Category:Basis Expansions
This category contains results about Basis Expansions.
Definitions specific to this category can be found in Definitions/Basis Expansions.
Positive Real Numbers
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.
Negative Real Numbers
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:
- $-\sqbrk {s . d_1 d_2 d_3 \ldots}_b$
Subcategories
This category has the following 3 subcategories, out of 3 total.
Pages in category "Basis Expansions"
The following 3 pages are in this category, out of 3 total.