Pythagoras was a well-known ancient Greek mathematician and philosopher. There are many legends about his life, and it is difficult to determine which are based on truth and which are not. All of the writers agree that Pythagoras was born on Samos, a Greek island off the coast of Asia Minor (now Turkey), around 580 B.C. Like Thales, he travelled and he probably visited both Egypt and Babylon as Thales did. He also studied at the Ionian school founded by Thales. It is unlikely that he studied directly under Thales though, because Thales would have been quite old by then.
Pythagoras later founded his own school/cult in the Greek colony of Croton in Magna Graecia (southern Italy). His school had about 300 members, divided into two groups: mathematikoi (Greek for "scientists"; this is from where English gets the word mathematics), the elite of the school, who were privy to Pythagoras' mathematical truths, and the akousmatikoi (Greek for "those who hear"), who followed the sect's rules but didn't know about the mathematical mysteries of the cult.
Pythagoras believed that the natural numbers were the basis of all reality. For example, he felt that musical harmonies were not merely related to ratios of whole numbers; they were ratios of whole numbers. Pythagoras said that "all is number", and he fostered a number mysticism which classified numbers as male (odd numbers), female (even numbers), triangular, square, prime, composite, perfect, deficient, abundant, amicable, or into other categories.
Pythagoras is most famous for his proof of the Pythagorean Theorem. The ancient Egyptians had known that a right triangle could be made with sides of lengths 3, 4, and 5, but were not aware of the general rule. The Babylonians knew that, in general, if a and b were the lengths of the legs of a right triangle and c the length of the hypotenuse, then a² + b² = c², but they never attempted to prove this theorem. Pythagoras was the first person to do so. This theorem led the school to a sensational discovery when applied to a right triangle whose two legs were of length 1.