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:
“Hidden away amongst Aschenbach’s writing was a passage directly asserting that nearly all the great things that exist owe their existence to a defiant despite: it is despite grief and anguish, despite poverty, loneliness, bodily weakness, vice and passion and a thousand inhibitions, that they have come into being at all. But this was more than an observation, it was an experience, it was positively the formula of his life and his fame, the key to his work.”
—Thomas Mann (18751955)
“Love’s secrets, being mysteries, ever pertain to the transcendent and the infinite; and so they are as airy bridges, by which our further shadows pass over into the regions of the golden mists and exhalations; whence all poetical, lovely thoughts are engendered, and drop into us, as though pearls should drop from rainbows.”
—Herman Melville (1819–1891)
“Official dignity tends to increase in inverse ratio to the importance of the country in which the office is held.”
—Aldous Huxley (1894–1963)
“No love is entirely without worth, even when the frivolous calls to the frivolous and the base to the base.”
—Iris Murdoch (b. 1919)
“I’m not even thinking straight any more. Numbers buzz in my head like wasps.”
—Kurt Neumann (1906–1958)
“A disposition to preserve, and an ability to improve, taken together, would be my standard of a statesman.”
—Edmund Burke (1729–1797)
“Dug from the tomb of taste-refining time,
Each form is exquisite, each block sublime.
Or good, or bad,—disfigur’d, or deprav’d,—
All art, is at its resurrection sav’d;
All crown’d with glory in the critic’s heav’n,
Each merit magnified, each fault forgiven.”
—Martin Archer, Sir Shee (1769–1850)