## Saturday, March 29, 2008

### More Math in the Head: Simplifying the Square of a number

The other night I was out in my garage and needed to know what 75*75 is, and couldn't remember. I could always work it out by multiplying columns by hand quickly (i. e. long multiplication) but I find that when I try and do it in my head I can't remember all the numbers I need, simultaneously. To multiply larger numbers in my head I've had to learn many different methods.

One of my favourite techniques for doing multiplications in my head is using the properties of a square polynomial expansion. For instance,

( x + a )2 = x2 + 2ax + a2

Thinking of 75*75 as an unknown square with some offset away from a square I do know, I can do an easier and quicker calculation than long multiplication. For instance, if I say in the equation above that x = 80 and a = -5, I can substitute (x + a) for 75:

( 75 * 75 ) = ( 75 )2 = (80 -5)2 = 802 + 2(-5)(80) + 52 = 6400 - 800 + 25 = 5625

For me, this method requires less short term memory space so I'm able to do it in my puny brain.

What if I don't know a perfect square really close? In that case, I just jump farther and farther to one I do know.

( 75 * 75 ) = ( 75 )2 = (100 -25)2 = 1002 + 2(-25)(100) + 252 = 10000 - 5000 + 625 = 5625

Very often you can find a nearby "nice" (easy-to-square) number allowing you to simplify the calculation in your head.

It works easily for rational reals as well! For instance, if I needed to square 0.75, just convert it to 75, do the same algorithm, and remember where to put the decimal place in the answer. ( 0.75 * 0.75 = 0.5625 )

In previous posts I have discussed topics largely overlapping with this here, here, and here. I also have some more unrelated math-in-your-head tricks here, here, and here.

If you go looking, there's lots of interesting alternate multiplication algorithms online here, here, and here, and a bunch of other places. I have no idea if they'd fit in your head. They're not in mine yet.

Good luck!/Bonne Chance! Happy Computing!

Burton MacKenZie www.burtonmackenzie.com