Main Question or Discussion Point
The following is a quote from Wikipedia on Irrational Numbers (the bold is mine):
If this is the case and there is no "small indivisible unit that could fit evenly into one of these lengths as well as the other", then how can Planck's Constant be true, unless it is a paradoxical number that is both odd and even at the same time, or neither odd nor even?The first proof of the existence of irrational numbers is usually attributed to a Pythagorean (possibly Hippasus of Metapontum), who probably discovered them while identifying sides of the pentagram. The then-current Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other. However, Hippasus, in the 5th century BC, was able to deduce that there was in fact no common unit of measure, and that the assertion of such an existence was in fact a contradiction. He did this by demonstrating that if the hypotenuse of an isosceles right triangle was indeed commensurable with an arm, then that unit of measure must be both odd and even, which is impossible. His reasoning is as follows:
* The ratio of the hypotenuse to an arm of an isosceles right triangle is c:b expressed in the smallest units possible.
* By the Pythagorean theorem: c2 = a2+b2 = 2b2. (Since the triangle is isosceles, a = b.)
* Since c2 is even, c must be even.
* Since c:b is in its lowest terms, b must be odd.
* Since c is even, let c = 2y.
* Then c2 = 4y2 = 2b2
* b2 = 2y2 so b2 must be even, therefore b is even.
* However we asserted b must be odd. Here is the contradiction.