Representations for 1 in Golden Mean Number System

Theorem

Then there are infinitely many ways to express the number $1$ in the golden mean number system.

We have that:

$\phi^0 = 1$

and so $1$ has the representation $S_1$ as:

$S_1 := \left[{1 \cdotp 00}\right]_\phi$

By inspection it is seen that $S_1$ is the simplest form of $1$.

$S_2 := \left[{0 \cdotp 11}\right]_\phi$

which can then be expressed as:

$S_2 := \left[{0 \cdotp 1100}\right]_\phi$

$S_3 := \left[{0 \cdotp 1011}\right]_\phi$

and so:

$S_4 := \left[{0 \cdotp 101011}\right]_\phi$

Continuing in this way we obtain an infinite sequence $\left\langle{S_n}\right\rangle$ all of which are a representation of $1$.

$\blacksquare$

$1$ can be represented in the golden mean number system as $\sqbrk {0 \cdotp 11}_\phi$.

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