'll say more about this later, as well, but for the moment let me just point to the savings in sheer verbiage involved in the shift from rhetorical to syncopated algebra - "2cb5sq3vr2" versus "two, and three times a number, and five times its square, and twice its cube".
After Diophantus number theory degenerated; by the time of St. Augustine, it was the province of numerologists and fortune-tellers. (When Augustine inveighs against "mathematicians", these are the people he's talking about.) The Muslim and Hindu civilizations were to make some advances, but primarily in the form of geometric algebra; algebra, pursued autonomously, was not to revive until the time of Fermat.
Oh, if you're wondering about the title, the answer is "an Elephantine equation". Math-geek humor, sorry.
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Diophantus' most important work was Arithmetica, a compendium of numerical problems. Here's a typical example, together with his method of solution. (My interpolations are in brackets.)
To find two numbers so that each, together with the square of the other, is a square. [In modern notation, find two rational numbers a and b so that a2+b and b2+a are squares.] Let one number be x and the other 2x+1. [This is rigged to make one of the two sums a square: x2+2x+1=(x+1)2.] Then (2x+1)2+x=4x2+5x+1. [Now Diophantus looks for another square beginning with 4x2.] Set this equal to (2x-2)2, or 4x2-8x+4; we find that x=3/13. Our two numbers are 3/13 and 19/13.Diophantus was not systematic in this work; each problem called forth a different trick. Note that he was generally satisfied with finding one solution.
Modern mathematicians use the phrase "Diophantine equations" to describe problems of this type, with the slightly stronger requirement that integer - whole-number - solutions are sought. It turns out that, given an equation or set of equations of the type Diophantus considered, it's possible to set up a (possibly different) set of equations with the property that rational solutions of the first set correspond to integer solutions of the second. I'll discuss this a bit more when we get to Fermat.
Before I can described Diophantus' innovations in notation, I have to say a few words about Greek numerals. The most common way of representing numbers in Greek, at this time, was as follows. The Greek alphabet had 24 letters; three archaic letters, which had been discarded from everyday use, were added, for a total of 27. The first nine of these were used to represent the numbers 1 through 9. The next nine represented 10 through 90; the last nine, 100 through 900. By way of comparison, think of adding the character * at the end of the English alphabet (to bring the total to 27). Then the number 243 would be represented by the string of letters "tmc" - "t" for 200, "m" for 40, "c" for 3. The largest number representable by this system would be 999, or "*ri". (The inflexibility of this system prompted Archimedes, in The Sand-Reckoner, to devise a method for representing very large numbers, coming very close to inventing logarithms.)
Prior to the time of Diophantus, numerical problems of this type were described purely in words, what historians of mathematics call "rhetorical algebra". Diophantus took the step forward to "syncopated algebra", in which certain commonly occurring words are replaced by abbreviations. In particular, he introduced a symbol for an unknown - like, but not the same in effect, as modern algebra's "x". He couldn't use a single Greek letter, since all of them already had specific numerical values; instead, he devised a symbol, resembling our lower-case "s", apparently derived from the first two letters of the Greek word "arithmos". To refer to the square of the unknown, he used the first two letters of "dynamis" ("power"); for the cube, the first letters of "kybos" ("cube"); and, for higher powers, up to about the sixth, various combinations of these. Addition was represented by concatenation, as if we were to write "2cb5sq3vr2" instead of 2x3+5x2+3x+2. For subtraction, he used a symbol derived from "leipsis" ("lacking"); all terms with negative coefficients were grouped together after this symbol.
Why is this significant? Because well-chosen notation, by downplaying some factors and emphasizing others, can make reasoning more or less efficient. I'll say more about this later, as well, but for the moment let me just point to the savings in sheer verbiage involved in the shift from rhetorical to syncopated algebra - "2cb5sq3vr2" versus "two, and three times a number, and five times its square, and twice its cube".
After Diophantus number theory degenerated; by the time of St. Augustine, it was the province of numerologists and fortune-tellers. (When Augustine inveighs against "mathematicians", these are the people he's talking about.) The Muslim and Hindu civilizations were to make some advances, but primarily in the form of geometric algebra; algebra, pursued autonomously, was not to revive until the time of Fermat.
Oh, if you're wondering about the title, the answer is "an Elephantine equation". Math-geek humor, sorry.
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