My group theory textbook gives the following definition for the real numbers mod 1
Let $G = \{x \in \mathbb{R} \mid 0 \leq x < 1\}$ and for $x, y \in G$ let $x + y$ be the fractional part of $x + y$ (i.e., $x + y = x + y - [x + y]$) where $[a]$ is the greatest integer less than or equal to $a$)
At first I thought this will just be $\mathbb{R}/[0, 1]$ but this is nonesense because $[0, 1]$ is not even a subgroup of $(\mathbb{R}, +)$. Is there any way to define the real numbers mod 1 in terms of quotient groups (or any other concept if it helps provide insight)?