They thought that numbers formed the basis of the universe. Initially, they believed that all numbers were rational, that is, they can be written as a fraction. One problem that they studied was the construction of the diagonal of a square. In fact you will know that the length of the diagonal of a square 1 cm long is 2 cm. 2 is a number whose decimal part continues forever with no pattern.
All rational numbers give decimal parts that either terminate or recur.
So 2 cannot be a rational number. This was a devastating discovery for the Pythagoreans who are reputed to have stoned one of their members to death for letting out this secret. It was Aristotle again who is first known to have shown that 2 is irrational. He used a method known as proof by contradiction. It works by first assuming that 2 is a rational number.