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:
“A song is no song unless the circumstance is free and fine. If a singer sing from a sense of duty or from seeing no way to escape, I had rather have none. Those only can sleep who do not care to sleep; and those only write or speak best who do not too much respect the writing or the speaking.”
—Ralph Waldo Emerson (1803–1882)
“With sweet May dews my wings were wet,
And Phoebus fir’d my vocal rage;
He caught me in his silken net,
And shut me in his golden cage.
He loves to sit and hear me sing,
Then, laughing, sports and plays with me;
Then stretches out my golden wing,
And mocks my loss of liberty.”
—William Blake (1757–1827)
“Official dignity tends to increase in inverse ratio to the importance of the country in which the office is held.”
—Aldous Huxley (1894–1963)
“It is a base thing for a man among the people not to obey those in command. Never in a state can the laws be well administered when fear does not stand firm.”
—Sophocles (497–406/5 B.C.)
“Out of the darkness where Philomela sat,
Her fairy numbers issued. What then ailed me?
My ears are called capacious but they failed me,
Her classics registered a little flat!
I rose, and venomously spat.”
—John Crowe Ransom (1888–1974)
“A disposition to preserve, and an ability to improve, taken together, would be my standard of a statesman.”
—Edmund Burke (1729–1797)
“For me, as a beginning novelist, all other living writers form a control group for whom the world is a placebo.”
—Nicholson Baker (b. 1957)