\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \)

The irrationality of \( \sqrt(2) \)

Consider a rational number fully reduced, \( \frac{p}{q}\) such that \(\frac{p^2}{q^2} = 2\). Investigate whether \(p\) or \(q\) are odd or even numbers leads to a contradiction which shows that \(2\) cannot be rational.

• \(p\) and \(q\) can't both be even, as [...].
• \(p\) and \(q\) must both be odd, as [...].
• \(p\) is even as [...].

These points lead to a contradiction.