The other two philosophers who were to influence Pythagoras, and to introduce him to mathematical ideas, were Thales and his pupil Anaximander who both lived on Miletus. In [8] it is said that Pythagoras visited Thales in Miletus when he was between 18 and 20 years old. By this time Thales was an old man and, although he created a strong impression on Pythagoras, he probably did not teach him a great deal. Thales
In about 535 BC Pythagoras went to Egypt. This happened a few years after the tyrant Polycrates seized control of the city of Samos. There is some evidence to suggest that Pythagoras and Polycrates were friendly at first and it is claimed [5] that Pythagoras went to Egypt with a letter of introduction written by Polycrates. In fact Polycrates had an alliance with Egypt and there were therefore strong links between Samos and Egypt at this time.
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. 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.
Number is the ruler of forms and ideas, and the cause of gods and demons. Iamblichus Every man has been made by God in order to acquire knowledge and contemplate. Geometry is knowledge of the eternally existent. Number is the within of all things. There is geometry in the humming of the strings. Time is the soul of this world. Quoted in Des Michael, Wisdom (London, 2002). Above the cloud with its shadow is the star with its light. Above all things reverence thyself Quotations by Pythagoras
Like Thales, Pythagoras is rather known for mathematics than for philosophy. Anyone who can recall math classes will remember the first lessons of plane geometry that usually start with the Pythagorean theorem about right-angled triangles: a²+b²=c². In spite of its name, the Pythagorean theorem was not discovered by Pythagoras. The earliest known formulation of the theorem was written down by the Indian mathematician Baudh ā yana in 800BC. The principle was also known to the earlier Egyptian and the Babylonian master builder.
In this example, the missing side is not the long one. But the theorem still works, as long as you start with the hypotenuse: In this example, the missing side is not the long one. But the theorem still works, as long as you start with the hypotenuse: 15 2 = x Simplifying the squares gives: 225 = x and then: = x = x = x = x 2 12 = x 12 = x
In the right triangle at the left, we know that: In the right triangle at the left, we know that: h 2 = Simplifying the squares gives: h 2 = h 2 = 149 h 2 = 149 This square root is not perfect. A calculator gives: h = 12.2 h = 12.2 (rounded to one decimal place)
How far up a wall will an 11m ladder reach, if the foot of the ladder must be 4m from the base of the wall? 112 = x = x = x = x = x = x = x 10.2 = x The ladder will reach 10.2 metres up the wall.
It is called "Pythagoras' Theorem" and can be written in one short equation: a2 + b2 = c2 c is the longest side of the triangle a and b are the other two sides
Pythagoras left Samos and went to southern Italy in about 518 BC (some say much earlier). Iamblichus [8] gives some reasons for him leaving. First he comments on the Samian response to his teaching methods:-... he tried to use his symbolic method of teaching which was similar in all respects to the lessons he had learnt in Egypt. The Samians were not very keen on this method and treated him in a rude and improper manner
The Pythagorean Theorem must work in any 90 degree triangle. This means that if we know two of the sides, we can always find the third one. In the right triangle at the left, we know that: h 2 = Simplifying the squares gives: h 2 = and then: h 2 = 100 h 2 = 100 h = 10 h = 10 (by doing the square root of 100)
DDDDefinition The longest side of the triangle is called the "hypotenuse", so the formal definition is: In a right angled triangle: the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Example: A "3,4,5" triangle has a right angle in it. L et's check if the areas are the same: = 5 2 Calculating this becomes: = 25 It works... like Magic
Example: Solve this triangle. a 2 + b 2 = c = c = c = c 2 c 2 = 169 c = √169 c = 13
Example: What is the diagonal distance across a square of size 1? a 2 + b 2 = c = c = c 2 2 = c 2 c 2 = 2 c = √2 =