Representations for 1 in Golden Mean Number System/Examples/0.01111...

Example of Representations for 1 in Golden Mean Number System

$1$ can be represented in the golden mean number system as $\left[{0 \cdotp 01111 \ldots}\right]_\phi$.

$\ds \sqbrk {0 \cdotp 01111 \ldots}_\phi$

$=$

$\ds \phi^{-2} + \phi^{-3} + \phi^{-4} + \phi^{-5} + \cdots$

$\ds $

$=$

$\ds \dfrac 1 {\phi^2} \paren {1 + \phi^{-1} + \dfrac 1 {\phi^2} + \dfrac 1 {\phi^3} }$

$\ds $

$=$

$\ds \dfrac 1 {\phi^2} \dfrac 1 {1 - \phi^{-1} }$

$\ds $

$=$

$\ds \dfrac 1 {\phi^2} \dfrac \phi {\phi - 1}$

$\ds $

$=$

$\ds \dfrac 1 \phi \dfrac 1 {\phi - 1}$

$\ds $

$=$

$\ds \dfrac \phi \phi$

Definition 2 of Golden Mean

$\ds $

$=$

$\ds 1$

$\blacksquare$