Closed Form of One Minus Golden Mean
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Theorem
- $\hat \phi = \dfrac {1 - \sqrt 5} 2$
where:
- $\hat \phi$ denotes one minus the golden mean: $\hat \phi = 1 - \phi$.
Proof
\(\ds \hat \phi\) | \(=\) | \(\ds 1 - \phi\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 - \dfrac {1 + \sqrt 5} 2\) | Definition 2 of Golden Mean | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {2 - \left({1 + \sqrt 5}\right)} 2\) | common denominator | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {1 - \sqrt 5} 2\) |
$\blacksquare$