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.