# Definition:Group of Rationals Modulo One

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## 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

## Sources

- 1974: Thomas W. Hungerford:
*Algebra*... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups