Pythagoras

Pythagoras was born about 569 BC in Samos, Ionia and died about 475 BC.

He was a Greek philosopher and mathematician, whose doctrines strongly influenced Plato. Born on the island of Sámos, Pythagoras was instructed in the teachings of the early Ionian philosophers Thales, Anaximander, and Anaximenes. Pythagoras is said to have been driven from Sámos by his disgust for the tyranny of Polycrates. About 530 BC Pythagoras settled in Crotona, a Greek colony in southern Italy, where he founded a movement with religious, political, and philosophical aims, known as Pythagoreanism.

Pythagoras argued that there are three kinds of men. The lowest consists of those who come to buy and sell, and next above them are those who come to compete. Best of all are those who simply come to look on.

These pages explain the famous mathematical result known as Pythagoras's Theorem, give you various interactive proofs of the theorem, problems based on the theorem, and mathematical things related to the theorem.
The areas of the two smaller squares add up to the area of the largest one. Check it out here:

To sum up: another way of stating the theorem is

The square on the hypotenuse is equal to the sum of the squares on the other two sides.
if the three sides of the right angled triangle are labelled a, b and c (with c the longest side or hypotenuse), then Pythagoras's theorem says that:
c² = a²+b²

Returning to the knotted rope triangle with sides of length 3, 4 and 5 units, we can see that

5² = 3²+4².

Triples like these (ie., 3,4,5) are called Pythagorean triples.

There remains one small problem. The Egyptians used knotted ropes to measure the dimensions of their fields. Further evidence from the The Rhind papyrus confirms this. The papyrus, a scroll about 6 metres long and 1/3 of a metre wide, was written around 1650 BC by the scribe Ahmes who is copying a document which is 200 years older. This makes the original papyrus date from about 1850BC.

One of the Babylonian tablets (Plimpton 322) which is dated from between 1900 and 1600 BC contains answers to a problem containing Pythagorean triples, i.e. numbers a, b, c with a² + b² = c². It is said to be the oldest number theory document in existence.

BUT Pythagoras was not born until about 1200 years later! He could not, therefore, have discovered the theorem that bears his name. There is a theory that many great inventions were not invented by the people they are named after. This is Dye's Theory. For example, On June 15th, 2002, the US Congress officially recognized that the italian inventor Antonio Meucci is to be credited for the invention of the telephone, and not Alexander G. Bell, as previously claimed.

Related things - Pythagorean triples
Triples like (3,4,5) have the property that 5² = 3²+4². They could be the measurements of the sides of a right-angled triangle and are called Pythagorean triples.
(6,8,10) is another such triple, as is (5,12,13). There are infinitely many of these.

Both Pythagoras and the Babylonians used the formulas 2m, m²-1, m²+1 to generate such triples.

A version known to the early Greeks was 2pq, p²+q² and p²-q² based on two different integers p and q. These three numbers will form a Pythagorean triple.
Enter two positive integers:p:q: a Pythagorean Triple:

If p and q are relatively prime (have no factors in common) and not both odd then the above method will produce a triple that is irreducable.
In any Pythagorean triple, one integer is always divisable by 3 and another (maybe the same one) by 5. The product of the two smaller integers is divisable by 12 and the product of all three is divisable by 60.
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