My student, of course, meant it as a dig at time she thought her high school had wasted, but I saw it only as a sad commentary on herself. High schools waste countless hours of teenagers’ time, I agree. But the time spent teaching algebra is not among the wasted hours. I don’t know if Paul Simon would agree, but algebra is not part of the crap. If my student spends day after day without using it, I can only say that she’s the poorer for it.
I use algebra almost every day of my life. I compare prices. I estimate fuel costs. I’ve enjoyed programming little computer games for about thirty years now, and the code to determine scores or the placement of objects on the screen consists of nothing but algebra. And of course I calculate grades, estimate grades, average grades, figure the minimum grade a panicked student needs on the final to get an A or a C. (Hardly any student worries about getting a B instead of a C.) I use algebra all the time.
But why should I even accept the terms of this silly quip? Why assume algebra is to be used, like a tissue you throw away once you blow your nose with it? To see that an algebraic equation works is to put into action that part of ourselves which was created in the image of God, and thus to see into God’s mind. And the vision of God is happiness.
Why do A? In order to do B. Why do B? In order to do C. But why do C? Aristotle points out that if this search for reasons has no end, I’ll never do A in the first place. There must exist some goal, he reasons, about which we do not ask, Why? And that goal is happiness. I do algebra because it brings me happiness.
I just finished book VIII of Euclid’s Elements yesterday, and I was surprised to see that, like the two or three chapters just before, it leaves rigorous geometry aside for a while longer to continue with number theory and algebra. At least, I can turn it into something that looks more like algebra to me. I wish I knew more about the history of mathematics. I wish I knew for sure whether Euclid could conceive of ratios (comparisons of whole numbers) as fractions (parts of units). And I wish I knew if algebra proper depends on the concept of fractions. But what I do know is that if I turn Euclid’s prose into algebraical symbols and use fractions to represent his ratios, things start to make sense to me.
For instance, one of the main points of book VIII is that between any two square numbers, say A and B, lies exactly one number C (one whole number, not a fraction or irrational number) such that C/A = B/C. And that number is found by multiplying the square roots of the two original numbers. As an example, take 4 (2 squared) and 9 (3 squared). Multiply 2 x 3 to get 6. And, sure enough, 6/4 equals 9/6. I hope I expressed it clearly; it’s clear to me, at least. But here’s part of Euclid’s proof:
Let A, B be square numbers, and let C be the side of A, and D of B. . . . Let C by multiplying D make E. Now since A is a square and C is its side, therefore C by multiplying itself has made A. For the same reason also, D by multiplying itself has made B. Since, then, C by multiplying the numbers C, D has made A, E respectively, therefore, as C is to D, so is A to E.And it goes on. In a geometrical proof with the handy diagram and its labeled points, I can usually follow Euclid’s prose. But in these algebraical sections, I get lost easily. So I translated some of the proofs of book VIII into terminology more familiar to me in order to see their methods and conclusions more clearly. This one could look something like this:
Given c • c = a, d • d = b, find e such that e/a = b/e.Reverse demonstration, or check proof, after e has been found:
1. e/a = b/e (Given)
2. e = (b • a)/e (Multiply both sides by a)
3. e • e = b • a (Multiply both sides by e)
4. e • e = d • d • c • c (3: Substitution with given facts concerning a and b.)
5. e • e = d • c • d • c (Commutative property of multiplication)
6. e • e = (d • c) • (d • c) (Associative property of multiplication)
7. e = d • c (Take the square root of both sides.)
1. e/a = b/e (Given)I know, I know. I paid attention forty years ago as almost none of my classmates did. But now, thanks to eighth-grade algebra class and Euclid, I see something I didn’t see before, and it brings me happiness.
2. a = c • c (Given)
3. b = d • d (Given)
4. e = d • c (Found above)
5. (d • c) / (c • c) = (d • d) / (d • c) (1, substituting 2, 3, and 4)
6. d / c = (d • d) / (d • c) (Reduce left fraction)
7. d / c = d / c (Reduce right fraction)