Definition:Basis Expansion/Recurrence/Non-Recurring Part
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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.