The son of Mnesarchus, Pythagoras is arguably the most interesting figure in the history of mathematics. He was probably born between the 50th and 52nd Olympiads, thus it is generally agreed upon he was born around 580 B.C.E. Although he is called a Samian, it is still uncertain whether he was born on the island of Samos, off the coast of Asia Minor. A story tells of his initiation by the master into the secrets of Zeus on Mount Ida and was told if he would gain further knowledge, he must seek it in Egypt. He spent much of his time not only in Egypt but also in Babylonia. He returned to Samos and attempted to find a school there but was unsuccessful. Around 530 B.C.E., he was forced to leave Samos and emigrated to Crotona, in Southern Italy.
In Crotona, he soon gained a leading position among his fellow townsmen. Among these disciples, he formed a brotherhood, united by common philosophical beliefs and pursuits. The Pythagoreans, as they were come to be known, was both a religious order and a philosophical school. They were bound by oath not to speak of the doctrines and discoveries of their school; the fact that no extant works of Pythagoras or the Pythagoreans exist, it is difficult to identify the contributor of any portion of their creed, thus it is necessary to acknowledge the Pythagoreans as a collective body and not as individual contributors. It should be noted that the Pythagoreans were in custom to attribute all their doctrines to their master.
Pythagoras was probably more of a mystic than a rational thinker, but he still commanded great respect from his followers. He was known as a performer of miracles. Xenophanes, a poet-singer told a story, regarding the Pythagorean doctrine of the transmigration of souls, about a scene where a little dog is being thrashed, Pythagoras says "Stop beating, for in this dog lives the soul of my friend; I recognize him by his voice." Tales of Pythagoras traveled, that one of his calves was made of gold, that he was seen in two places at the same time, and that once when he crossed a small stream, the stream rose out of its bed, greeted him and said "Hail! Pythagoras." The Pythagoreans were ridiculed on stage as superstitious, filthy vegetarians, yet never as mathematicians. The Pythagoreans exercised ascetic, monastic living, vegetarianism and common-ownership of belongings. They believed the elevation of the soul to God was to be attainable by means of mathematics. Their belief was that "number was the substance of all things", positive integers formed the basic organizing principles of the universe. God is unity, and all on earth is plurality, consisting of contrasting elements. Harmony is divine, as it is a ratio of numbers.
Musical harmonies are numerical ratios. A string or flute shortened to half of its original length produces a tone which is one octave higher. Ratios of 3 : 2 give a fifth and 4 : 3 give a fourth. The ratio of 3 : 4 : 5 gives the sides of a right-angled triangle, thus establishing a connection of numbers to angles. All of these are of which the Pythagoreans had knowledge. They classified numbers in to categories of odd, even, prime, composite, perfect and amicable numbers. They used stones or pebbles, in groups to form different patterns of which they classified figurate, triangular and square numbers. The first n natural numbers, ½*n(n+1) formed a triangular number while the first n odd numbers (1+3+5+...+(2*n-1)) forms a square number. Aside, the Pythagoreans used the symbol of heath to distinguish them from others.