If anybody was wondering how to prove this method of multiplying two integers... here it is:
Pretend the numbers you want to multiply are x and y. To find there distance from 100, all you have to do is subtract the number from 100. So the table he wrote would look like this:
x 100 - x
y 100 - y
The difference between x and the difference of 100 and y, and the difference between y and the difference between 100 and x are the same. In other words,
x - (100 - y) = x + y - 100
y - (100 - x) = x + y - 100
Secondly, he multiplied the numbers on the right by each other: (100 - x)(100 - y). Because this product has to be less than 99 for the method to work correctly, the method is limited to numbers close to 100.
By writing (x + y - 100) in the thousands and hundreds place, the expression is in effect being multiplied by 100. The sum of this and (100 - x)(100 - y) is xy as shown below:
100(x + y - 100) + (100 - x)(100 - y) =
(100x + 100y - 10000) + (-100x - 100y + 10000) =
By substituting 100 for c throughout the method, one could show numbers close to a 10, 1000, 10,000, etc. could be multiplied together with a slightly altered form of the method.
c(x + y - c) + (c - x)(c - y) =
(cx + cy - cc) + (-cx -cy + cc + xy) =
Well, there you have it, this method proved in an extremely geeky fashion! :)
Hilarious commentary on an "average" grad-student's day.