Golden Ratio Base - Writing Golden Ratio Base Numbers in Standard Form

Writing Golden Ratio Base Numbers in Standard Form

211.01φ is not a standard base-φ 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 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.

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.

Read more about this topic:  Golden Ratio Base

Famous quotes containing the words writing, golden, ratio, base, numbers, standard and/or form:

    “... no writing is a waste of time,—no creative work where the feelings, the imagination, the intelligence must work.”
    —Brenda Ueland (1891–1985)

    “The golden age, when rambunctious spirits were regarded as the source of evil.”
    —Friedrich Nietzsche (1844–1900)

    “Official dignity tends to increase in inverse ratio to the importance of the country in which the office is held.”
    —Aldous Huxley (1894–1963)

    “In the land of turkeys in turkey weather
    At the base of the statue, we go round and round.
    What a beautiful history, beautiful surprise!
    Monsieur is on horseback. The horse is covered with mice.”
    —Wallace Stevens (1879–1955)

    “He bundles every forkful in its place,
    And tags and numbers it for future reference,
    So he can find and easily dislodge it
    In the unloading. Silas does that well.
    He takes it out in bunches like birds’ nests.”
    —Robert Frost (1874–1963)

    “Neither I nor anyone else knows what a standard is. We all recognize a dishonorable act, but have no idea what honor is.”
    —Anton Pavlovich Chekhov (1860–1904)

    “Here the term ‘language-game’ is meant to bring into prominence the fact that the speaking of language is part of an activity, of a form of life.”
    —Ludwig Wittgenstein (1889–1951)