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.
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.