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:
“Historians desiring to write the actions of men, ought to set down the simple truth, and not say anything for love or hatred; also to choose such an opportunity for writing as it may be lawful to think what they will, and write what they think, which is a rare happiness of the time.”
—Sir Walter Raleigh (1552–1618)
“Fair Hope! our earlier Heaven! by thee
Young Time is taster to Eternity.
The generous wine with age grows strong, not sour,
Nor need we kill thy fruit to smell thy flower.
Thy golden head never hangs down
Till in the lap of Love’s full noon
It falls and dies: Oh no, it melts away
As doth the dawn into the day,
As lumps of sugar lose themselves, and twine
Their subtle essence with the soul of wine.”
—Abraham Cowley (1618–1667)
“Official dignity tends to increase in inverse ratio to the importance of the country in which the office is held.”
—Aldous Huxley (1894–1963)
“Yet herein will I imitate the sun,
Who doth permit the base contagious clouds
To smother up his beauty from the world,
That when he please again to be himself,
Being wanted, he may be more wondered at
By breaking through the foul and ugly mists
Of vapors that did seem to strangle him”
—William Shakespeare (1564–1616)
“Think of the earth as a living organism that is being attacked by billions of bacteria whose numbers double every forty years. Either the host dies, or the virus dies, or both die.”
—Gore Vidal (b. 1925)
“An indirect quotation we can usually expect to rate only as better or worse, more or less faithful, and we cannot even hope for a strict standard of more and less; what is involved is evaluation, relative to special purposes, of an essentially dramatic act.”
—Willard Van Orman Quine (b. 1908)
“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.”
—Charles Dickens (1812–1870)