Golden Mean by One Minus Golden Mean equals Minus 1
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Theorem
- $\phi \hat \phi = -1$
where:
- $\phi$ denotes the golden mean
- $\hat \phi := 1 - \phi$
Proof
\(\ds \phi\) | \(=\) | \(\ds \frac 1 {\phi - 1}\) | Definition 3 of Golden Mean | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \phi \paren {\phi - 1}\) | \(=\) | \(\ds 1\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \phi \paren {1 - \phi}\) | \(=\) | \(\ds -1\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \phi \hat \phi\) | \(=\) | \(\ds -1\) | by definition |
$\blacksquare$