This involved firing up some rather long dormant mental machinery and saying "uhhh" for protracted periods, thus being unable to concentrate on rooting for the Suns and thereby possibly contributing to their loss. But, to a level of astonishment that even caused the game to pause and Amare Stoudemire to look through the screen at me and say "Whoa!", I did it!
I will delete the various durations of inarticulate hemming that came between each step:
- "So, if the square root of two was a fraction, it would be equal to x/y where both x and y would be integers."
- "That would mean that the square of x was twice the square of y."
- "So what I need to do is prove that a number that is twice the square of an integer can't be the square of another integer."
- "Well, any integer can be expressed as its prime factors, and the prime factors of the square of an integer are just going to be those prime factors twice. So the square of an integer always has an even number of prime factors"
- "Twice the square of an integer has to have an odd number of prime factors. Because its prime factors are going to be the prime factors of the square of the integer--an even number--and two."
- "If twice the square of an integer has an odd number of prime factors, then it isn't itself the square of any integer. So there are no integers x and y for which the square of x is twice the square of y. Q-E-[expletive deleted]-D."