TITLE: Is it in any way possible to work with reals modulo integer values?
QUESTION [10 upvotes]: I'm working with a strange and perhaps (mathematically) nonsensical realm.
I'd like to know if we can work with $\mathbb{R}/ \mathbb{Z}$.
For example, if we take $(\pi \bmod 3)$ we get $(\pi - 3)$.
I'm wondering if I always end up with the same equivalence class modulo 3 if I truncate the reals in this way.
In general, I'd like to know if I can work this way modulo any integer.  I want to know if this will work for addition, subtraction and multiplication.

REPLY [7 votes]: $\mathbb Z$ is a normal subgroup of $\mathbb R$ (as a group under addition), so the quotient group $\mathbb R/\mathbb Z$ makes sense in the usual way in group theory.  
$\mathbb Z$ is not an ideal of $\mathbb R$ (as a ring with addition and multiplication), so $\mathbb R/\mathbb Z$ does not make sense in the usual way in ring theory.