Non-trivial examples of irrational numbers $x, y$ such that $x - y$ is rational.

Question

If we factor the additive group $\mathbb R$ by $\mathbb Q$ we will have the quotient group $\mathbb R / \mathbb Q$ where $\mathbb Q$ is the identity element. And I try to find non-trivial examples of irrational numbers which are in the same coset (i.e $x - y \in \mathbb Q$). Yet I am not able to come with one.

What i mean when i say "trivial" is like $x = y$ or $x = \sqrt{2}$ and $y = \sqrt{2} - 2$.

Answer 1

There aren't going to be any "non-trivial" examples. Whatever we try, we will have $y= x-q$ for some rational number $q$. We might use two different ways of describing $x$ to camouflage this fact (e.g. $x=\sqrt2, y=\frac{\sqrt8}{2}-2$ is a simple example), or we can cleverly hide $q$ (an example given in the comments above), but ultimately we can't change this fact.

Answer 2

You can find many examples also using appropriately the identity $\cos^2{x}+\sin^2{x}=1$.

Or the identity $\ln{\frac{x}{y}}=\ln{x}-\ln{y}$

For instance take $x=2e^3$ and $2e^2$