## 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 simplifiedThen

\[2= \frac{a^2}{b^2} \rightarrow 2b^2 =a^2\]

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.