Fractional and Non-Integer Number Bases

Recall that a number in base $b$ is an expression

$$a_ka_{k-1}\ldots a_1a_0a_{-1}a_{-2}\ldots,$$

where the $a_i$ are nonnegative integers less than $b$. So, for instance, the number $25 = 2\cdot 3^2 + 2\cdot 3 + 1\cdot 3^0$ is represented as $221$ in base $3$.

Although most applications of bases restrict $b$ to be a positive integer, it is worth considering what happens when $b$ is, more generally, a positive real number.

Let $b > 1$ be a real number. The $b$-expansion of a nonnegative real number $x$ is a representation

$$x = a_k b^k +a_{k-1}b^{k-1} + \cdots + a_0 + a_{-1}b^{-1} + \cdots,$$

where the $a_i$ are nonnegative integers less than $b$.

Note that $b > 1$ is required because there need to be some positive integers less than $b$ (otherwise there are no nonzero choices for the $a_i$).

From this definition, it is straightforward to deduce the following fact:

For any $b > 1$, every nonnegative real number has a $b$-expansion.

The algorithm to generate a $b$-expansion of a real number $x$ is the familiar one: first let $k$ be the integer satisfying $b^k \le x < b^{k+1}$, and then let $a_k =\left \lfloor\frac{ x}{b^k}\right\rfloor.$ Then $a_k < b$ and $x-a_kb^k < b^k$, so the process can be iterated: let $x' = x-a_kb^k$ and let $a_{k-1} = \left\lfloor \frac{x'}{b^{k-1}}\right \rfloor,$ and so on.

Let $\phi$ be the golden ratio $\frac{1+\sqrt{5}}2$. Find a $\phi$-expansion of $6$.

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First, $\phi^3 \le 6 < \phi^4$, so the expansion starts $6 = \phi^3+(6-\phi^3)$. Now $6-\phi^3 = 6-(2\phi+1) = 5-2\phi \approx 1.8$, so the expansion starts $6 = \phi^3 + \phi + (5-3\phi)$. Since $5-3\phi \approx 0.15$, this will need to be expressed using negative coefficients. In fact, $\phi^{-2} = 2-\phi$ so $\phi^{-4} = 4-4\phi+\phi^2 = 5-3\phi$, so the $\phi$-expansion of $6$ is $6 = \phi^3+\phi+\phi^{-4}$. In base notation, this is

$$6_{10} = 1010.0001_{\phi}.$$

Note that this representation is not unique: for example, since $\phi = 1 +\phi^{-1}$, it is also true that $6_{10} = 1001.1001_{\phi}.$ $_\square$

This illustrates a situation that is different for non-integer bases: finite (terminating) integer-base expansions are always unique $($but integer-base expansions are not unique, e.g. $1 = 0.9999\ldots$ in base $10$, as discussed here$).$ However, finite base-$b$ expansions may not be unique for algebraic integers $b$.