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Wednesday, March 7, 2012

Mental Math - Multiplying two digit numbers in your head

I learned a cool trick recently that I wanted to share with you. I found a really easy way to multiply any set of two digit numbers in your head.

I have always been good at math, but traditionally rather poor at mental calculation. I'm trying to change that, since I found that my peers who were good at mental calculation had a lot less trouble with complex mathematical concepts. While it won't make you a math savant by itself, with practice it will give you a serious edge over your peers in math class.
Given a set of two digit numbers, say

ab * cd

Where a is the tens digit of the first number, b is the ones digit of the first number, and c and d are the corresponding places for the second number.

Because of the natural of numbers, this can be written another way. The first number is really 10 * a + b. The second number can be expanded in a similar way. So we have the following

(10a + b) * (10c + d)

Using the foil method this reduces to

100ac + 10ad + 10bc + bd
100ac + 10(ad + bc) + bd

We should rearrange this so that the middle term comes first, taking advantage that our memory will tend to work better if we do this calculation first. Our mind for numbers typically will just hold around 7 things.

10(ad + bc) + 100ac + bd

This final equation can now be used to quickly calculate the multiplication as a series of much easier multiplications and a few sums. Lets do a quick example.

76 * 34
10(7*4 + 6*3) + 100(7*3) + 6*4

First lets figure out the first part

10(28 + 18)

Now lets figure out the rest

460 + 100(21) + 24
460 + 2100 + 24
484 + 2100

If you use your calculator you will see that you now have the correct answer. With practice this technique becomes easier and easier to do. Because of the way we broke up the numbers, the sums will be pretty easy generally. There is also a technique for doing sums left to right in your head that I will share in another future post that makes them even easier to do in your head.
Now get to practicing, with persistence you might eventually be able to do this faster then someone can put the numbers in a calculator.

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