Golden ratio base

Golden ratio base is a non-integer positional numeral system that uses the golden ratio (the irrational number 1 + √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 φ + 1 = φ2. For instance, 11φ = 100φ. Golden ratio base is a non-integer positional numeral system that uses the golden ratio (the irrational number 1 + √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 φ + 1 = φ2. 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; 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 (mentioned above) with a terminating expansion also have a non-terminating expansion, as they do in base-10; for example, 1 = 0.99999…. In the following example the notation 1 is used to represent −1. 211.01φ is not a standard base-φ numeral, since it contains a '11' and a '2', which isn't a '0' or '1', and contains a 1 = −1, which isn't a '0' or '1' either. To 'standardize' a numeral, we can use the following substitutions: 011φ = 100φ, 0200φ = 1001φ, 010φ = 101φ and 110φ = 001φ. We can apply the substitutions in any order we like, as the result is the same. Below, the substitutions applied to the number on the previous line are on the right, the resulting number on the left. Any positive number with a non-standard terminating base-φ representation can be uniquely standardized in this manner. If we get to a point where all digits are '0' or '1', except for the first digit being negative, then the number is negative. (The exception to this is when the first digit is negative one and the next two digits are one, like 1111.001=1.001.) This can be converted to the negative of a base-φ representation by negating every digit, standardizing the result, and then marking it as negative. For example, use a minus sign, or some other significance to denote negative numbers. If the arithmetic is being performed on a computer, an error message may be returned.