So, I know how the sum of 2 irractional numbers as a whole can be rational. For example, an irrational number $a$ and it's negative counterpart $-a$ have a sum of zero and so the sum is rational. But what if both of the terms had to be positive? Is there a way to make it rational?
My own sort of answer: I think that you can split a number into something like $\sqrt{2}$ and $2 - \sqrt{2}$ and those 2 would sum up to a rational number $2$. Would that be an acceptable answer though, and if so how would we prove that the difference of a rational number and an irrational number is irrational?
Let $r_1$ and $r_2$ be two rational numbers and let $i$ be an irrational number. If possible, let $$r_1-i=r_2$$ This implies $$r_1-r_2=i$$ But, we know the difference of two rationals must be a rational number. This gives us a contradiction.