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:
“Romance reading and writing might be seen ... as a collectively elaborated female ritual through which women explore the consequences of their common social condition as the appendages of men and attempt to imagine a more perfect state where all the needs they so intensely feel and accept as given would be adequately addressed.”
—Janice A. Radway (b. 1949)
“I had a little nut-tree, nothing would it bear
But a golden nutmeg and a silver pear;”
—Mother Goose (fl. 17th–18th century. I had a little nut-tree, nothing would it bear (l. 1–2)
“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)
“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)
“All experience teaches that, whenever there is a great national establishment, employing large numbers of officials, the public must be reconciled to support many incompetent men; for such is the favoritism and nepotism always prevailing in the purlieus of these establishments, that some incompetent persons are always admitted, to the exclusion of many of the worthy.”
—Herman Melville (1819–1891)
“The art of advertisement, after the American manner, has introduced into all our life such a lavish use of superlatives, that no standard of value whatever is intact.”
—Wyndham Lewis (1882–1957)
“At daybreak, on the bleak sea-beach,
A fisherman stood aghast,
To see the form of a maiden fair,
Lashed close to a drifting mast.”
—Henry Wadsworth Longfellow (1809–1882)