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 $011$.


The validity of the material on this page is questionable.
In particular: Except it doesn't, necessarily -- for example it converts $01011$ to $01100$. Something more sophisticated is needed.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues.
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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:

\(\ds y\) \(=\) \(\ds \phi^n \ds \sum_{k \mathop \ge 0} \phi^{-2 k}\)
\(\ds \) \(=\) \(\ds \phi^n \dfrac 1 {1 - \phi^{-2} }\) Sum of Infinite Geometric Sequence
\(\ds \) \(=\) \(\ds \phi^n \dfrac {\phi^2} {\phi^2 - 1}\)
\(\ds \) \(=\) \(\ds \phi^n \dfrac {\phi^2} {\paren {\phi + 1} - 1}\) Square of Golden Mean equals One plus Golden Mean
\(\ds \) \(=\) \(\ds \phi^{n + 1}\) 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 $100$.

$\Box$


This needs considerable tedious hard slog to complete it.
In particular: so far so good, but I'm going to the beach.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Finish}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.


Sources

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