I am trying to better understand the group $\mathbb{R/Q}$. It's unclear to me when two irrational numbers will give the same coset of $\mathbb{Q}$, but I know that this must happen since, for example $\pi+1 = (\pi-1)+2$, meaning the cosets $\pi\mathbb{Q}$ and $(\pi−1)\mathbb{Q}$ share an element and thus are equivalent. Can we describe a set of irrational numbers that give each coset of $\mathbb{Q}$ exactly once in a way that, given an irrational number, we would be able to say whether or not it's in the set?
Edit: I am also interested in understanding this group in other ways. What is it's order? Are there any groups it's isomorphic to?
$\mathbb{R/Z}$ is isomorphic to the unit circle $S^1$, the set of all complex numbers of absolute value $1$, seen as a multiplicative group.
$\mathbb{Q/Z}$ is isomorphic to the torsion subgroup of $S^1$, formed by the elements of finite order, that is, all roots of unity.
$\mathbb{R/Q}$ is thus isomorphic to $S^1/tor(S^1)$, a very large and complicated group. For instance, the powers of every element form a dense subset.
Another approach is to consider $\mathbb{R}$ as a vector space over $\mathbb{Q}$. The dimension here is the cardinality of $\mathbb{R}$ and so $\mathbb{R/Q}$ is essentially the same as $\mathbb{R}$.