The irrationality of the square root of 2

Question

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?

Answer 1

This question has been asked before. Please search Math.SE for an answer to your question before asking.

Regardless, yes, there are lots of ways to prove the irrationality of $\sqrt{2}$. Here are some pertinent resources:

• https://mathoverflow.net/questions/32011/direct-proof-of-irrationality/32017#32017 (That's a really neat proof of the type for which you are looking.)
• http://en.wikipedia.org/wiki/Square_root_of_2#Proofs_of_irrationality
• Direct proof for the irrationality of $\sqrt 2$.
• Irrationality proofs not by contradiction

Answer 2

Here is an overkill proof: $x^2-2$ is irreducible over $\mathbb{Q}$ by Eisenstein's =].

Answer 3

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