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:
“Scott took LITERATURE so solemnly. He never understood that it was just writing as well as you can and finishing what you start.”
—Ernest Hemingway (1899–1961)
“Such is always the pursuit of knowledge. The celestial fruits, the golden apples of the Hesperides, are ever guarded by a hundred-headed dragon which never sleeps, so that it is an Herculean labor to pluck them.”
—Henry David Thoreau (1817–1862)
“Official dignity tends to increase in inverse ratio to the importance of the country in which the office is held.”
—Aldous Huxley (1894–1963)
“Were I as base as is the lowly plain,
And you, my Love, as high as heaven above,
Yet should the thoughts of me, your humble swain,
Ascend to heaven in honour of my love.”
—Joshua Sylvester (1561–1618)
“The principle of majority rule is the mildest form in which the force of numbers can be exercised. It is a pacific substitute for civil war in which the opposing armies are counted and the victory is awarded to the larger before any blood is shed. Except in the sacred tests of democracy and in the incantations of the orators, we hardly take the trouble to pretend that the rule of the majority is not at bottom a rule of force.”
—Walter Lippmann (1889–1974)
“This unlettered man’s speaking and writing are standard English. Some words and phrases deemed vulgarisms and Americanisms before, he has made standard American; such as “It will pay.” It suggests that the one great rule of composition—and if I were a professor of rhetoric I should insist on this—is, to speak the truth. This first, this second, this third; pebbles in your mouth or not. This demands earnestness and manhood chiefly.”
—Henry David Thoreau (1817–1862)
“To do the opposite of something is also a form of imitation, namely an imitation of its opposite.”
—G.C. (Georg Christoph)