Squaring two digit numbers in your head

Squaring two digit numbers in your head

[W]hat's 27². Of course, the brute force way to do that is to calculate 27 x 27 which is a bit of a pain because it involves doing something like 27 x 20 + 27 x 7 = 540 + 189 = 729. But there's a much faster way.

Observe that 27² = 30 x 24 + 3². Since you probably know that 3² = 9 this means you have to calculate 30 x 24 + 9 which is relatively easy because the multiplication involves a multiple of ten which means it's really 3 x 24 and then add a zero.

So the rule is that if you want to square number X you first round it to the nearest multiple of 10, called that X + r, and then calculate X - r (i.e. round the same amount in the opposite direction). You calculate (X + r) x (X - r) and add back the square of the amount you rounded by, r², which will be 1, 4, 9, 16 or 25.

This works because ( X + r ) x ( X - r ) + r² = X² - rX + rX - r² + r² = X².

Squaring two digit numbers in your head