Proof (Intuition) Math Example 2

Solution

1. 1
   Intuition: if $\sqrt{2} = p/q$ in lowest terms, then $p^2 = 2q^2$ — the left side and the right side must match exactly.
2. 2
   The right side is 'doubly even' — it has at least two factors of 2. So $p^2$ must be divisible by 4.
3. 3
   This forces $p$ itself to be even. But then $q^2 = p^2/2$ is also even, making $q$ even too.
4. 4
   Core contradiction: $p$ and $q$ are both even, contradicting 'lowest terms.' The assumption breaks.