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:
“All the critics who could not make their reputations by discovering you are hoping to make them by predicting hopefully your approaching impotence, failure and general drying up of natural juices. Not a one will wish you luck or hope that you will keep on writing unless you have political affiliations in which case these will rally around and speak of you and Homer, Balzac, Zola and Link Steffens.”
—Ernest Hemingway (1899–1961)
“Behold that great Plotinus swim
Buffeted by such seas;
Bland Rhadamanthus beckons him,
But the Golden Race looks dim,
Salt blood blocks his eyes.”
—William Butler Yeats (1865–1939)
“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)
“Things base and vile, holding no quantity,
Love can transpose to form and dignity.
Love looks not with the eyes, but with the mind,
And therefore is winged Cupid painted blind.”
—William Shakespeare (1564–1616)
“All experience teaches that, whenever there is a great national establishment, employing large numbers of officials, the public must be reconciled to support many incompetent men; for such is the favoritism and nepotism always prevailing in the purlieus of these establishments, that some incompetent persons are always admitted, to the exclusion of many of the worthy.”
—Herman Melville (1819–1891)
“Gentlemen, those confederate flags and our national standard are what has made this union great. In what other country could a man who fought against you be permitted to serve as judge over you, be permitted to run for reelection and bespeak your suffrage on Tuesday next at the poles.”
—Laurence Stallings (1894–1968)
“Now, what I want is, Facts. Teach these boys and girls nothing but Facts. Facts alone are wanted in life. Plant nothing else, and root out everything else. You can only form the minds of reasoning animals upon Facts: nothing else will ever be of any service to them. This is the principle on which I bring up my own children, and this is the principle on which I bring up these children. Stick to Facts, sir!”
—Charles Dickens (1812–1870)