Sunday, August 19, 2007

An Introduction to Abstract Math

My son Nicholas (soon to be a 7th grader) and I have been reading together Mathematics: A Very Short Introduction by Timothy Gowers, winner of the Fields prize. The book is not a primer but a wonderful, basic introduction to the idea of mathematics as an exercise in abstract, rigorous reasoning.

Gowers uses a series of intriguing yet simple examples to explain how mathematicians think about what they do. I love the questions he addresses: For example, how can one prove, from first principles, that 0 x 0 = 0, or that 2 to the power of 1/2 equals the square root of two? (If questions like these strike you as nonsensical, that just goes to show, Gowers might say, that you have not thought about the issues in a sufficiently fundamental way.) Or how can one prove that the square root of two is an irrational number?

Here is a fun problem from the book: Consider an NxN square grid made up of 1x1 unit squares. Now take away from this grid two corner squares from diagonally opposite corners. The resulting board, which is now a square with two notches, has NxN-2 unit squares. You are then given dominoes with dimensions of 2x1. For what N can you completely cover the board with these dominoes?

Think about the problem on your own. I will post the answer (or at least part of it) as the first comment.