A trick for squaring numbers near 50

Did you know, there is a neat trick for working out the square of numbers near 50?

Lets say, you want to find 53²

Start with 2500 (the square of 50), add on 100 times the difference between your number and 50 (3×100 = 300 in this example), and finally add on the square of the difference between your number and 50 (3² = 9 in this example).

So the answer to our example is 2809. Can you see why this works??

Hint: let the difference between your number and 50 be x

Hans Bethe taught this trick to Richard Feynman as a quick way to calculate 48² whilst they were working together at Los Alamos . Read about it in Feynman’s book Surely You’re Joking Mr. Feynman!:

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.

 

 

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