Why is the diagonal of a unit square equal to the irrational number √2?

The diagonal of a unit square is equal to $\sqrt{2}$ because of the Pythagorean theorem, which states that for a right triangle with legs of length 1, the hypotenuse $d$ must satisfy the equation $d^2=1^2+1^2=2$.1

It is a common misconception that all lengths can be expressed as a ratio of two integers. Ancient Pythagoreans initially believed that all quantities were commensurable, meaning they could be measured by integer multiples of a common unit. However, the diagonal of a unit square cannot be expressed as such a ratio. If one assumes the diagonal is a rational number $p/q$ in simplest form, the equation $(p/q{)}^2=2$ leads to a logical contradiction where both $p$ and $q$ must be even, violating the assumption that they share no common factors.12

Because this ratio cannot be represented by a fraction, it is defined as an irrational number, represented by an endless, nonrepeating decimal series.12