Definition:Golden Mean Number System

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Definition

The golden mean number system is a system for representing a non-negative real number $x$ by a sequence of zeroes and ones using the golden mean $\phi$ as a number base.


Equivalent Representations

Let $x \in \R_{\ge 0}$ have two representations $S_1$ and $S_2$ in the golden mean number system.

Then $S_1$ and $S_2$ are equivalent representations.


Simplest Form

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

Then $S$ is the simplest form for $x$ if and only if:

$(1): \quad$ No two adjacent $1$s appear in $S$
$(2): \quad S$ does not end with the infinite sequence $\cdotp \ldots 010101 \ldots$


Simplification

Let $x \in \R_{\ge 0}$ have a representation which includes the string $011$, say:

$x = p011q$

where $p$ and $q$ are strings in $\left\{ {0, 1}\right\}$.


From 100 in Golden Mean Number System is Equivalent to 011, $x$ can also be written as:

$x = p100q$


The expression $p100q$ is a simplification of $p011q$.


Expansion

Let $x \in \R_{\ge 0}$ have a representation which includes the string $100$, say:

$x = p100q$

where $p$ and $q$ are strings in $\left\{ {0, 1}\right\}$.


From 100 in Golden Mean Number System is Equivalent to 011, $x$ can also be written as:

$x = p011q$


The expression $p011q$ is an expansion of $p011q$.


Examples

Example: $100 \cdotp 1$

The number expressed in the golden mean number system as $100 \cdotp 1$ is:

\(\ds \left[{100 \cdotp 1}\right]_\phi\) \(=\) \(\ds \phi^2 + \phi^{-1}\)
\(\ds \) \(\approx\) \(\ds 3 \cdotp 236\)


Sources

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