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:
“When all things are equal, translucence in writing is more effective than transparency, just as glow is more revealing than glare.”
—James Thurber (1894–1961)
“For they had golden earrings, because they were Ishmaelites.”
—Bible: Hebrew Judges 8:24.
“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)
“What if it tempt you toward the flood, my lord,
Or to the dreadful summit of the cliff
That beetles o’er his base into the sea,
And there assume some other horrible form
Which might deprive your sovereignty of reason,
And draw you into madness?”
—William Shakespeare (1564–1616)
“I’m not even thinking straight any more. Numbers buzz in my head like wasps.”
—Kurt Neumann (1906–1958)
“I find it interesting that the meanest life, the poorest existence, is attributed to God’s will, but as human beings become more affluent, as their living standard and style begin to ascend the material scale, God descends the scale of responsibility at a commensurate speed.”
—Maya Angelou (b. 1928)
“Fantasy is a product of thought, Imagination of sensibility. If the thinking, discursive mind turns to speculation, the result is Fantasy; if, however, the sensitive, intuitive mind turns to speculation, the result is Imagination. Fantasy may be visionary, but it is cold and logical. Imagination is sensuous and instinctive. Both have form, but the form of Fantasy is analogous to Exposition, that of Imagination to Narrative.”
—Sir Herbert Read (1893–1968)