Definition:Golden Mean Number System

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:

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