# The irrationality of the square root of 2 [duplicate]

Is there a proof to the irrationality of the square root of 2 besides using the argument that a rational number is expressed to be p/q?

## marked as duplicate by egreg, Dan Rust, Nick Peterson, Daniel Fischer, user1620696Nov 6 '13 at 1:09

Regardless, yes, there are lots of ways to prove the irrationality of $\sqrt{2}$. Here are some pertinent resources:
Here is an overkill proof: $x^2-2$ is irreducible over $\mathbb{Q}$ by Eisenstein's =].
If $\sqrt 2=p/q$ where $gcd(p,q)=1$ then $2=p^2/q^2$ <=>$p^2=2q^2$ and thus $p^2$ is even and thus $p$ is even let's say $p=2m$. Then $4m^2=2q^2$<=> $q^2=2m^2$ and thus $q$ is also even. So $gcd(p,q)>1$ which is false