Definition:Group of Rationals Modulo One

Definition
Define a relation $\sim$ on $\Q$ such that:


 * $\forall p, q \in \Q: p \sim q \iff p - q \in \Z$

Then $\left({\Q / \sim, +}\right)$ is a group referred to as the group of rationals modulo one.

That is, it is the quotient group $\Q / \Z$.

Also see

 * Group of Rationals Modulo One is Group, where this is proven to be a group.
 * Real Numbers under Addition Modulo 1 form Group