Closed Form of One Minus Golden Mean

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

\(\ds \)

\(=\)

\(\ds \dfrac {1 - \sqrt 5} 2\)

$\blacksquare$