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.

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

This article, or a section of it, needs explaining. In particular: $\Q$ and $\Z$ themselves are not groups as such until you give them an operation. Also, it needs to be demonstrated. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

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