Foundations of the golden ratio base

Positional numeration systems have come to dominate mathematics, with the ubiquitous base-ten number system used nearly universally. In addition to base-ten, other bases such as base-two and base-sixteen have found widespread usage (for example in computer engineering). We review a particularly novel take on the positional numeration system: the golden ratio base, first introduced by George Bergman in 1957, who was a 12 year old junior high student at the time. We shall prove that the number system is correct, starting with basic properties of the golden ratio up to proofs of the existence and uniqueness of representations for certain classes of numbers, which rely on algebraic number theory. In addition we will introduce simple algorithms for performing arithmetic in the system.

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