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Concept
Golden ratio base
Summary
Golden ratio base is a non-integer positional numeral system that uses the golden ratio (the irrational number 1 + /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 φ1 + φ0 = 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[1 + /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 base-10.
In the following example of conversion from non-standard to standard form, the notation 1 is used to represent the signed digit -1.
211.01φ is not a standard base-φ numeral, since it contains a "11" and additionally a "2" and a "1" = −1, which are not "0" or "1".
To put a numeral in standard form, we may use the following substitutions: , , , . The substitutions may be applied in any order we like, as the result will be 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.
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