Reciprocal of Golden Mean
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Theorem
| \(\ds \dfrac 1 \phi\) | \(=\) | \(\ds \phi - 1\) | where $\phi$ denotes the Golden Mean | |||||||||||
| \(\ds \) | \(\approx\) | \(\ds 0 \cdotp 61803 \, 3988 \ldots\) |
This sequence is A094214 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
| \(\ds \phi\) | \(=\) | \(\ds \dfrac 1 {\phi - 1}\) | Definition 3 of Golden Mean | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \dfrac 1 \phi\) | \(=\) | \(\ds \phi - 1\) | taking reciprocals of both sides |
The numeric approximation follows from the decimal expansion of the Golden Mean:
The decimal expansion of the golden mean starts:
- $\phi \approx 1 \cdotp 61803 \, 39887 \, 49894 \, 84820 \, 45868 \, 34365 \, 63811 \, 77203 \, 09179 \, 80576 \ldots$
$\blacksquare$
Sources
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,61803 3988 \ldots$