Determining equivalence classes on $\Bbb{R}$.

Question

Say we have the following equivalence relation on $\Bbb{R}$: $$a\sim b\iff a-b\in\Bbb{Q}$$ What do the equivalence classes look like? On a preliminary investigation I got the following equivalence classes: $\Bbb{Q}$ and $\Bbb{Q}+r$, where $r$ is any irrational number. Are these the only equivalence classes?

Answer 1

Any equivalent class is of the form $r+\mathbb Q$ where $r\in \mathbb R$. But, of course, many equivalence classes are thus repeated. What you describe still contains duplications. For instance, $\sqrt2+\mathbb Q$ and $(\sqrt 2 + 2)+ \mathbb Q$ are both listed in what you give, since the representatives are both irrational, but they are equal equivalence classes, since the difference between the representatives is rational.

So, the answer to "how does a typical equivalence class look like" is simply: $r+\mathbb Q$ where $r\in \mathbb R$. If you want to know what do all equivalences classes, as a single object, look like, then that's much more complicated.

Basically, you are looking at the quotient $\mathbb R/\mathbb Q$. If you consider these as group, then the quotient is, of course, a group. Are you interested in the properties of this group?