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)
“The golden mean in ethics, as in physics, is the centre of the system and that about which all revolve, and though to a distant and plodding planet it be an uttermost extreme, yet one day, when that planet’s year is completed, it will be found to be central.”
—Henry David Thoreau (1817–1862)
“People are lucky and unlucky not according to what they get absolutely, but according to the ratio between what they get and what they have been led to expect.”
—Samuel Butler (1835–1902)
“Time, force, and death
Do to this body what extremes you can,
But the strong base and building of my love
Is as the very centre of the earth,
Drawing all things to it.”
—William Shakespeare (1564–1616)
“One murder makes a villain, millions a hero. Numbers sanctify, my good fellow.”
—Charlie Chaplin (1889–1977)
“As long as male behavior is taken to be the norm, there can be no serious questioning of male traits and behavior. A norm is by definition a standard for judging; it is not itself subject to judgment.”
—Myriam Miedzian, U.S. author. Boys Will Be Boys, ch. 1 (1991)
“Every neurosis is a primitive form of legal proceeding in which the accused carries on the prosecution, imposes judgment and executes the sentence: all to the end that someone else should not perform the same process.”
—Lionel Trilling (1905–1975)