Quick: What's 19²... without your calculator? (361.) Squaring complex, two-digit numbers in your head isn't that difficult if you know a simple trick, according to the Secrets of Mental Math author Michael Shermer.
Photo by farnea.
Arithmetic doesn't always have to be done from right-to-left — in fact plenty of other people do it the other way around. If you see a problem like 32+18, adding it in your head from left to right lets you see 40 first (30 + 10), and then the 10 (2 + 8). A similar process that can be used for multiplication. But what if you want to square numbers like 33²? It's a little bit more complex, but not so much more so:
[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.
I'm adding this tip to my mental repertoire, right next to my mental trick for quickly calculating a 20 per cent tip quickly when eating out (move the digit over one place to the left, and double it). If you've got any useful maths tips for the rest of us who haven't taken maths classes in years, we'd love to hear it in the comments.
Squaring two digit numbers in your head [John Graham-Cumming]