# Definition:Basis Expansion/Recurrence/Period

## Definition

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

Let $x$ be a rational number.

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

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

From Basis Expansion of Rational Number, $x$ either terminates or recurs.

The period of recurrence is the number of digits in the recurring part after which it repeats itself.