Golden ratio base

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Golden ratio base is a non-integer positional numeral system that uses the golden ratio (the irrational number ${\textstyle {\frac {1+{\sqrt {5}}}{2}}}$ ≈ 1.61803399 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 properties of the base φ — most notably that φⁿ + φⁿ⁻¹ = φⁿ⁺¹. For instance, 11_φ = 100_φ.

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[${\textstyle {\frac {1+{\sqrt {5}}}{2}}}$]; 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. For example, 1 = 0.1010101… in base-φ just as 1 = 0.99999… in decimal.