Direct proof for the irrationality of $\sqrt 2$.

Question

Prove that $\sqrt 2$ is irrational using direct proof.

I have seen TONS indirect proofs (e.g. proof by contradiction) for it, and people say that it's difficult to proof this directly. So is this impossible?

Thank you.

Answer 1

What do you exactly mean by a "direct proof"?

The most direct argument I can think of for showing that $\sqrt{2}$ is irrational uses continued fractions. $\sqrt{2}$ has an infinite continued fraction (namely: $[1,2,2,2,...,]$) and can as such not be rational.

Answer 2

1) wikipedia has given a constructive proof, see http://en.wikipedia.org/wiki/Square_root_of_2

2) all rational numbers have a finite continued fraction expression, but $\sqrt2$ doesn't

Answer 3

It has an infinite continued fraction.

$$ \!\ \sqrt{2} = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}. $$