Algodoo

In a number system with base φ (the golden ratio), the way numbers are expressed and converted follows a unique set of rules, mainly using the digits 0 and 1 and avoiding consecutive '11's.

This system is also called the Golden ratio base, or phinary.

In the golden ratio base, to make sure each number has a unique representation, we use what's called the 'standard form'. The rules are pretty simple:

1. Only use the digits 0 and 1.
2. Never have consecutive '11's anywhere, like in 10.01.

Any representation that has '11' or other digits (like 2) can be converted into the standard form using algebraic rules.

The idea behind this rule comes from a basic property of the golden ratio φ itself: φ squared equals φ plus 1. So we can turn consecutive '11's into '100'.

The picture on the cover shows the numbers 0 to 10 represented in both decimal and golden ratio base.