Why \(\sqrt{2}\) is Irrational.
Working towards a contradiction, we suppose that \(\sqrt{2}\) is rational.
So $$ \sqrt{2} = \frac{p}{q} $$ for some \(p, q \in \mathbb{Z}\).
Yeah, then we get a contradiction after we see some fun stuff with \(p\) and \(q\) one being even and the other odd which makes our initial assumption false, thus proving that \(\sqrt{2}\) is not rational, and thus has to be irrational.
Decimal Expansion
\(\sqrt{2} \approx 1.4142135623730950488016887242096980785697\)
Binary Expansion of \(\sqrt{2}\)
\(\sqrt{2} \approx 1.011010100000…\)