Numerical proof that √2 is irrational

The following beautiful proof that √2 is irrational is due to Robert Gauntt, a first year uni student at the time (American Mathematical Monthly 63 (1956), p247). It uses the idea of writing whole numbers in base three instead of our usual base ten. As an illustration, the number fifteen is normally written as 15, amounting to 1 x 10 + 5 x 1. In base three, this same number would be written as 120, amounting to 1 x 9 + 2 x 3 + 0 x 1.

The important thing to note is: written in base three, the first non-zero digit of a square number is always a 1. For example, the square of fifteen is two hundred and twenty-five, which we would normally write as 15² = 225. However, in base three, this is written as 120² = 22100, with 22100 coming from 2 x 81 + 2 x 27 + 1 x 9 + 0 x 3 + 0 x 1. In this example, the 1 in the "nines place" is the first non-zero digit.

This fact about squares written in base three follows easily from the base three products 1 x 1 = 1 and 2 x 2 = 11, combined with the normal rules for multiplication.

With that background, Gauntt's proof is now really easy. The statement that √2 is irrational (that is, it's not a fraction) is the claim that there are no whole numbers M and N with 2 = (N/M)². Multiplying through by M ², this means we want to show that the equation N² = 2M² is impossible. But suppose now that M and N are written in base three. Then M² and N² both end in a 1. That means 2M² ends in a 2, and so can't possibly equal N².

Here is a PDF containing two proofs that √2 is irrational: the proof above and a second, geometric proof. You can also find many more proofs here.