Examples
Writing golden ratio base numbers in standard form
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 additionally a "2" and a "1" = −1, which are not "0" or "1". 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, anRepresenting 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 11φ = 100φ, 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 1000.1001φ.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 = 0.11φ = 0.1011φ = 0.101011φ = ... = 0.10101010....φ *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.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[]. 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[].
Representing irrational numbers of note as golden ratio base numbers
The base-φ representations of some interesting numbers: * ≈ 100.0100 1010 1001 0001 0101 0100 0001 0100 ...φ * ≈ 100.0000 1000 0100 1000 0000 0100 ...φ * ≈ 1.0100 0001 0100 1010 0100 0000 0101 0000 0000 0101 ...φ * φ = = 10φ * = 10.1φAddition, subtraction, and 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
ForAvoid 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.Division
No non-integer rational number can be represented as aRelationship with Fibonacci coding
Fibonacci coding is a closely related numeration system 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 ''F''''k''+1 = ''F''''k'' + ''F''''k''−1. For example, :30 = 1×21 + 0×13 + 1×8 + 0×5 + 0×3 + 0×2 + 1×1 + 0×1 = 10100010fib.Practical usage
It is possible to mix base-φ arithmetic with Fibonacci integer sequences. The sum of numbers in a General Fibonacci integer sequence that correspond with the nonzero digits in the base-φ number, is the multiplication of the base-φ number and the element at the zero-position in the sequence. For example: *product 10 (10100.0101 base-φ) and 25 (zero position) = 5 + 10 + 65 + 170 = 250 *:base-φ: 1 0 1 0 0. 0 1 0 1 *:partial sequence: ... 5 5 10 15 ''25'' 40 65 105 170 275 445 720 1165 ... *product 10 (10100.0101 base-φ) and 65 (zero position) = 10 + 25 + 170 + 445 = 650 *:base-φ: 1 0 1 0 0. 0 1 0 1 *:partial sequence: ... 5 5 10 15 25 40 ''65'' 105 170 275 445 720 1165 ...See also
* Beta encoder – Originally used golden ratio base * Ostrowski numerationNotes
References
* * *External links