# 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.