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Katanga wrote:In short form, a2 + b2 = c2. That’s the Pythagorean theorem. In one example, a=3, b=4, and c=5. 3,4,5 is a Pythagorean triple. A and B represent the two legs of the triangle (the line segments that form a right angle) and C represents the hypotenuse of the triangle (the line segment opposite of the right angle). Here are some more:

(3, 4, 5) (5, 12, 13) (7, 24, 25) (8, 15, 17) (9, 40, 41) (11, 60, 61) (12, 35, 37) (13, 84, 85) (16, 63, 65) (20, 21, 29) (28, 45, 53) (33, 56, 65) (36, 77, 85) (39, 80, 89) (48, 55, 73) (65, 72, 97)

Those are the sixteen primitive Pythagorean triple under 100. Now you are probably wondering what a primitive Pythagorean triple is. To answer, if A, B, and C are coprime, meaning that they do not have a common factor; they are then a primitive Pythagorean triple.

Euclid came up with the formula for generating a Pythagorean Triple (around 300 B.C.) in his book Elements and is often referred to as Euclid’s formula.

Before Pythagoras discovered the Pythagorean theorem, others had come close to the discovery, like the Egyptians. For example, when Hammurbi was in power, the Mesopotamian tablet called ‘Plimpton 322’, which was written between 1790 and 1750 BC, contained entries related to the Pythagorean triples.

This formula is known as the Pythagorean theorem here, however it has different names in other places. For example, it is known in China as the Gougu theorem and in India as the Bhaskara Theorem.

Now I think it is time to learn a little bit about Pythagoras and the Pythagoreans themselves. Pythagoras was the founder of Pythagoreanism, and he is known for the discovery of the Pythagorean theorem. However, it could have been any other Pythagorean. But Pythagoras got the credit, and for that his name will never be forgotten.

Pythagoras was Greek, and is believed to have lived between aproximetely 569 and 500 B.C. When he was a young boy, he lived in an atmosphere full of culture and art, at just the beginning of the golden age for Greece. In his early years, Pythagoras traveled a lot in the countries bordering the Mediterranean Sea. In 509 B.C., he settled in Crotona, a Greek city in southern Italy. There he organized the secret society of brotherhood. They were most interested in moral reform, but they also had scientific discussions. These led to the development of important mathematical and astronomical ideas.

Science of music was a big interest to the Pythagoreans. For example, they proved that if two strings under the same tension have length in the ratio of 2 to 1, then when the strings are plucked, the resulting musical notes are exactly one octave apart.

The Pythagoreans associated numbers with life. For example, 1 stood for reason since it was unchangeable, 2 stood for masculine, 3 stood for feminine, 4 stood for justice since it was the first product of equals (2x2=4), and 5 stood for marriage since it was the first union of masculine and feminine numbers (2+3=5).
So now you know a bit about Pythagoras and his companions. But now it is time for proof. Pythagoras, you see, was not the only one that discovered proofs for this theorem. Euclid was also a big part of it, which you already know since he discovered the formula or generating a Pythagorean triple.