# Pythagoras

**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 in 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 *n*th triangular number is
*n*(*n*+1)/2.

This entry edited from an entry contributed by Gary Spencer.