\begin{definition}[Definition:Basis Expansion/Recurrence/Non-Recurring Part]
Let $b \in \N: b \ge 2$.
Let $x$ be a real number.
Let the basis expansion of $x$ in base $b$ be recurring:
:$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots d_r d_{r + 1} d_{r + 2} \ldots d_{r + p} d_{r + 1} d_{r + 2} \ldots d_{r + p } d_{r + 1} d_{r + 2} \ldots d_{r + p} d_{r + 1} \ldots}_b$
The '''non-recurring part''' of $x$  is:
:$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots d_r}$
\end{definition}