The other two philosophers who were to influence The other two philosophers who were to influence Pythagoras, and to introduce him to Pythagoras, and to introduce him to mathematical ideas, were Thales and his pupil mathematical ideas, were Thales and his pupil Anaximander who both lived on Miletus. In [8] it Anaximander who both lived on Miletus. In [8] it is said that Pythagoras visited Thales in Miletus is said that Pythagoras visited Thales in Miletus when he was between 18 and 20 years old. By when he was between 18 and 20 years old. By this time Thales was an old man and, although this time Thales was an old man and, although he created a strong impression on Pythagoras, he created a strong impression on Pythagoras, he probably did not teach him a great deal.he probably did not teach him a great deal.

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 Baudhāyana in 800BC. The principle was also known to the earlier Egyptian and the Babylonian master builder.

In this example, the missing side is In this example, the missing side is not the long one. But the theorem not the long one. But the theorem still works, as long as you start with still works, as long as you start with the hypotenuse:the hypotenuse:

151522 = x = x22 + 9 + 922

Simplifying the squares gives: Simplifying the squares gives:

225 = x225 = x22 + 81 + 81

and then: and then:

225 - 81 = x225 - 81 = x22                                    

144 = x144 = x22                  

12 = x    12 = x            

In the right triangle at the left, we know that: In the right triangle at the left, we know that:

hh22 = 7 = 722 + 10 + 1022

Simplifying the squares gives: Simplifying the squares gives:

hh22 = 49 + 100 = 49 + 100

hh22 = 149            = 149           

This square root is not perfect. A calculator gives: This square root is not perfect. A calculator gives:

h = 12.2h = 12.2                

(rounded to one decimal place) (rounded to one decimal place)

How far up a wall will an 11m ladder reach, if How far up a wall will an 11m ladder reach, if the foot of the ladder must be 4m from the the foot of the ladder must be 4m from the base of the wall? base of the wall?

112 = x112 = x22 + 4 + 422

121 = x121 = x2 2 + 16+ 16

121 - 16 = x121 - 16 = x22                                      

105 = x105 = x22                      10.2 = x               10.2 = x               The ladder will reach 10.2 metres up the The ladder will reach 10.2 metres up the

wall. wall.

It is called "Pythagoras' Theorem" and It is called "Pythagoras' Theorem" and can be written in one short equation:can be written in one short equation:

a2 + b2 = c2a2 + b2 = c2

c is the longest side of the triangle

a and b are the other two sides

Pythagoras left Samos and Pythagoras left Samos and went to southern Italy in went to southern Italy in about 518 BC (some say about 518 BC (some say much earlier). Iamblichus [8] much earlier). Iamblichus [8] gives some reasons for him gives some reasons for him leaving. First he comments leaving. First he comments on the Samian response to on the Samian response to his teaching methods:- his teaching methods:-

... he tried to use his symbolic ... he tried to use his symbolic method of teaching which method of teaching which was similar in all respects to was similar in all respects to the lessons he had learnt in the lessons he had learnt in Egypt. The Samians were Egypt. The Samians were not very keen on this not very keen on this method and treated him in a method and treated him in a rude and improper mannerrude 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 In the right triangle at the left, we know that: left, we know that: hh22 = 6 = 622 + 8 + 822

Simplifying the squares Simplifying the squares gives: gives: hh22 = 36 + 64 = 36 + 64and then: and then: hh22 = 100      = 100     hh = 10 = 10                (by doing the square root (by doing the square root of 100) of 100)

DefinitionDefinition

The longest side of the triangle is The longest side of the triangle is called the "hypotenuse", so the called the "hypotenuse", so the formal definition is:formal definition is:

In a right angled triangle:In a right angled triangle:the square of the hypotenuse is the square of the hypotenuse is equal toequal tothe sum of the squares of the the sum of the squares of the other two sides. other two sides.

Example: A "3,4,5" triangle has a right angle in it.

Let's check if the areas are the same:

32 + 42 = 52

Calculating this becomes: 9 + 16 = 25 It works ... like Magic

Example: Solve this triangle. a2 + b2 = c2

52 + 122 = c2

25 + 144 = c2

169 = c2

c2 = 169c = √169 c = 13

Example: What is the Example: What is the diagonal distance across a diagonal distance across a square of size 1?square of size 1?

a2 + b2 = c2 12 + 12 = c2 1 + 1 = c2

2 = c2

c2 = 2c = √2 = 1.4142...