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:
“You may be used to a day that includes answering eleven phone calls, attending two meetings, and writing three reports; when you are at home with an infant you will feel you have accomplished quite a lot if you have a shower and a sit-down meal in the same day.”
—Anne C. Weisberg (20th century)
“Ah, Sun-flower, weary of time,
Who countest the steps of the Sun,
Seeking after that sweet golden clime
Where the traveller’s journey is done:
Where the Youth pined away with desire,
And the pale Virgin shrouded in snow
Arise from their graves, and aspire
Where my Sun-flower wishes to go.”
—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)
“You see how this House of Commons has begun to verify all the ill prophecies that were made of it—low, vulgar, meddling with everything, assuming universal competency, and flattering every base passion—and sneering at everything noble refined and truly national. The direct tyranny will come on by and by, after it shall have gratified the multitude with the spoil and ruin of the old institutions of the land.”
—Samuel Taylor Coleridge (1772–1834)
“Green grow the rushes-O
What is your one-O?”
—Unknown. Carol of the Numbers (l. 2–3)
“Any honest examination of the national life proves how far we are from the standard of human freedom with which we began. The recovery of this standard demands of everyone who loves this country a hard look at himself, for the greatest achievments must begin somewhere, and they always begin with the person. If we are not capable of this examination, we may yet become one of the most distinguished and monumental failures in the history of nations.”
—James Baldwin (1924–1987)
“I am a communist because I believe that the Communist idea is a state form of Christianity.”
—Alexander Zhuravlyov (b. 1924)