Golden Ratio Base

Golden ratio base is a non-integer positional numeral system that uses the golden ratio (the irrational number  ≈ 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 (φ¹) + 1 (φ⁰) = φ². 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 /a> 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.

Examples

Writing golden ratio base numbers in standard form

In the following example the notation 1 is used to represent −1.

211.01_φ 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 "standardize" a numeral, we can use the following substitutions: 011_φ = 100_φ, 0200_φ = 1001_φ, 010_φ = 101_φ and 110_φ = 001_φ. 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.

Representing integers as golden ratio base numbers

We can either consider our integer to be the (only) digit of a nonstandard base-φ numeral, and standardize it, or do the following:

1 × 1 = 1, φ × φ = 1 + φ and = −1 + φ. Therefore, we can compute

: (''a'' + ''b''φ) + (''c'' + ''d''φ) = ((''a'' + ''c'') + (''b'' + ''d'')φ),
: (''a'' + ''b''φ) − (''c'' + ''d''φ) = ((''a'' − ''c'') + (''b'' − ''d'')φ)

and

: (''a'' + ''b''φ) × (''c'' + ''d''φ) = ((''ac'' + ''bd'') + (''ad'' + ''bc'' + ''bd'')φ).

So, using integer values only, we can add, subtract and multiply numbers of the form (''a'' + ''b''φ), and even represent positive and negative integer powers of φ.

(''a'' + ''b''φ) > (''c'' + ''d''φ) if and only if 2(''a'' − ''c'') − (''d'' − ''b'') > (''d'' − ''b'') × . If one side is negative, the other positive, the comparison is trivial. Otherwise, square both sides, to get an integer comparison, reversing the comparison direction if both sides were negative. On squaring both sides, the is replaced with the integer 5.

So, using integer values only, we can also compare numbers of the form (''a'' + ''b''φ).

# To convert an integer ''x'' to a base-φ number, note that ''x'' = (''x'' + 0φ).
# Subtract the highest power of φ, which is still smaller than the number we have, to get our new number, and record a "1" in the appropriate place in the resulting base-φ number.
# Unless our number is 0, go to step 2.
# Finished.

The above procedure will never result in the sequence "11", since 11_φ = 100_φ, so getting a "11" would mean we missed a "1" prior to the sequence "11".

Start, e.g., with integer = 5, with the result so far being ...00000.00000..._φ

Highest power of φ ≤ 5 is φ³ = 1 + 2φ ≈ 4.236067977

Subtracting this from 5, we have 5 − (1 + 2φ) = 4 − 2φ ≈ 0.763932023..., the result so far being 1000.00000..._φ

Highest power of φ ≤ 4 − 2φ ≈ 0.763932023... is φ⁻¹ = −1 + 1φ ≈ 0.618033989...

Subtracting this from 4 − 2φ ≈ 0.763932023..., we have 4 − 2φ − (−1 + 1φ) = 5 − 3φ ≈ 0.145898034..., the result so far being 1000.10000..._φ

Highest power of φ ≤ 5 − 3φ ≈ 0.145898034... is φ⁻⁴ = 5 − 3φ ≈ 0.145898034...

Subtracting this from 5 − 3φ ≈ 0.145898034..., we have 5 − 3φ − (5 − 3φ) = 0 + 0φ = 0, with the final result being 1000.1001_φ.

Non-uniqueness

Just as with any base-n system, numbers with a terminating representation have an alternative recurring representation. In base-10, this relies on the observation that 0.999...=1. In base-φ, the numeral 0.1010101... can be seen to be equal to 1 in several ways:

*Conversion to nonstandard form: 1 = 0.11_φ = 0.1011_φ = 0.101011_φ = ... = 0.10101010...._φ
*Geometric series: 1.0101010..._φ is equal to
:$\sum_^\infty \varphi^=\frac = \varphi$
*Difference between "shifts": φ² ''x'' − ''x'' = 10.101010..._φ − 0.101010..._φ = 10_φ = φ so that ''x'' = = 1

This non-uniqueness is a feature of the numeration system, since both 1.0000 and 0.101010... are in standard form.

In general, the final 1 of any number in base-φ can be replaced with a recurring 01 without changing the value of that number.

Representing rational numbers as golden ratio base numbers

Every non-negative rational number can be represented as a recurring base-φ expansion, as can any non-negative element of the field Q[] = Q + Q, the field generated by the rational numbers and . Conversely any recurring (or terminating) base-φ expansion is a non-negative element of Q[]. For recurring decimals, the recurring part has been overlined:

* ≈ 0.00101000_φ
* = 10.1_φ
*2 + ≈ 10.01long division