Beyond Pythagoras .

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Ali     10N                Mathematics GCSE Coursework





Pythagoras of Samos is often described as the first pure mathematician. He is an extremely important figure in the development of mathematics yet we know relatively little about his mathematical achievements. Unlike many later Greek mathematicians, where at least we have some of the books, which they wrote, we have nothing of Pythagoras's writings. The society which he led, half religious and half scientific, followed a code of secrecy which certainly means that today Pythagoras is a mysterious figure.

As a child Pythagoras spent his early years in Samos but travelled widely with his father although little is known of his childhood. All accounts of his physical appearance are likely to be fictitious except the description of a striking birthmark that Pythagoras had on his thigh. It is probable that he had two brothers although some sources say that he had three. He was well educated, learning to play the lyre, learning poetry and to recite Homer, a poet who wrote the Iliad and Odyssey among his work. There were, among his teachers, three philosophers who were to influence Pythagoras while he was a young man. One of the most important was Pherekydes who many describe as the teacher of Pythagoras. Another teacher of his was Thales who was said to have first introduced him to mathematical ideas. Although he created a strong impression on Pythagoras, he probably did not teach him a great deal. However he did contribute to Pythagoras's interest in mathematics and astronomy, and advised him to travel to Egypt to learn more of these subjects. Pythagoras became a frequent visitor to Egypt where he gave lectures about Astronomy, Mathematics and also Geology and Cosmology.

Pythagoras founded a philosophical and religious school in Croton (on the east of the heel of southern Italy) that had many followers. Pythagoras was the head of the society with an inner circle of followers known as mathematikoi. The mathematikoi lived permanently with the Society, had no personal possessions and were vegetarians. They were taught by Pythagoras himself and obeyed strict rules.

Although Pythagoras is best known for creating the most famous formula in mathematics, it is said that it was one of his students that actually discovered the formula that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides but Pythagoras took the formula and claimed that it was himself that found it.

        The problem of this investigation is to find a general formula that can generate all Pythagorean triples. I will start by looking at triples where the smallest side has an odd number, and to try and find a pattern, formula or just to make a useful and interesting observation. I hope this task shall prove o be enlightening and interesting.

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3,4,5 are called Pythagorean triples because shortest2 + middle2  = longest2.

a2 + b2  = c2

32 + 42 = 52

9 + 16 = 25                                       5 X 5 = 25 



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