# Fractional and Non-Integer Number Bases

Recall that a number in base \( b\) is an expression \[ a_ka_{k-1}\cdots a_1a_0a_{-1}a_{-2}\cdots \] 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-expansionof 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 \( a_k = \lfloor x/b^k\rfloor \). 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 \( a_{k-1} = \lfloor x'/b^{k-1} \rfloor\), and so on.

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

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 \( 6_{10} = 1001.1001_{\phi}. \)

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. \( 1 = 0.9999\cdots \) in base \( 10 \), as discussed here). However, finite base-\(b\) expansions may not be unique for algebraic integers \( b \).

Let \( \phi = \dfrac{1+\sqrt{5}}2 \). Which of these is a representation of \( \dfrac \phi2 \) in base \( \phi \)?

**Cite as:**Fractional and Non-Integer Number Bases.

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