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

by Steven H. Strogatz

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.