The Joy of X: A Guided Tour of Mathematics, from One to Infinity

by Steven H. Strogatz

instead of adding just two odd numbers together, suppose we add all the consecutive odd numbers, starting from 1: 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 = 16 1 + 3 + 5 + 7 + 9 = 25. The sums above, remarkably, always turn out to be perfect squares.

instead of adding just two odd numbers together, suppose we add all the consecutive odd numbers, starting from 1: 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 = 16 1 + 3 + 5 + 7 + 9 = 25. The sums above, remarkably, always turn out to be perfect squares.

In a balanced triangle, the sign of the product of any two sides, positive or negative, always agrees with the sign of the third.

Back in Ms. Stanton’s class, what stopped us from looking at decimals that neither terminate nor repeat periodically? It’s easy to cook up such stomach-churners. Here’s an example: 0.12122122212222 . . . By design, the blocks of 2 get progressively longer as we move to the right. There’s no way to express this decimal as a fraction. Fractions always yield decimals that terminate or eventually repeat periodically — that can be proven — and since this decimal does neither, it can’t be equal to the ratio of any whole numbers. It’s irrational.

Here’s one that impressed the physicist Richard Feynman, no slouch himself at mental maths: When I was at Los Alamos I found out that Hans Bethe was absolutely topnotch at calculating. For example, one time we were putting some numbers into a formula, and got to 48 squared. I reach for the Marchant calculator, and he says, “That’s 2,300.” I begin to push the buttons, and he says, “If you want it exactly, it’s 2,304.” The machine says 2,304. “Gee! That’s pretty remarkable!” I say. “Don’t you know how to square numbers near 50?” he says. “You square 50 — that’s 2,500 — and subtract 100 times the difference of your number from 50 (in this case it’s 2), so you have 2,300. If you want the correction, square the difference and add it on. That makes 2,304.” Bethe’s trick is based on the identity (50 + x)2 = 2,500 + 100x + x2. He had memorized that equation and was applying it for the case where x is –2, corresponding to the number 48 = 50 – 2.

the fundamental theorem of algebra says that the roots of any polynomial are always complex numbers.

If the cold-water faucet can fill the tub in a half-hour, and the hot-water faucet can fill it in an hour, how long will it take to fill the tub when they’re running together?

Suppose three men can paint three fences in three hours. How long would it take one man to paint one fence?

Jefferson revered Euclid. A few years after he finished his second term as president and stepped out of public life, he wrote to his old friend John Adams on January 12, 1812, about the pleasures of leaving politics behind: “I have given up newspapers in exchange for Tacitus and Thucydides, for Newton and Euclid; and I find myself much the happier.”

The eerie point is that light behaves as if it were considering all possible paths and then taking the best one.

you may be wondering if geodesics have anything to do with reality. Of course they do. Einstein showed that light beams follow geodesics as they sail through the universe. The famous bending of starlight around the sun, detected in the eclipse observations of 1919, confirmed that light travels on geodesics through curved space-time, with the warping being caused by the sun’s gravity.

the theory of infinite sets. Cantor used it to prove that there are exactly as many positive fractions (ratios p/q of positive whole numbers p and q) as there are natural numbers (1, 2, 3, 4, . . .). That’s a much stronger statement than saying both sets are infinite. It says they are infinite to precisely the same extent, in the sense that a one-to-one correspondence can be established between them.