It was one of the most surprising discoveries of the Pythagorean School of Greek mathematicians that there are irrational numbers. According to Courant and Robbins in "What is Mathematics": This revelation was a scientific event of the highest importance. Quite possibly it marked the origin of what we consider the specifically Greek contribution to rigorous procedure in mathematics. Certainly it has profoundly affected mathematics and philosophy from the time of the Greeks to the present day.
Specifically, the Greeks discovered that the diagonal of a square whose sides are 1 unit long has a diagonal whose length cannot be rational. By the Pythagorean Theorem, the length of the diagonal equals the square root of 2. So the square root of 2 is irrational!
The following proof is a classic example of a proof by contradiction: We want to show that A is true, so we assume it's not, and come to contradiction. Thus A must be true since there are no contradictions in mathematics!
Fine print, your comments, more links, Peter Alfeld, PA1UM