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Golden ratio base
The Golden mean (φ) can be used as a number base. It is known as the golden mean base, or colloquially, phinary (since the symbolism for the golden mean is the Greek letter "phi"). Any real number has a standard representation as a base-φ numeral in which only the digits 0 and 1 are used, and the digit sequence "11" is avoided. A nonstandard base-φ numeral with this digit sequence (or with other digits) can always be rewritten in standard form, relying on algebraic properties of the number φ -- most notably that φ+1 = φ2. For instance 11φ = 100φ. Despite using an irrational base, it is a remarkable fact that all integers have a unique representation as a terminating (or finite) base-φ expansion. 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. (As they do in base 10: 2.2=2.199999...)
|Decimal||Powers of φ||Base φ|
Writing a φ-base number in standard form
211.01φ is not a standard φ-base 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 and 010φ = 101φ. We can apply the substitutions in any order we like, as the result is the same. Below, the substitutions used are on the right, the resulting number on the left.
211.01φ 300.01φ 011φ → 100φ 1101.01φ 0200φ → 1001φ 10001.01φ 011φ → 100φ (again) 10001.101φ 010φ → 101φ 10000.011φ 010φ → 101φ (again) 10000.1φ 011φ → 100φ (again)
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. This can be converted to the negative of a base-φ representation by negating every digit, standardizing the result, and then mark 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.
Note that when adding the digits "9" and "1", the result is a single digit "(10)", "A" or similar, as we are not working in decimal.
Representing integers as Golden mean base numbers
We can either consider our integer to be the (only) digit of a nonstandard φ-base numeral, and standardize it, or do the following:
Note that 1×1 = 1, φ × φ = 1 + φ and 1/φ = -1 + φ. Therefore, we can compute
- (a + bφ) + (c + dφ) = ((a + c) + (b + d)φ), (a + bφ) - (c + dφ) = ((a - c) + (b - d)φ)
- (a + bφ) × (c + dφ) = ((a × c + b × d) + (a × d + b × c + b × d)φ).
So, using integer values only, we can add, subtract and multiply numbers of the form (a + bφ), and even represent positive and negative integer powers of φ. (Note that φ-1 = 1/φ.)
(a + bφ) > (c + dφ) if and only if 2(a - c) - (d - b) > (d - b) × √5. 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 √5 is replaced with the integer 5.
So, using integer values only, we can also compare numbers of the form (a + bφ).
- To convert an integer x to a φ-base number, note that x = (x + 0φ).
- Subtract the highest power of φ, which is still smaller than the number we have, to get out new number, and record a "1" in the appropriate place in the resulting φ-base number.
- Unless our number is 0, go to step 2.
The above procedure will never result in the sequence "11", since 11φ = 100φ, so getting a "11" would mean we missed a "1" prior to the sequence "11".
e.g Start 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..., with 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..., with 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 1000.1001φ.
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 if x = 0.9999... then 10x = 9.99999... so that 9x = 9 and x=1. In base φ, the numeral 0.1010101... can be seen to be equal to 1 in several ways:
- Conversion to nonstandard form: 1 = 0.11φ = 0.1011φ = 0.101011φ = ... = 0.10101010....φ
- Geometric series: 1.0101010...φ is equal to
- Difference between "shifts": φ2 x - x = 10.101010...φ - 0.101010...φ = 10φ = φ so that x = φ/(φ2-1) = 1
This non-uniqueness is a feature of the numeration system, since both 1.0000 and 0.101010... are in standard form.
Representing Rationals as Golden mean base numbers
Every rational number can be represented as a recurring base φ expansion, as can any element of the field Q[√5] = Q + √5Q, the field generated by the rational numbers and √5. Conversely any recurring (or terminating) base φ expansion is an element of Q[√5]. Some examples (with spaces added for emphasis):
- 1/2 = 0.010 010 010 ... φ
- 1/3 = 0.00101000 00101000 00101000... φ
- √5 = 10.100000φ
- 2+(1/13)√5 = 10.010 1000100010101000100010000000 1000100010101000100010000000 1000100010101000100010000000 ...φ
The justification that a rational gives a recurring expansion is analogous to the equivalent proof for a base-n numeration system (n=2,3,4,...). Essentially in base-φ long division, there are only a finite number of possible remainders, and so once there must be a recurring pattern. For example with 1/2 = 1/10.01φ = 100φ/1001φ long division looks like this (note that subtraction is a bit freaky)
.0 1 0 0 1 ------------------------ 1 0 0 1 ) 1 0 0.0 0 0 0 0 0 0 0 1 0 0 1 trade: 10000 = 1100 = 1011 ------- so 10000-1001 = 1011-1001 = 10 1 0 0 0 0 1 0 0 1 ------- etc
The converse is also true, in that a number with a recurring base-φ representation is an element of the field Q[√5]. This follows from the observation that a recurring representation with period k involves a geometric series with ratio φ-k, which will sum to an element of Q[√5].
Addition, subtraction, multiplication
It is possible to adapt all the standard algorithms of base-10 arithmetic to base-φ arithmetic. There are two approaches to this:
Calculate then convert to standard form
To add two φ-base numbers, add each pair of digits, without carry, and then convert the numeral to standard form. To subtract, subtract each pair of digits without borrow (borrow is a negative amount of carry), and then convert the numeral to standard form. To multiply, multiply how you normally multiply, without carry, and then convert the numeral to standard form.
- 2+3 = 10.01 + 100.01 = 110.02 = 110.1001 = 1000.1001
- 2×3 = 10.01 × 100.01 = 1000.1 + 1.0001 = 1001.1001 = 1010.0001
- 7-2 = 10000.0001 - 10.01 = 10010.0101 = 1110.0101 = 1001.0101 = 1000.1001
Avoiding digits other than 0 and 1
A more "native" approach is to avoid having to add digits 1+1 or to subtract 0-1. This is done by reorganising the operands into nonstandard form so that these combinations do not occur. For example
- 2+3 = 10.01 + 100.01 = 10.01 + 100.0011 = 110.0111 = 1000.1001
- 7-2 = 10000.0001 - 10.01 = 1100.0001 - 10.01 = 1011.0001 - 10.01 = 1010.1101 - 10.01 = 1000.1001
The subtraction seen here uses a modified form of the standard "trading" algorithm for subtraction.
No fractions (a/b, where a and b are integers, a not divisible by b) can be represented as a finite φ-base number, in other words, all finitely representable φ-base numbers are either integers or (more likely) an irrational in the field Q[√5]. Due to long division having only a finite number of possible remainders, a division of two integers (or other numbers with finite base-φ representation) will have a recurring expansion, as demonstrated above.
A close relation: Fibonacci Representation
A closely related numeration system is Fibonacci representation used for integers. In this system, only digits 0 and 1 are used and the place values of the digits are the Fibonacci numbers. As with base-φ, the digit sequence "11" is avoided by rearranging to a standard form, using the Fibonacci recurrence relation Fk+1 = Fk + Fk-1. For example
- 30 = 1×21 + 0×13 + 1×8 + 0×5 + 0×3 + 0×2 + 1×1 + 0×1 = 10100010fib.
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