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:
“... in writing you cannot possibly be interesting if what you say is not true, if it is what I call “a true lie,” i.e., a truth which gives the wrong impression. For no matter how subtly you lie in writing, people know it and don’t believe you, and the whole secret of being interesting is to be believed.”
—Brenda Ueland (1891–1985)
“Life has no other discipline to impose, if we would but realize it, than to accept life unquestioningly. Everything we shut our eyes to, everything we run away from, everything we deny, denigrate or despise, serves to defeat us in the end. What seems nasty, painful, evil, can become a source of beauty, joy and strength, if faced with an open mind. Every moment is a golden one for him who has the vision to recognize it as such.”
—Henry Miller (1891–1980)
“Official dignity tends to increase in inverse ratio to the importance of the country in which the office is held.”
—Aldous Huxley (1894–1963)
“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 but three chairs in my house; one for solitude, two for friendship; three for society. When visitors came in larger and unexpected numbers there was but the third chair for them all, but they generally economized the room by standing up.”
—Henry David Thoreau (1817–1862)
“... the meanest life, the poorest existence, is attributed to God’s will, but as human beings become more affluent, as their living standard and style begin to ascend the material scale, God descends the scale of responsibility at a commensurate speed.”
—Maya Angelou (b. 1928)
“Being cultured is the least expensive form of respectability.”
—Mason Cooley (b. 1927)