Two specific irrational numbers summing to rational number that is not zero?

Question

There already exist pages that show how two irrational numbers can sum to a rational number. However, is there an actual example of this? What are some of irrational numbers $x$ and $y$ such that $x + y$ is rational and $x ≠ -y$ (for an example $\sqrt{2}$ and -$\sqrt{2}$ doesn't count)?

If there is an example of an irrational number $y$ for every irrational number $x$ that will sum to a rational number, does this property apply to all irrational numbers?

Answer 1

For any $x \in \mathbb R$ and any $q \in \mathbb Q$, since $\mathbb R$ contains $\mathbb Q$, $y = q-x$ will satisfy $y+x\in \mathbb Q$.

For any algebraic number, it is possible to find the minimal polynomial over $\mathbb Q$ and thus the algebraic conjugates of the number, a pile of numbers that, along with $x$, both add up to a rational number and multiply together to a rational number!

Bad news though: the algebraic conjugate of $\sqrt{2}$ specifically is $-\sqrt{2}$.

Answer 2

Here is a simple example: $$\frac 1{\sqrt n-1}-\frac1{\sqrt n+1}=\frac 2{n-1},$$ so choose $n$ such that $\sqrt n$ is irrational.