Constructing a square with twice more area than another square can be done with a simple “Geometrical construction”. We mean to be used only a ruler and a compasses

The square that has side the diagonal ΒΔ, as shown in the figure, has area twice more than the initial square However, the side β, in the square with the double area, is not calculated from the side α multiplying it by a rational number

 That means that there is no line segment (as unit of measurement) with which we can measure exactly these two segments, in the side and the diagonal of a square.

 Until then they had a deep faith that always two segments have a common measure.  That`s why the discovery of the Pythagoreans was not just an interesting mathematical proposition, but meant the overthrow fundamental philosophical conceptions about the world and the nature.

 «Everything we know can be represented with a natural number, otherwise it would be impossible to understand it with logic. The “1” is the beginning of everything.»

 There are legends surrounding this happening. One of them says that irrational numbers were discovered by Ipassos, while he was travelling by ship. As soon as it was learned, the rest of the Pythagorians plotted to murder him and keep it as a secret.

 Overcoming the 'difficulties‘, that the existence of irrational numbers brings in Mathematics, was made possible by Eudoxus (360p.Ch.) with the ingenious “Theory of Ratios“. The proof that a particular number is irrational is a problem that often requires complex syllogisms.