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 think it’s the real world. The people we’re writing about in professional sports, they’re suffering and living and dying and loving and trying to make their way through life just as the brick layers and politicians are.”
—Walter Wellesley (Red)
“Care keeps his watch in every old man’s eye,
And where care lodges, sleep will never lie;
But where unbruisèd youth with unstuffed brain
Doth couch his limbs, there golden sleep doth reign.”
—William Shakespeare (1564–1616)
“Personal rights, universally the same, demand a government framed on the ratio of the census: property demands a government framed on the ratio of owners and of owning.”
—Ralph Waldo Emerson (1803–1882)
“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)
“Out of the darkness where Philomela sat,
Her fairy numbers issued. What then ailed me?
My ears are called capacious but they failed me,
Her classics registered a little flat!
I rose, and venomously spat.”
—John Crowe Ransom (1888–1974)
“This unlettered man’s speaking and writing are standard English. Some words and phrases deemed vulgarisms and Americanisms before, he has made standard American; such as “It will pay.” It suggests that the one great rule of composition—and if I were a professor of rhetoric I should insist on this—is, to speak the truth. This first, this second, this third; pebbles in your mouth or not. This demands earnestness and manhood chiefly.”
—Henry David Thoreau (1817–1862)
“Driving is a spectacular form of amnesia. Everything is to be discovered, everything to be obliterated.”
—Jean Baudrillard (b. 1929)