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 a performance but a generosity.”
—Brenda Ueland (1891–1985)
“Life has no other discipline to impose, if we would but realize it, than to accept life unquestioningly. Everything we shut our eyes to, everything we run away from, everything we deny, denigrate or despise, serves to defeat us in the end. What seems nasty, painful, evil, can become a source of beauty, joy and strength, if faced with an open mind. Every moment is a golden one for him who has the vision to recognize it as such.”
—Henry Miller (1891–1980)
“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 (1862–1933)
“No love is entirely without worth, even when the frivolous calls to the frivolous and the base to the base.”
—Iris Murdoch (b. 1919)
“What culture lacks is the taste for anonymous, innumerable germination. Culture is smitten with counting and measuring; it feels out of place and uncomfortable with the innumerable; its efforts tend, on the contrary, to limit the numbers in all domains; it tries to count on its fingers.”
—Jean Dubuffet (1901–1985)
“If the Revolution has the right to destroy bridges and art monuments whenever necessary, it will stop still less from laying its hand on any tendency in art which, no matter how great its achievement in form, threatens to disintegrate the revolutionary environment or to arouse the internal forces of the Revolution, that is, the proletariat, the peasantry and the intelligentsia, to a hostile opposition to one another. Our standard is, clearly, political, imperative and intolerant.”
—Leon Trotsky (1879–1940)
“I always say, my motto is “Art for my sake.” If I want to write, I write—and if I don’t want to, I won’t. The difficulty is to find exactly the form one’s passion—work is produced by passion with me ... Mwants to take.”
—D.H. (David Herbert)