Squaring Numbers Near Fifty

And now, a brief break from non-blogging:

Today, I’d like to start with a specific example and move on to a general point. The specific example is a way to approximate the squares of numbers and then refine those approximations to get exact answers, and the general point concerns the place such techniques should have in mathematics education.

My last calculator broke years ago, so when I have to do a spot of ciphering, I have to work the answer out in my head or push a pencil. (If the calculation involves more numbers than can fit on the back of an envelope, then it’s probably a data-analysis job which is being done on a computer anyway.) Every once in a while, the numbers teach you a lesson, in their own sneaky way.

It’s easy to square a smallish multiple of 10. We all learned our times tables, so squaring a number from 1 to 9 is a doddle, and the two factors of 10 just shift the decimal point over twice. Thus, 502 is 2500, no thinking needed.

Now, what if we want to square an integer which is near 50? We have a trick for this, a stunt which first yields an answer “close enough for government work,” and upon refinement gives the exact value. (I use the “close enough for government” line advisedly, as this was a trick Richard Feynman learned from Hans Bethe while they were calculating the explosive power of the first atomic bomb, at Los Alamos.) To get your first approximation, find the difference between your number and 50, and add that many hundreds to 2500. The correction, if you need it, is to add the difference squared. Thus, 482 is roughly 2300 and exactly 2304, while 532 is roughly 2800 and exactly 2809.

I wouldn’t advise teaching this as “the way to multiply,” first because its applicability is limited and second because it’s, well, arcane. What a goofy sequence of steps! Surely, if we’re drilling our children on an algorithm, it should be one which works on any numbers you give it. The situation changes, though, after you’ve seen a little algebra, and you realize where this trick comes from. It’s just squaring a binomial:
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