Are there any non-trivial counterexamples to the non-closure of the irrational numbers over addition? It is trivial to show that the set of irrational numbers is not closed under addition. Just choose an irrational number $p$ and add it to its additive inverse $-p$ to get $0\in\mathbb{Q}$. However, I have yet to see a (non-trivial) example of a rational sum $(p + q) \in \mathbb{Q}$ of two irrational numbers $p$ and $q$ where $q \ne -p$. Can anyone provide such an example, or is this not possible? A similar question might be whether the set of positive irrational numbers is closed under addition.

$$
\underbrace{{}\quad\pi\quad{}}_{\text{an irrational number}} + \underbrace{\Big( 10-\pi\Big)}_{\text{an irrational number}} \text{is rational.}
$$
Any time you have $x$ and $y$ both positive irrational numbers and $x+y=r$ rational, then clearly you have $y = r-x$, so necessarily your pair is $x$ and $r-x$.
