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:
“Even in writing an annual report, the unconscious plays a role.”
—Mason Cooley (b. 1927)
“Beneath the azure current floweth;
Above, the golden sunlight glows.
Rebellious, the storm it wooeth,
As if the storms could give repose.”
—Mikhail Lermontov (1814–1841)
“Official dignity tends to increase in inverse ratio to the importance of the country in which the office is held.”
—Aldous Huxley (1894–1963)
“No love is entirely without worth, even when the frivolous calls to the frivolous and the base to the base.”
—Iris Murdoch (b. 1919)
“The principle of majority rule is the mildest form in which the force of numbers can be exercised. It is a pacific substitute for civil war in which the opposing armies are counted and the victory is awarded to the larger before any blood is shed. Except in the sacred tests of democracy and in the incantations of the orators, we hardly take the trouble to pretend that the rule of the majority is not at bottom a rule of force.”
—Walter Lippmann (1889–1974)
“When Freedom, from her mountain height,
Unfurled her standard to the air,
She tore the azure robe of night,
And set the stars of glory there;”
—Joseph Rodman Drake (1795–1820)
“The Virgin filled so enormous a space in the life and thought of the time that one stands now helpless before the mass of testimony to her direct action and constant presence in every moment and form of the illusion which men thought they thought their existence.”
—Henry Brooks Adams (1838–1918)