# Definition:Basis Expansion/Termination

## Definition

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

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

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

Let it be the case that:

$\exists m \in \N: \forall k \ge m: d_k = 0$

That is, every digit of $x$ in base $b$ after a certain point is zero.

Then $x$ is said to terminate.