Golden Mean by One Minus Golden Mean equals Minus 1

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

$\blacksquare$