I think there are multiple ways of doing this, however I like proof by contradiction, I'll outline it as best I can below:
Let us suppose that sqrt(2) is rational. Therefore it can be written as x/y where x and y have no common factors (because if they did, the fraction could be cancelled).
Squaring both sides, this means [tex]2= \frac{x^2}{y^2} [/tex]
Rearranging, we can say [tex]x^2=2y^2[/tex]
Then there are a few logical steps.
Since y is multiplied by 2, x^2 must be even. So the only way this can be the case is if x is even (since squaring an odd number still gives an odd number).
But since it is squared, that also means it is therefore divisible by 4, and so this means that y^2 and hence y must also be even.
This contradicts our assumption that they have no common factors, which means it cannot be rational.
