The Pythagoreans were the first to contemplate sums of series, but their most important discovery was that off irrational numbers-  numbers that cannot be expressed as a ration of integers. This was developed from the Pythagorean theorem (a^2 + b^2 = c^2) because if a triangle had two sides with length one, the hypotenuse would be length square root of two, which has an infinite number of decimals.

A common representation of the Pythagorean theorem.

This discovery rocked the number oriented religion of the society and later, the world. The Pythagoreans initially tried to keep this discovery a secret and Pythagoras himself called it “unspeakable” because of the implications it had for their priorly developed system.