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:
\(\ds \left[{100 \cdotp 1}\right]_\phi\) | \(=\) | \(\ds \phi^2 + \phi^{-1}\) | ||||||||||||
\(\ds \) | \(\approx\) | \(\ds 3 \cdotp 236\) |
Sources
- 1957: George Bergman: Number System with an Irrational Base (Math. Mag. Vol. 31, no. 2: pp. 98 – 110) www.jstor.org/stable/3029218
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers: Exercise $35$