Pythagoras (circa 580 - 500 B.C.) was a native of the Aegean Island of Samos and founded a school in southern Italy. This school, now called the Pythagoreans, was a secret society and would most likely be label as a cult today. The symbol of the Pythagoreans was a pentagram star.
A central belief of Pythagoras and his followers was that "everything is number." To the Pythagoreans a number was a quantity that could be expressed as a ratio of two integers (a rational number). The Pythagoreans used music as an example of this belief. They demonstrated that pitch could be represented as a simple ratios that came from the length of equally tight strings that could be plucked.
Perhaps the most famous of the Pythagoreans mathematical results is the Pythagorean theorem: the square of the length of the hypotenuse of a right triangle is the sum of the squares of the lengths of the two sides, usually expressed a2+b2 = c2. Integer triples which satisfy this equation are Pythagorean triples. For example (3,4,5) and (5,12,13).
The Pythagoreans later disproved the belief that "everything is [a rational] number" using their own theorems. They were able to show the square root of two (the length of the hypotenuse of a triangle with sides one and one) is irrational (not a rational number). The Pythagoreans vowed to keep this discovery a secret, but the secret was later revealed by one of the members.
It is believed that Pythagoras (or at least the Pythagoreans since they had a habit of attributing all their discoveries to Pythagoras) also developed several of the figurate numbers: numbers derived from arranging dots in regular patterns. For example, the square numbers n2, are the numbers of dots that can be arrange in a square. The triangular numbers: 1, 1+2, 1+2+3, ... are the number that can be arrange in a triangle where each row of the triangle has one more dot than the previous row. The nth triangular number is n(n+1)/2.
This entry edited from an entry contributed by Gary Spencer.