# Definition:Basis Expansion/Recurrence/Notation

## Definition

Let $b \in \N: b \ge 2$.

Let $x$ be a real number.

Let the basis expansion of $x$ in base $b$ be:

$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots}_b$

such that $x$ is recurring.

Let the non-recurring part of $x$ be:

$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots d_r}_b$

Let the recurring part of $x$ be:

$\sqbrk {\ldots d_{r + 1} d_{r + 2} \ldots d_{r + p} \ldots}_b$

Then $x$ is denoted:

$x = s.d_1 d_2 d_3 \ldots d_r \dot d_{r + 1} d_{r + 2} \ldots \dot d_{r + p}$

That is, a dot is placed over the first and last digit of the first instance of the recurring part.