Greece

The people of Greece started to ask "why?" which lead to them developing the idea of logic. One mathematician, Thales of Miletus, proved various theorems that some we have seen in previous math classes such as:

• Two angles between two intersecting straight lines are equal   

•  An angle in a semi-circle is a right angle

Another mathematician, Pythagoras, started a school in Greece and helped distinguished odd and even numbers using pebbles as well as identify square numbers and triangular numbers. Pythagoras and his students believed that "numbers was the basis of the universe, everything could be counted, including lengths" and to the Greeks, a number was a 'multitude composed of units.' However that belief was broken with the discovery of the Pythagorean triple. Using the knowledge that the difference of 2 consecutive square numbers is an odd number, Pythagoras was able to construct the Pythagorean triple.

Pythagorean triple

• n is odd, the triple was : ( n, [(n^2)+1]/2 , [(n^2)-1]/2 )
• n is even, the triple was: (n, ((m/2)^2)-1, ((m/2)^2)+1 )

The numbers that are given from these triples satisfy the Pythagorean theorem( which, interestingly, has be known to other cultures long before Pythagoras lived). From these triples they encounter a triangle that did not up hold their belief:

The Pythagoreans assumed that one could always find a common measure, the unit, between two length; however upon looking closely they could not find a unit between the leg,1, and the hypotenuses (they were incommensurable), the square root of 2 and thus breaking their belief. The idea that their were numbers that could not be measured open up the possibility to new mathematical theories. The Pythagorean theorem would lead to the discovery of irrational numbers.

note: 1 was not a number to the Greeks; it was a unit. 2,3,4,5, etc were consider numbers(composed of unit)

Unanswered question

1. Why did the Greeks did not consider 1 as a number? Did they think fractions as numbers or units since multiple fractions can make whole numbers?
2. If the Pythagorean theorem, as it is know now,  was known to others cultures before Pythagoras birth why was it named after him and his students?

Evaluation
Interesting  7
quality       6
complexity 8