## Sunday, June 11, 2006

### HOWTO: Squares of Numbers Ending in a Five in your Head

After I mentioned how to square a number near fifty (50) in your head, a friend of mine taught me how to square numbers ending in five (5) in your head.

The numbers meant by "ending in five" are 15, 25, 35, 45, etc. (5 itself is a degenerate case that need not be included) These numbers can be represented as x = 10*a + 5, where a=1, 2, 3, 4, etc for the series listed above. To get the square, x^2 = (10*a + 5)^2 = 100 * a * (a+1) + 25. All you have to remember is that the square of one of these numbers has 25 as the lower two digits, followed by the original tens digit multiplied by the original tens digit plus one.

For example, 75^2 = [(7 * 8)][25] = [56][25] = 5625. (where [] represents literal digits)

In practice, your only limit for how high you can go is how big a value of 'a' you can multiply in your head with 'a+1'. (or alternately, a^2 + a, as below)

This can also be combined with other math tricks, like the aforementioned squares near fifty in your head to do much bigger number crunching in your noggin.

For example, 535^2 = [(53 * 54)][25] = [(53*53 + 53)][25] = [((2500 + 300 + 9) + 53)][25] = [(2809 + 53)][25] = [2862][25] = 286225

Whee! I'm learning to square three digit numbers in my head!

Burton MacKenZie www.burtonmackenzie.com

Note: I changed an example above (for clarity) from 45^2 to 75^2.