So far I that for any irrational number without a real part (that $-n=\overline{n}$) plus/minus any irrational number with the same restrictions equals another irrational number. However, I want to know if there is a concrete rule that states what the properties of a sum/difference made up of irrational numbers. My intuition says that two irrational numbers will only add up to a rational number if their irrational parts are opposites (for instance $\sqrt{2}$ and $-\sqrt{2}$), but I'm not really sure.
Thank you!
MAJOR EDIT: I'm sorry, I forgot to mention something very important. The irrational numbers in this problem are all in the form $a+\sqrt{b}$ where a and b are both positive integers and $\sqrt{b}$ is irrational.