Why did the ratio between a square’s diagonal and side challenge the followers of Pythagoras?

The followers of Pythagoras believed that reality was fundamentally mathematical and could be described entirely through integers and their ratios. They held the conviction that any two lengths were commensurable, meaning they could be measured by a common unit.1

The ratio between a square’s diagonal and its side challenged this belief because it resulted in an irrational number. Using the Pythagorean theorem, a square with sides of length 1 has a diagonal $d$ where $d^2=2$. If one assumes $d$ can be expressed as a ratio of two integers ($p/q$) in simplest form, the equation $p^2=2q^2$ leads to a logical contradiction: both $p$ and $q$ must be even, which violates the assumption that they share no common factors.1

This discovery of incommensurable quantities undermined the Pythagorean worldview, as it proved that rational numbers were insufficient to describe all geometric lengths. This realization created a significant impasse in ancient Greek mathematical reasoning that persisted until later mathematicians developed new definitions of proportion.1