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

< Definition:Basis Expansion | Recurrence(Redirected from Definition:Non-Recurring Part of Recurring Basis Expansion)

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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**.