Irrationality of the square root of 2

Statement

The square root $\sqrt{2}$ of $2$ is an irrational number. That is, there are no integers $a$ and $b$ with $b \neq 0$ and $(a / b)^2 = 2$.

Remarks

A classical proof: suppose that $a / b = \sqrt{2}$ in lowest terms. Then $a^2 = 2 b^2$, thus $a$ is even. Then $b$ is also even, which is a contradiction. The theorem is a special case of the irrationality of the square root of a prime.

History

The ancient Greeks knew this result. Tradition attributes its discovery to the Pythagoreans.

Explanations

L1 human

No fraction squared gives exactly $2$. Fractions such as $7/5$ and $141/100$ come close, but there is always an error. This means that the diagonal of a square with side $1$ has a length that is not a ratio of whole numbers.