## Fermat's Method of Infinite Descent

Fermat's 'method of infinite descent' is used to prove the irrationality of square roots.
Suppose for example that
$\sqrt{2}$
is rational so that
$\sqrt{2} = \frac{a}{b}$
and cannot be simplified
Then
$2= \frac{a^2}{b^2} \rightarrow 2b^2 =a^2$

This means that
$a$
is even so we can write
$a=2x$

Hence
$2b^2 =(2x)^2=4x^2$

We can cancel by 2 to give
$b^2=2x^2 \rightarrow 2=\farc{b^2}{z^2} \rightarrow \sqrt{2} = \frac{b}{x}$

This contradicts the initial assumption that
$\frac{a}{b}$
cannot be simplified, hence
$\sqrt{2}$
is irrational.