Monday, January 19, 2009

numbers to die for

The Pythagorans wanted to believe that everything could be expressed in terms of whole numbers. I was wondering about that. For example, how could you prove that square root (2) could not be somehow expressed in terms of whole numbers?

Hippasus, one of the Pythagoreans, did prove that such an assumption did lead to a logical contradiction. This so upset his group that they killed him. This history gives the expression "irrational number" some real bite.

Here is the brilliant reasoning of Hippasus in the 5th Century BC, which cost him his life:

Start with a right angled isosceles triangle, assume all the numbers are whole:
  • The ratio of the hypotenuse to an arm of an isosceles right triangle is a:b expressed in the smallest units possible.
  • By the Pythagorean theorem: a2 = 2b2
  • 2b2 must be even, since anything multiplied by 2 is even
  • Hence a2 is even and furthermore a must be even as the square of an odd number is odd.
  • Since a:b is in its lowest terms (first assumption above), then b must be odd (otherwise the ratio could be further simplified by dividing by 2)
  • Since a is even, let a = 2y
  • Then a2 = 4y2 = 2b2
  • b2 = 2y2 so b2 must be even, therefore b is even
  • However we asserted b must be odd. Here is the contradiction.
The story goes that Hippasus made this discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans “…for having produced an element in the universe which denied the…doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.”

I've taken this mainly from wikipedia where there is much more detail about irrational number but have added a few extra bits of explanation to the reasoning.

1 comment:

Richard said...

When I was in grade school, I was trying to understand the proof of the irrationality of the square root of 2 in the textbook written by my grandfather. I got stuck at the greater and than less than signs so I asked my teacher who didn't understand them either. While I now believe either of my parents could have explained it completely, I just gave up. Now, some sixty odd years later I have gotten much better at trying repeatedly before giving up, but I hadn't yet learned that either.

Nobody threw me overboard, so I consider myself lucky. Still, we have things like the HIV fraud and the low-fat diet fraud and the presidential voting frauds of four and eight years ago which still get the truth tellers thrown overboard even in these allegedly enlightened times. Not too much progress is obvious to me, even after Karl Popper clarified the import of falsification to science.