**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 15^{2} = 225. However, in base three, this is written as 120^{2} = 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 *N*^{2} = 2*M*^{2} is impossible. But suppose now that *M* and *N* are written in base three. Then *M*^{2} and *N*^{2} both end in a 1. That means 2*M*^{2} ends in a 2, and so can't possibly equal *N*^{2}.

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.