Equivalence relation by mapping/R/Floor/Digits/Example

We consider the floor of a real number, that is, the mapping

$\lfloor \,\,\rfloor \colon \mathbb {R} \longrightarrow \mathbb {Z} ,t\longmapsto \lfloor t\rfloor .$

A real number ${}t$ is mapped to the largest integer number that is smaller or equal ${}t$. Here, the interval ${}[n,n+1)$ (with integer bound ${}n$, closed on the left, open on the right) is mapped to ${}n\in \mathbb {Z}$. With respect to this mapping, two real numbers are equivalent if and only if they lie in the same interval of this type. In the decimal expansion, this means that their digits before the point are identical.

We can also consider the digits after the point. This corresponds to the mapping

$\mathbb {R} \longrightarrow [0,1),t\longmapsto t-\lfloor t\rfloor .$

Under the equivalence relation defined by this mapping, two real numbers are equivalent if their digits after the point are identical. This is the case if and only if their difference is an integer number.