Square of Golden Mean equals One plus Golden Mean
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Theorem
- $\phi^2 = \phi + 1$
where $\phi$ denotes the golden mean.
Decimal Expansion
The decimal expansion of $\phi^2$ is given as:
- $\phi^2 \approx 2 \cdotp 61803 \, 39887 \, 49894$
Thus the square of the golden mean is the unique number $n$ such that:
- $\sqrt n = n - 1$
Proof
\(\ds \phi\) | \(=\) | \(\ds \frac 1 {\phi - 1}\) | Definition 3 of Golden Mean | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \phi \paren {\phi - 1}\) | \(=\) | \(\ds 1\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \phi^2\) | \(=\) | \(\ds \phi + 1\) |
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1 \cdotp 61803 \, 39887 \, 49894 \, 84820 \, 45868 \, 34365 \, 63811 \, 77203 \, 09179 \, 80576 \ldots$
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.1$: Mathematical Induction
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1 \cdotp 61803 \, 39887 \, 49894 \, 84820 \, 45868 \, 34365 \, 63811 \, 77203 \, 09179 \, 80576 \ldots$