Mental Arithmetic: Squaring Near 50

What is 47²? The answer is 2209, and if you didn’t know that almost immediately, by the end of this post and with a modicum of practice I think you’ll be able to perform that calculation and others like it extremely quickly.

We’re interested in squaring whole numbers near 50. The strategy logically works for any number, but in practice is only useful from between 40 and 60 (each of which are fairly trivial to square). The method is:

  1. Take the difference from 50, multiply it by a hundred (should give either a positive or a negative number.
  2. Add/subtract that value to 2500.
  3. Add the difference squared to that.

So in the case of 47, the difference is -3, so we multiply that by a hundred to get -300, add that to 2500 to get 2200, and then add 3² = 9 to get 2209. If instead we had 51², we would go 2500 + 100 + 1 = 2601.

The proof for this method is merely a binomial expansion:

\displaystyle (50 \pm x)^2 = 2500 \pm 100x + x^2

Try a few out. If the \displaystyle x is greater than 10 or less than -10, the method still works but you start having to square and add numbers greater than 100 which makes it more difficult to do the calculations rapidly in your head.

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2 Responses to Mental Arithmetic: Squaring Near 50

  1. Pilly says:

    I think it’s easier to just think of (50-3)(50-3) and using FOIL.

  2. Lee Pavelich says:

    That is in effect what was done. Except I used FOIL on the general case (50-x)(50+x) which comes out to a nice round equation, and if you memorize *that* you can perform the calculations quickly without having to do an explicit FOIL (the FOIL is already baked in to the algorithm).

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