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:
“In the learned journal, in the influential newspaper, I discern no form; only some irresponsible shadow; oftener some monied corporation, or some dangler, who hopes, in the mask and robes of his paragraph, to pass for somebody. But through every clause and part of speech of the right book I meet the eyes of the most determined men; his force and terror inundate every word: the commas and dashes are alive; so that the writing is athletic and nimble,—can go far and live long.”
—Ralph Waldo Emerson (1803–1882)
“How doth the little crocodile
Improve his shining tale,
And pour the waters of the Nile
On every golden scale!”
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“People are lucky and unlucky not according to what they get absolutely, but according to the ratio between what they get and what they have been led to expect.”
—Samuel Butler (1835–1902)
“They must to keep their certainty accuse
All that are different of a base intent;
Pull down established honour; hawk for news
Whatever their loose phantasy invent
And murmur it with bated breath....”
—William Butler Yeats (1865–1939)
“One murder makes a villain, millions a hero. Numbers sanctify, my good fellow.”
—Charlie Chaplin (1889–1977)
“[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 (1809–1865)
“Polarized light showed the secret architecture of bodies; and when the second-sight of the mind is opened, now one color or form or gesture, and now another, has a pungency, as if a more interior ray had been emitted, disclosing its deep holdings in the frame of things.”
—Ralph Waldo Emerson (1803–1882)