Prove that sqrt(2) is not rational, that is, it cannot be
written a/b for any integers a,b.
assume a/b is in lowest terms. in particular, a and b are not both even.
now,
a^2/b^2 = 2
a^2 = 2 b^2
showing that a is even. let a = 2c, a^2 = 4 c^2. then
4 c^2 = 2 b^2
2 c^2 = b^2
showing that b is even, a contradiction!