Heres two problems from an A Level related paper: prove that if pq is irrational then atleast one of p or q is irrational. Also prove that if if p + q is irrational then atleast one of p or q is irrational. These two proofs are trivial proof by contradiction problems but it got me thinking more about cases where operations on two irrational numbers result in irrational or rational numbers Now whats interesting are the inverses of the statements, i.e. if atleast one of p or q is irrational then pq is irrational and if atleast one of p or q is irrational then p+q is irrational. I can think of two seemingly counter examples to both but they still leave some questions. To the first clearly anything of the form (sqrt(a))^2 is counterexample and the second something like (a + irrational number)+(a - irrational number). The question that is remaining is: Other than the case of square roots squared and any other special case, can two normal different irrational numbers multiply to give a rational number? And can two irrational numbers add or subtract to give a irrational number which does not involve cancellation of the irrational parts? The answer seems to be an intuitive "no" because both cant involve cancellation but heres a way of looking at things that just makes everything confusing, consider the special case of like (sqrt2)^2, we can break it into the sum of a rational and irrational part namely (1+ir)^2 where ir is the irrational part then (1+ir)^2 = 1 +2ir + ir^2 = 2 Now heres two seemingly irrational numbers 2ir +ir^2 that give a rational, whats going on? What about subtraction and division of irrationals?