How can the side and diagonal of a square show that some lengths are incommensurable?

The discovery of incommensurable lengths challenged the early Pythagorean belief that all quantities could be expressed as ratios of integers. Euclid provided a logical proof demonstrating that the side and diagonal of a square are incommensurable, meaning their ratio cannot be expressed as a fraction.

Using the Pythagorean theorem, a square with a side of 1 unit has a diagonal $d$ where $d^2=1^2+1^2=2$. If we assume $d$ is a rational number, it can be written as a reduced fraction $\frac{p}{q}$. This leads to the equation:

$$\frac{p^2}{q^2}=2⟹p^2=2q^2$$

This implies $p$ must be an even number. If we substitute $p=2r$ into the equation, we find $q^2=2r^2$, meaning $q$ must also be even. This contradicts the initial assumption that $\frac{p}{q}$ is in simplest form, as both numbers share a common factor. Therefore, no ratio of integers can represent the diagonal, identifying it as an irrational number.1