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.