Prove that the sum of two irrational numbers need not be irrational and the product of two irrational numbers need not be irrational.
Proof. Let be irrational. Then, by an argument we made in I.3.12, Exercise #7, we know that and are both irrational. However,
Therefore, the sum and product of two irrationals need not be irrational
(Note: we could also just say is not irrational, but we have not yet established that is irrational. In fact, I’m not sure we’ve even established that an irrational number actually exists yet.)