Definition:Basis Expansion

Definition

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:

$\forall n \in \N: n \ge 1: \left \langle {f_n}\right \rangle = \begin{cases} b \left({x - \left \lfloor {x} \right \rfloor}\right) & : n = 1 \\ b \left({f_{n-1} - \left \lfloor {f_{n-1}} \right \rfloor}\right) & : n > 1 \end{cases}$

Then we define:

$\forall n \in \N: n \ge 1: \left \langle {d_n}\right \rangle = \left \lfloor {f_n} \right \rfloor$

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 $x = \left \lfloor {x} \right \rfloor + \sum_{j \ge 1} \frac {d_j} {b^j}$ as:

$\left[{s . d_1 d_2 d_3 \ldots}\right]_b$

where $s = \left \lfloor {x} \right \rfloor$.

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.

If it is the case, for a given $x$ and $b$, that $\exists m \in \N: d_m = 0$ then the expansion is said to terminate.

Note that it is far from guaranteed that the sequence $\left \langle {d_n}\right \rangle$ will actually terminate.

Negative Real Numbers

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

We take the absolute value $y$ of $x$, i.e. $y = \left|{x}\right|$.

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

$\left[{s . d_1 d_2 d_3 \ldots}\right]_b$

where $s = \left \lfloor {x} \right \rfloor$.

Then:

$y = -\left[{s . d_1 d_2 d_3 \ldots}\right]_b$

Also see

• The Existence of Base-N Representation‎ for a rigorous proof that this expansion always exists and (except in a particular case) is unique.

• The Basis Representation Theorem for the equivalent theorem for integers.