From a young age his abilities were apparent. He travelled to Tyre with his father where he learnt astronomy from Chaldean and Syrian teachers. When he was older, he decided to leave Samos because he hated the dictatorship and tyranny of the ruler, Polycrates. He studied in Miletus under the mathematician and philosopher, Thales who was impressed enough to advise Pythagoras to go and study in Memphis (Egypt). It was here that he encountered the geometry of the Egyptians and Babylonians. The beauty and elegance of mathematics fascinated him so much that he came to see it as the explanation of every phenomena known to humanity.
He spent five years in Babylon, learning more from the Magi (followers of Zarathustra). When he returned to Samos (now ruled by Persia) he attempted to set up a school but it was not well received. The local government tried to get him to perform diplomatic and political duties but he was not happy doing this.
Eventually, Pythagoras moved to the town of Crotone (a Greek colony in southern Italy) where he established a school to investigate philosophy and mathematics. It was a semi-religious school based on the cult of Orpheus. Pythagoras included cultural and religious ideas in his teachings and quickly established a circle of followers around himself.
His school became a community for both male and female students - unusual at the time. There were strict rules governing conduct. He had an inner circle of followers who called themselves the Mathematikoi ("those who know") from which the word "mathematics" comes from. They lived at the school, were not allowed personal possessions, wore their hair long and had to be vegetarians. Other students could attend the school during the day (called Akousmatics - "listeners") and they could live normal lives outside.
The community ate together, sang hymns to the Sun god, Apollo, played the lyre and studied music, geometry and philosophy. They had a strict rule of silence - the secrets of the school and its discovereies could not be disclosed to outsiders. The punishment was death.
The theorem named after Pythagoras was known much earlier in Egypt, Mesopotamia and India. Pythagoras may have been the first to prove it. The first association of the theorem with Pythagoras occured several hundred years after his death.
The school did produce original work with musical scales (basing octaves on whole number ratios), number theory (odd, even, prime, square) and irrational numbers (like the square root of two). These latter numbers could not be written down as a fraction and caused great controversy among Pythagoras and his followers. One of their beleifs was that numbers constitute the true nature of things. Another was that the Earth was a sphere - this was after studying the shape of the Earth's shadow on the Moon during lunar eclipses.
Pythagoras and his ideas influenced Plato and his name is still mentioned wherever mathematics is taught.
Pythagoras and a number of his inner circle were murdered by an ex-disciple. The school was continued by his wife Theano and their two daughters. Theano was a mathematician in her own right. She is credited with having written text books on mathematics (including important discoveries on the golden mean), physics, medicine, and child psychology. Unfortunately none of her writings survive.
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Biography of Pythagoras.
Pythagoras' Theorem and its proofs
Proofs of the theorem associated with Pythagoras.