Representations for 1 in Golden Mean Number System/Examples/0.11
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Example of Representations for 1 in Golden Mean Number System
$1$ can be represented in the golden mean number system as $\sqbrk {0 \cdotp 11}_\phi$.
Proof
| \(\ds \sqbrk {0 \cdotp 11}_\phi\) | \(=\) | \(\ds \phi^{-1} + \phi^{-2}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 \phi + \dfrac 1 {\phi^2}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\phi^2 + \phi} {\paren {\phi^2} \phi}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\phi \paren {\phi + 1} } {\paren {1 + \phi} \phi}\) | Square of Golden Mean equals One plus Golden Mean | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1\) |
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers: Exercise $35$