1. The problem statement, all variables and given/known data Prove that if a and b are rational numbers with a≠b then a+(1/√2)(b-a) is irrational. 2. Relevant equations 3. The attempt at a solution Assume that a+(1/√2)(b-a) is rational. then by definition of rationality a+(1/√2)(b-a) =p/q for some integers p&q so a+(b-a)/√2 =p/q a(1+(b-1)/√2) = p/q so (p/qa) -1= (b-1)/√2 qa/p-1= √2/(b-1) so √2 = (b-1)((qa/p) -1) but (b-1)((qa/p) -1) is rational since b,1,q,a,p are all integers and the sum, difference and products of integers are integers. but √2 is not an integer. Contradiction a+(1/√2)(b-a) must be irrational.