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<caldwell@utm.edu> 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 a^{2}+b^{2} = c^{2}. 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 is regular patterns. For example, the square numbers n^{2}, 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.
Chris K. Caldwell © 19992014 (all rights reserved)
