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:
“Writing is not like painting where you add. It is not what you put on the canvas that the reader sees. Writing is more like a sculpture where you remove, you eliminate in order to make the work visible. Even those pages you remove somehow remain.”
—Elie Wiesel (b. 1928)
“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)
“I think it is worse to be poor in mind than in purse, to be stunted and belittled in soul, made a coward, made a liar, made mean and slavish, accustomed to fawn and prevaricate, and “manage” by base arts a husband or a father,—I think this is worse than to be kicked with hobnailed shoes.”
—Frances Power Cobbe (1822–1904)
“I’m not even thinking straight any more. Numbers buzz in my head like wasps.”
—Kurt Neumann (1906–1958)
“The urge for Chinese food is always unpredictable: famous for no occasion, standard fare for no holiday, and the constant as to demand is either whim, the needy plebiscite of instantly famished drunks, or pregnancy.”
—Alexander Theroux (b. 1940)
“Upon the whole, necessity is something, that exists in the mind, not in objects; nor is it possible for us ever to form the most distant idea of it, consider’d as a quality in bodies. Either we have no idea of necessity, or necessity is nothing but that determination of thought to pass from cause to effects and effects to causes, according to their experienc’d union.”
—David Hume (1711–1776)