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.

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 $x$ is rational.