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