# Addition of 1 in Golden Mean Number System

## Algorithm

Let $x \in \R$ be a real number.

The following algorithm performs the operation of addition of $1$ to $x$ in the golden mean number system.

Let $S$ be the representation of $x$ in the golden mean number system in its simplest form.

**Step $1$**: Is the digit immediately to the left of the radix point a zero?

- If
**Yes**, replace that $0$ with $1$. Go to**Step $4$**.

- If

- If
**No**, set $m = 2$ and go to**Step $2$**.

- If

**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

- If
**No**, add $2$ to $m$. Repeat**Step $2$**.

- If

**Step $3$**: Is $m = 0$?

- If
**Yes**, set the digit immediately to the left of the radix point from $0$ to $1$. Go to**Step $4$**.

- If

- If
**No**, go to**Step $2$**.

- If

**Step $4$**: Convert $S$ to its simplest form.**Stop**.

## Proof

The above constitutes an algorithm, for the following reasons:

### Finiteness

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.

### Definiteness

Each step can be seen to be precisely defined.

### Inputs

The only input to the algorithm is the representation $S$ of $x$.

### Outputs

The only output from the algorithm is the representation $S$ of $x + 1$.

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 $3$**.

- In either step, it happens only once, after which the algorithm terminates.

### Effective

Each step is basic enough to be done exactly and predictably.

$\blacksquare$

## Sources

- 1957: George Bergman:
*Number System with an Irrational Base*(*Math. Mag.***Vol. 31**,*no. 2*: pp. 98 – 110) www.jstor.org/stable/3029218