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:
“I don’t really think that writers, even great writers, are prophets, or sages, or Messiah-like figures; writing is a lonely, sedentary occupation and a touch of megalomania can be comforting around five on a November afternoon when you haven’t seen anybody all day.”
—Angela Carter (1940–1992)
“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)
“For those who are base in judgement do not know the good they hold in their hands until they cast it off.”
—Sophocles (497–406/5 B.C.)
“I had a feeling that out there, there were very poor people who didn’t have enough to eat. But they wore wonderfully colored rags and did musical numbers up and down the streets together.”
—Jill Robinson (b. 1936)
“This unlettered man’s speaking and writing are standard English. Some words and phrases deemed vulgarisms and Americanisms before, he has made standard American; such as “It will pay.” It suggests that the one great rule of composition—and if I were a professor of rhetoric I should insist on this—is, to speak the truth. This first, this second, this third; pebbles in your mouth or not. This demands earnestness and manhood chiefly.”
—Henry David Thoreau (1817–1862)
“Ambivalence reaches the level of schizophrenia in our treatment of violence among the young. Parents do not encourage violence, but neither do they take up arms against the industries which encourage it. Parents hide their eyes from the books and comics, slasher films, videos and lyrics which form the texture of an adolescent culture. While all successful societies have inhibited instinct, ours encourages it. Or at least we profess ourselves powerless to interfere with it.”
—C. John Sommerville (20th century)