# Diagonal of a Rectangle [duplicate]

The Pythagoreans proved that the length of the diagonal of a square with side length 1 is not a rational number. Prove that the length of the diagonal of a rectangle with sides length 1 and 2 is not a rational number. When trying this problem, I did the pythagorean theorem. I think that is too simple.

## marked as duplicate by Claude Leibovici, JonMark Perry, user91500, José Carlos Santos, TomGrubbJun 17 '17 at 23:33

What you did is correct. For a rectangle with sides of length $1$ and $2$, the Pythagorean theorem states that the diagonal of that rectangle is $\sqrt {1^2+2^2}=\sqrt 5$. Since $\sqrt 5$ is a real number that cannot be expressed as a ratio of two integers where the dominator is nonzero, the diagonal of this rectangle is an irrational number.