Sum of two irrational numbers being rational or irrational

Question

I am currently doing a project on irrational and transcendental numbers and part of this project requires me to look at sums and products of irrational numbers.

I am aware that the sum of 2 irrational numbers can be rational or irrational but was wondering if anyone knew of a definite way to look at the numbers and say if their sum/product will be rational/irrational. Is there some sort of theorem than can be applied or is the only way of knowing just working it out?

Thanks in advance for any help.

Answer 1

No, there is not. If there was, we would know whether $e+\pi$ is rational or not. But, in fact, that's an open problem.

Answer 2

"requires me to look at sums and products of irrational numbers." so you don't ask for transcendental?

The sum or product of two irrational algebraic numbers is not necessarily irrational. Counterexamples:

$$ (2 +\sqrt 2)+(2 -\sqrt 2)= 4\\ (2 +\sqrt 2)(2 -\sqrt 2)= 2 $$