1. The problem statement, all variables and given/known data Prove: For every rational number z, there exists irrational numbers x and y such that x + y = z. 2. Relevant equations by definition, a rational number can be represented by ratio of two integers, p/q. 3. The attempt at a solution Is there a way to do this by contraposition? Would the contraposition be, For all rational numbers x and y, there does not exist an irrational number z such that x + y = z? I can handle from there, but I don't think my contraposition is correct.