tldr version: The numerator must be even and the denominator must be odd. The square of this taken modulo 4 must therefore have numerator = 0 mod 4, denominator = 1 mod 4 (1x1 and -1x-1 both become 1 mod 4). 2 times 1 mod 4 cannot be 0 mod 4. Therefore sqrt(2) cannot be rational.