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:
“There is a difference between dramatizing your sensibility and your personality. The literary works which we think of as classics did the former. Much modern writing does the latter, and so has an affinity with, say, night-club acts in all their shoddy immediacy.”
—Paul Horgan (b. 1904)
“Firm in our beliefs without dismay,
In any game the nations want to play.
A golden age of poetry and power
Of which this noondays the beginning hour.”
—Robert Frost (18741963)
“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 (18621933)
“Music is of two kinds: one petty, poor, second-rate, never varying, its base the hundred or so phrasings which all musicians understand, a babbling which is more or less pleasant, the life that most composers live.”
—Honoré De Balzac (17991850)
“The land cannot be cleansed of the blood that is shed therein, but by the blood of him that shed it.”
—Bible: Hebrew Numbers 35:33.
“[The Declaration of Independence] meant to set up a standard maxim for free society, which should be familiar to all, and revered by all; constantly looked to, constantly labored for, and even though never perfectly attained, constantly approximated, and thereby constantly spreading and deepening its influence, and augmenting the happiness and value of life to all people of all colors everywhere.”
—Abraham Lincoln (18091865)
“Who among us has not, in moments of ambition, dreamt of the miracle of a form of poetic prose, musical but without rhythm and rhyme, both supple and staccato enough to adapt itself to the lyrical movements of our souls, the undulating movements of our reveries, and the convulsive movements of our consciences? This obsessive ideal springs above all from frequent contact with enormous cities, from the junction of their innumerable connections.”
—Charles Baudelaire (18211867)