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
\[b^2=2x^2 \rightarrow 2=\frac{b^2}{z^2} \rightarrow \sqrt{2} = \frac{b}{x}\]
This contradicts the initial assumption that
\[\frac{a}{b}\]
cannot be simplified, hence \[\sqrt{2}\]
is irrational.