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.