Suppose that m/n+√2(p/q) is rational.
Then the following can be written
m/n+√2(p/q) = r/s
where m, n, p, q, r and s are all integers, and n, q and s are non-zero.
Rearranging the equation gives
√2 = q(rn−ms)/(pns).
√2 is irrational and q(rn−ms)/(pns) is rational q(rn−ms) and (pns) are integers.
So the hypothesis that m/n+√2(p/q) was rational is wrong.
