Golden ratio base

Golden ratio base is a non-integer positional numeral system that uses the golden ratio (the irrational number

1

2

(

1

+

5

)

{\displaystyle {\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}{\bigr )}}

≈ 1.61803... symbolized by the Greek letter φ) as its base. It is sometimes referred to as base-φ, golden mean base, phi-base, or, colloquially, phinary. Any non-negative real number can be represented as a base-φ numeral using only the digits 0 and 1, and avoiding the digit sequence "11" – this is called a standard form. A base-φ numeral that includes the digit sequence "11" can always be rewritten in standard form, using the algebraic property of the base φ that φn + φn − 1 = φn + 1. For instance, 11φ = 100φ.

Despite using an irrational number base, when using standard form, all non-negative integers have a unique representation as a terminating (finite) base-φ expansion. The set of numbers which possess a finite base-φ representation is the ring Z[φ] of golden integers; it plays the same role in this numeral systems as dyadic rationals play in binary numbers, providing a possibility to multiply.

Other numbers have standard representations in base-φ, with rational numbers having recurring representations. These representations are unique, except that numbers with a terminating expansion also have a non-terminating expansion.