If x+y is rational,must x and y be rational?

Question

Must x and y be rational

Answer 1

Let x = 2+√7 and y = 10-√7

 

Then x+y = 12 (a rational number), but x and y are both irrational.   

Answer 2

If y = -x, then x+y = 0 = 0/1 so that's rational, but both x and y could be irrational, as for example, when x =√2.

 

With a decent set of restrictions on x and y, such as x>y>0, it does seem likely that the claim is true.