Square root of 2 is not rational

  The Pythagorean school of mathematics originally insisted that only natural and rational numbers exist. The discovery of the irrationality of $\sqrt{2}$, the ratio of the diagonal of a square to its side (around 5th century BC), was a shock to them which they only reluctantly accepted. $^1$
  From the Pythagorean theorem we have that 

 \[ \alpha^2 = \beta^2 + \beta^2 \Leftrightarrow \] \[   \alpha^2 = 2\beta^2  \Leftrightarrow \] \[ \frac{\alpha^2}{\beta^2} = 2 \] and setting $ \frac{\alpha}{\beta} = \rho $ we end in the relation \[ \rho^2 = 2. \] So we have to proove that:

There is no rational number $ \rho$, with $ \rho^2 = 2 $.

Proof