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 can hardly bring myself to caution you against drinking, because I am persuaded that I am writing to a rational creature, a gentleman, and not to a swine. However, that you may not be insensibly drawn into that beastly custom of even sober drinking and sipping, as the sots call it, I advise you to be of no club whatsoever.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“I’d let my golden chances pass me by.”
—Oscar Hammerstein II (1895–1960)
“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)
“Who is here so base that would be a bondman? If any, speak, for him have I offended. Who is here so rude that would not be a Roman? If any, speak, for him have I offended. Who is here so vile that will not love his country? If any, speak, for him have I offended.”
—William Shakespeare (1564–1616)
“Individually, museums are fine institutions, dedicated to the high values of preservation, education and truth; collectively, their growth in numbers points to the imaginative death of this country.”
—Robert Hewison (b. 1943)
“Where shall we look for standard English but to the words of a standard man?”
—Henry David Thoreau (1817–1862)
“Touch me not.”
—Bible: New Testament Jesus, in John, 20:17.
Spoken to Mary Magdalene, after Jesus has risen from the dead and made himself known to her. The words are best known in the Latin form in which they appear in the Vulgate: Noli me tangere.