Question

Prove that root 2 is not a rational number

Answer 1

Assume that sqrt(2) is a rational number, i.e. where a and b are integers and a/b is irreducible. Squaring both sides, this implies that since the LHS is even, then the RHS is also even, and a is a multiple of 2. We can write 2k instead of a: Similarly, we can write b as 2m for some integer m. This contradicts our original statements, because this would imply a = 2k, b = 2m and they are not in simplest form (plus, we can apply this technique infinitely many times -- also not good). Hence we have a contradiction and sqrt(2) is irrational.