Addition of 1 in Golden Mean Number System
Let $x \in \R$ be a real number.
- If Yes, replace that $0$ with $1$. Go to Step $4$.
- If No, set $m = 2$ and go to Step $2$.
- Step $2$: Does the $m$th place after the radix point contain $0$?
- If Yes, expand the $100$ in the $3$ places ending in the $m$th place with $011$. Subtract $2$ from $m$. Go to Step $3$.
- If No, add $2$ to $m$. Repeat Step $2$.
- Step $3$: Is $m = 0$?
- If No, go to Step $2$.
- Step $4$: Convert $S$ to its simplest form. Stop.
The above constitutes an algorithm, for the following reasons:
The only case in which it is possible for the process not to terminate is if the $m$th place never contains $0$.
This can only happen if $S$ ends in an infinite string $01010101 \ldots$
But if this is the case, $S$ is not in its simplest form.
Each step can be seen to be precisely defined.
All operations that change $S$ are of the following nature:
- $(1): \quad$ Simplification of $S$, which does not change $x$, which happens if at all in Step $4$.
- $(2): \quad$ Expansion of $S$, which does not change $x$, which happens if at all in Step $2$.
- $(3): \quad$ Setting the digit corresponding to $\phi^0$ to $1$ from $0$, which happens either in Step $1$ or in Step $3$.
Each step is basic enough to be done exactly and predictably.