Definition:Basis Expansion/Positive Real Numbers
Definition
Let $x \in \R$ be a real number such that $x \ge 0$.
Let $b \in \N: b \ge 2$.
Let us define the recursive sequence:
- $\forall n \in \N: n \ge 1: \sequence {f_n} = \begin {cases} b \paren {x - \floor x} & : n = 1 \\ b \paren {f_{n - 1} - \floor {f_{n - 1} } } & : n > 1 \end{cases}$
Then we define:
- $\forall n \in \N: n \ge 1: \sequence {d_n} = \floor {f_n}$
It follows from the method of construction and the definition of the floor function that:
- $\forall n: 0 \le f_n < b$ and hence $\forall n: 0 \le d_n \le b - 1$
- $\forall n: f_n = 0 \implies f_{n + 1} = 0$ and hence $d_{n + 1} = 0$.
Hence we can express $x = \floor x + \displaystyle \sum_{j \mathop \ge 1} \frac {d_j} {b^j}$ as:
- $\sqbrk {s \cdotp d_1 d_2 d_3 \ldots}_b$
where:
- $s = \floor x$
- it is not the case that there exists $m \in \N$ such that $d_M = b - 1$ for all $M \ge m$.
(That is, the sequence of digits does not end with an infinite sequence of $b - 1$.)
This is called the expansion of $x$ in base $b$.
The generic term for such an expansion is a basis expansion.
It follows from the Division Theorem that for a given $b$ and $x$ this expansion is unique.
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