How does the diagonal of a square with side length 1 lead to the irrational number √2?

To determine the length of the diagonal $d$ of a square with side length 1, we apply the Pythagorean theorem, which states that $d^2=1^2+1^2=2$.1

If we assume the diagonal can be expressed as a ratio of two integers, $\frac{p}{q}$ (where $p$ and $q$ are relatively prime), then $\frac{p^2}{q^2}=2$, or $p^2=2q^2$. This implies that $p$ must be an even number. By substituting $p=2r$ into the equation, we find that $q^2=2r^2$, which means $q$ must also be even. This result contradicts the initial assumption that $p$ and $q$ are relatively prime, as both share a common factor of 2.1

Because no ratio of integers can satisfy this relationship, the length of the diagonal is an irrational number, specifically $\sqrt{2}$. This discovery of incommensurable quantities challenged the early Pythagorean belief that all lengths could be described by integer ratios.1