Definition:Basis Expansion/Recurrence
Definition
Let $b \in \N: b \ge 2$.
Let $x$ be a real number.
Let the basis expansion of $x$ in base $b$ be:
- $\sqbrk {s \cdotp d_1 d_2 d_3 \ldots}_b$
Let there be a finite sequence of $p$ digits of $x$:
- $\tuple {d_{r + 1} d_{r + 1} \ldots d_{r + p} }$
such that for all $k \in \Z_{\ge 0}$ and for all $j \in \set {1, 2, \ldots, p}$:
- $d_{r + j + k p} = d_{r + j}$
where $p$ is the smallest $p$ to have this property.
That is, let $x$ be of the form:
- $\sqbrk {s \cdotp d_1 d_2 d_3 \ldots d_r d_{r + 1} d_{r + 2} \ldots d_{r + p} d_{r + 1} d_{r + 2} \ldots d_{r + p} d_{r + 1} d_{r + 2} \ldots d_{r + p} d_{r + 1} \ldots}_b$
That is, $\tuple {d_{r + 1} d_{r + 2} \ldots d_{r + p} }$ repeats from then on, or recurs.
Then $x$ is said to recur.
Non-Recurring Part
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:
- $\sqbrk {s \cdotp d_1 d_2 d_3 \ldots d_r}$
Recurring Part
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 recurring part of $x$ is:
- $\sqbrk {d_{r + 1} d_{r + 2} \ldots d_{r + p}}$
Period
The period of recurrence is the number of digits in the recurring part after which it repeats itself.
Notation
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.
Also known as
Such a recurring basis expansion, when in the conventional base $10$ representation, is often called a recurring decimal.
Also see
- Basis Expansion of Rational Number: $x$ either recurs or terminates if and only if $x$ is rational.