Is it in any way possible to work with reals modulo integer values?
Solution 1:
$\mathbb Z$ 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.