\begin{definition}[Definition:Basis Expansion/Recurrence/Notation]
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.
\end{definition}