An Elegant New(ish) Proof

Tom Apostol’s square root of 2 proof is beautiful

This proof was published in 2000 by Tom Apostol. It just might become your new favourite proof that the square root of 2 is irrational. Here’s how it goes.

Figure 1

You have a right isosceles triangle, ABC (Fig. 1). Suppose sides AC and BC have integer length. Their ratio will be the square root of 2. Suppose further…