Golden Ratio Base - Representing Integers As Golden Ratio Base Numbers

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 = −1 + φ. Therefore, we can compute

(a + bφ) + (c + dφ) = ((a + c) + (b + d)φ),
(a + bφ) − (c + dφ) = ((a − c) + (b − d)φ)

and

(a + bφ) × (c + dφ) = ((ac + bd) + (ad + bc + bd)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).

  1. To convert an integer x to a base-φ number, note that x = (x + 0φ).
  2. 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.
  3. Unless our number is 0, go to step 2.
  4. 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.

Read more about this topic:  Golden Ratio Base

Famous quotes containing the words representing, golden, ratio, base and/or numbers:

    “... today we round out the first century of a professed republic,—with woman figuratively representing freedom—and yet all free, save woman.”
    —Phoebe W. Couzins (1845–1913)

    “I will find out where she has gone,
    And kiss her lips and take her hands;
    And walk among long dappled grass,
    And pluck till time and times are done
    The silver apples of the moon,
    The golden apples of the sun.”
    —William Butler Yeats (1865–1939)

    “A magazine or a newspaper is a shop. Each is an experiment and represents a new focus, a new ratio between commerce and intellect.”
    —John Jay Chapman (1862–1933)

    “The base of all artistic genius is the power of conceiving humanity in a new, striking, rejoicing way, of putting a happy world of its own creation in place of the meaner world of common days, of generating around itself an atmosphere with a novel power of refraction, selecting, transforming, recombining the images it transmits, according to the choice of the imaginative intellect. In exercising this power, painting and poetry have a choice of subject almost unlimited.”
    —Walter Pater (1839–1894)

    “The only phenomenon with which writing has always been concomitant is the creation of cities and empires, that is the integration of large numbers of individuals into a political system, and their grading into castes or classes.... It seems to have favored the exploitation of human beings rather than their enlightenment.”
    —Claude Lévi-Strauss (b. 1908)