Beyond Pythagoras

Pythagoras was born on the island Samos . he 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. The society which he led, half religious and half scientific, followed a code of secrecy. It is probable that he had two brothers. He was well educated andThere 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. The other two philosophers who were to influence Pythagoras, and to introduce him to mathematical ideas, were and his pupil who both lived on Miletus. Pythagoras is believed to have been born on the Greek Island of Samos. Little is known of his early years, except that from around 545 BC he travelled widely around the known world. He went to Egypt and Persia, learning about their religious and philosophical beliefs. He returned briefly to Samos, aged around fifty, but left shortly afterwards, possibly because he was uncomfortable under the yoke of Polycrates, the tyrant of Samos who ruled in luxury. He arrived in Croton in around 532 BC, where he began teaching and soon had a clutch of students.

In approximately 500 BC there appears to have been an uprising against the power of the Pythagoreans. A mob cruised the city of Croton burning, looting and killing. Pythagoras fled and died one of two alternative deaths. The first version saw him chased by rebels until he was caught because he refused to cross a bean field, and then killed. The second had him fleeing to Metapontum, a city up the coast of Italy, where he took refuge in the Temple of the Muses and died of starvation.

Pythagoras believed that the area of the hypotenuse side square should be the same as the area of the adjacent side square and the area of the opposite side square add together. There is an example:

The hypotenuse line (line labeled C) is 5 cm long and 5² equals 25, the adjacent line (line labeled B) is 4cm long and 4² equals 16 and the opposite line (line labeled A) is 3cm long and 3² equals 9. If you add the area of the adjacent line and the opposite line (lines labeled A & B) you get the area of the hypotenuse line (line labeled C). 9+16=25 so this equation follows the rule a2+b2=c2

I will investigate whether the theory a² + b² = c² works for every single right angled triangles. I will then see if there are any rules for each side and including the perimeter and area. I will then compare them and see if I could find out the x term which will work for any right angled triangles.

Here is a 3 equations I was given and I had to see if the Pythagoras theorem worked for them. The three equations I was given were 3 4 5, 5 12 13 and 7 24 25 and I also had to find out the area and the perimeter of the right angled triangle. Here is my working out:

3² = 9

4² =16

5² = 25

16 + 9 =25 C=5

A=3

Area = 3 x 4 = 12 2 = 6cm²

Perimeter = 3 + 4 + 5 = 12cm

B=4

5² = 25

12² = 144

13² = 169

25 + 144 = 169 C=13