Why is $\mathbb{Q}/ \mathbb{Z}$ called the groups of rationals modulo one?

Question

In the book of Algebra by Hungerford, at page 27, it is given that

The following relation on $(\mathbb{Q}, +)$ $$aR b \quad iff \quad a-b \in \mathbb{Z},$$ defined a congruence relation, and the set of equivalence classes $\mathbb{Q}/ \mathbb{Z}$ is a group called the groups of rationals modulo one.

Question:

Why is $\mathbb{Q}/ \mathbb{Z}$ called the groups of rationals modulo one ?

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Is it because we can identify all the equivalence classes with the rationals between $(0,1)$ plus the number $1$ ? (of course if this is true)

Answer 1

If you make an analogy between this and modulo in integers, recall that modulo $n$ means $\Bbb Z$ is taken quotient over the the (normal) subgroup generated by the element $n$.

So in this case of rational numbers, you take quotient of $\Bbb Q$ over the (normal) subgroup generated by the element $1$. This subgroup generated by $1$ is precisely $\Bbb Z$.