The Pythagorean Theorem: Its application and discovery of Irrational Numbers!

The Pythagorean Theorem,  is probably one of the earliest theorems known to our civilizations. The theorem is named after Greek mathematician and philosopher, Pythagoras. He is credited with many a contributions to mathematics, it is quite likely some of the work may have, actually, been the work of his students.

The Pythagorean Theorem is a statement about triangles containing a right angle. The Pythagorean Theorem states that:

"The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides." 
                                      

The point which needs to be understood clearly is the fact that during the era of Pythagoras the concept of Algebra didn't exist. The theorem was proved by using Geometry, the Geometrical representation used was something similar to the one shown above.

Applying the Pythagorean Theorem some of the problems solved are: 

If A= 3;  B = 4;   then C= 5, similarly, some more are
5 12 13
7 24 25
9 40 41
11 60 61

  (These are known as Pythagorean Triples all of them are whole Numbers, positive integers)

Pythagoras later discovered that the Square Root of 2 cannot be expressed as a ratio of two integers. This fact disturbed Pythagoras and his followers to a great extent. For them mathematics was a sacred issue, they were possessed , in a way, with the belief that any two lengths were integral multiple of some Unit-Length. They considered this as the threat to their belief and made many attempts to suppress the fact that the Square Root of 2 was an Irrational Number.

The importance of Pythagorean Theorem cannot be overlooked. We must realise that during that era, when it was discovered,  it must have enabled people solve problems of heights and distance.  The development of the concept of Trigonometry, later took over, as a tool for solving heights & Distance problems thus making Pythagorean Theorem redundant, so to say. 

The discoveries of Pythagoras laid the foundation for development of the concept of Rational and Irrational Numbers to start with and then moving on to development of the concept of Imaginary Numbers. The students will appreciate the importance of these developments as we go forward and unravel the utility of the above concepts and principles.