The proof of the irrationality of root 2 is often attributed to Hippasus of Metapontum, a member of the Pythagorean cult. He is said to have been murdered for his discovery (though historical evidence is rather murky) as the Pythagoreans didn't like the idea of irrational numbers.
Here’s one of the most elegant proofs in the history of maths. It shows that is an irrational number, in other words, that it cannot be written as a fraction where and are whole numbers.
We start by assuming that can be written as a fraction and that and have no common factor — if they did, we could simply cancel it out. In symbols,
Squaring both sides gives
and multiplying by gives
This means that is an even number: it’s a multiple of . Now if is an even number, then so is (you can check for yourself that the square of an odd number is odd). This means that can be written as for some other whole number . Therefore,
Dividing through by gives
This means that is even, which again means that is even. But then, both and are even, which contradicts the assumption that they contain no common factor: if they are both even, then they have a common factor of . This contradiction implies that our original assumption, that can be written as a fraction must be false. Therefore, is irrational.