Definition:Basis Expansion/Recurrence/Non-Recurring Part

Definition
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:
 * $\sqbrk {s \cdotp d_1 d_2 d_3 \ldots d_r}$

Also known as
The non-recurring part is also known as the non-periodic part and can also be seen presented unhyphenated: nonperiodic.

Also see

 * Definition:Recurring Part of Recurring Basis Expansion