Proof that the square root of any non-square number is irrational. First let's look at the proof that the square root of 2 is irrational. First, let's suppose that the square root of two is rational. Therefore, it can be expressed as a fraction: . Then let's suppose that is in lowest terms, meaning are relative primes, meaning their greatest common factor is 1. So far, . Let's square both sides. . Then multiply both sides by , meaning, and you can test this out, has to be divisible by two, or in other words, it is even.