assume 3/sqrt(2) is rational, by doing this we can express it as p/q (q≠0) (p and q have no common factors except 1)
3/sqrt(2) = p/q
if we square both sides we get 9/2 = p^2/q^2
cross multiply to get 9q^2 = 2p^2
by doing this, it is implied that 9q^2 is even, as any odd number squared is also odd. therefore q^2 is also even, so q must also be even.
let q = 2k, where k is an integer. we can substitute back to get 9(2k^2) = 2p^2 so 36k^2 = 2p^2
18k^2 = p^2
this implies that p^2 is even so p must also be even. however, if both p and q are even, then they have a common factor of 2 therefore the fraction p/q can be simplified, which contradicts the original assumption that p and q have no common factors except 1.
since assuming 3/sqrt(2) is rational leads to a contradiction, it must be irrational
