Definition:Basis Expansion/Negative Real Numbers

Definition

Let $x \in \R: x < 0$.

We take the absolute value $y$ of $x$, that is:

$y = \size x$

Then we take the expansion of $y$ in base $b$:

$\size {s . d_1 d_2 d_3 \ldots}_b$

where $s = \floor y$.

Finally, the expansion of $x$ in base $b$ is defined as:

$-\sqbrk {s . d_1 d_2 d_3 \ldots}_b$

Termination

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.

Recurrence

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.

Sources

• 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: The Continuum