The excellent Steven Strogatz, writing in last month’s New York Times Opinionator, gave another elegant lecture on high-school maths and algebra problems – including a fancy trick for squaring larger numbers.
The God of Whizz is behind on his reading, so I’ve come to this post late. But the trick was enough to impress the late, eminent, Richard Feynman, and will be enough to impress any internationally famous physicists that you know:
“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 2300.’ I begin to push the buttons, and he says, ‘If you want it exactly, it’s 2304.’
The machine says 2304. ‘Gee! That’s pretty remarkable!’ I say.
‘Don’t you know how to square numbers near 50?’ he says. ‘You square 50 — that’s 2500 — and subtract 100 times the difference of your number from 50 (in this case it’s 2), so you have 2300. If you want the correction, square the difference and add it on. That makes 2304.’ â€
[From “Surely You’re Joking, Mr. Feynman!â€, by R.P. Feynman]
Strogatz goes on to explain to us mortals how it works:
Bethe’s trick is based on the identity
(50 + x)2 = 2500 + 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.
For an intuitive proof of this formula, imagine a square patch of carpet that measures 50 + x on each side.
Then its area is (50 + x) squared, which is what we’re looking for. But the diagram [below] shows that this area is made of a 50 by 50 square (this contributes the 2500 to the formula), two rectangles of dimensions 50 by x (each contributes an area of 50x, for a combined total of 100x), and finally the little x by x square gives an area of x squared, the final term in Bethe’s formula.
Geeky anecdote alert: As well as being a rather top scientist, Hans Bethe was also the missing link in a famous physics paper about the origin of elements in the early universe.
When Ralph Alpher and George Gamow co-authored a paper on the subject they decided, a little cheekily, to add the eminent Bethe to the list of authors, making it the Alpher, Bethe, Gamow paper about the very conditions after the Big Bang.
The joke is somewhat lost if you’ve not been up on your Greek alphabet (for shame!), which makes it sound as though the paper was written by the first three letters – alpha, beta, gamma.
Given that God tends to describe himself as the alpha and the omega, this would seem a pretty impressive lineup on a modest science paper…
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