The side \(a\) and the diagonal \(d\) in any square are incommensurable. \[d=\sqrt{2}\cdot a.\]
In particular and for \(a=1\), there are no positive integers \(q,p\) such that \[q\cdot\sqrt 2=p\quad\text{or}\quad\sqrt 2=\frac pq.\]
Definitions: 1
Parts: 2
Proofs: 3