\begin{definition}[Definition:Decimal Expansion]
Let $x \in \R$ be a real number.
The '''decimal expansion''' of $x$ is the expansion of $x$ in base $10$.
$x = \floor x + \ds \sum_{j \mathop \ge 1} \frac {d_j} {10^j}$:
:$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots}_{10}$
where:
:$s = \floor x$, the floor of $x$
:it is not the case that there exists $m \in \N$ such that $d_M = 9$ for all $M \ge m$.
(That is, the sequence of digits does not end with an infinite sequence of $9$s.)
\end{definition}