Conversion of Number in Golden Mean Number System to Simplest Form

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Theorem

Let $x \in \R_{\ge 0}$ have a representation $S$ in the golden mean number system.

Then $S$ can be converted to its simplest form as follows:

$(1): \quad$ Replace any infinite string on the right hand end of $S$ of the form $010101 \ldots$ with $100$
$(2): \quad$ Repeatedly replace the leftmost instance of $011$ with $100$.


Proof

Note that step $(2)$ is an instance of a simplification of $S$.

From 100 in Golden Mean Number System is Equivalent to 011, it has been established that $011$ is equivalent to $100$.


The following need to be established:

$010101 \ldots$ is equivalent to $100$
Replacing the leftmost $011$ with $100$ reduces the overall number of instances of $011$.



Let the first $1$ of $010101 \ldots$ represent the instance of $\phi^n$ for some $n \in \Z$.

It follows that the first $0$ of $010101 \ldots$ represents the instance of $\phi^{n + 1}$ for some $n \in \Z$.


Thus $010101 \ldots$ represents the real number $y$ where:

$y = \phi^n + \phi^{n - 2} + \phi^{n - 4} + \cdots$

and so:

\(\displaystyle y\) \(=\) \(\displaystyle \phi^n \displaystyle \sum_{k \mathop \ge 0} \phi^{-2 k}\)
\(\displaystyle \) \(=\) \(\displaystyle \phi^n \dfrac 1 {1 - \phi^{-2} }\) Sum of Infinite Geometric Progression
\(\displaystyle \) \(=\) \(\displaystyle \phi^n \dfrac {\phi^2} {\phi^2 - 1}\)
\(\displaystyle \) \(=\) \(\displaystyle \phi^n \dfrac {\phi^2} {\left({\phi + 1}\right) - 1}\) Square of Golden Mean equals One plus Golden Mean
\(\displaystyle \) \(=\) \(\displaystyle \phi^{n + 1}\) after simplification

This can represented in the golden mean number system by $100$, where the $1$ corresponds to the first instance of $\phi^{n + 1}$.

Hence $010101 \ldots$ is equivalent to $100$.

$\Box$



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