Representing Rational Numbers As Golden Ratio Base Numbers
Every non-negative rational number can be represented as a recurring base-φ expansion, as can any non-negative element of the field Q = Q + √5Q, the field generated by the rational numbers and √5. Conversely any recurring (or terminating) base-φ expansion is a non-negative element of Q. Some examples (with spaces added for emphasis):
- 1/2 ≈ 0.010 010 010 010 ... φ
- 1/3 ≈ 0.00101000 00101000 00101000... φ
- √5 = 10.1φ
- 2+(1/13)√5 ≈ 10.010 1000100010101000100010000000 1000100010101000100010000000 1000100010101000100010000000 ...φ
The justification that a rational gives a recurring expansion is analogous to the equivalent proof for a base-n numeration system (n=2,3,4,...). Essentially in base-φ long division there are only a finite number of possible remainders, and so once there must be a recurring pattern. For example with 1/2 = 1/10.01φ = 100φ/1001φ long division looks like this (note that base-φ subtraction may be hard to follow at first):
.0 1 0 0 1 ________________________ 1 0 0 1 ) 1 0 0.0 0 0 0 0 0 0 0 1 0 0 1 trade: 10000 = 1100 = 1011 ------- so 10000-1001 = 1011-1001 = 10 1 0 0 0 0 1 0 0 1 ------- etc.The converse is also true, in that a number with a recurring base-φ; representation is an element of the field Q. This follows from the observation that a recurring representation with period k involves a geometric series with ratio φ-k, which will sum to an element of Q.
Read more about this topic: Golden Ratio Base
Famous quotes containing the words representing, rational, numbers, golden, ratio and/or base:
“Brave people may be persuaded to an action by representing it as being more dangerous than it really is.”
—Friedrich Nietzsche (1844–1900)
“The poet makes himself a seer by a long, prodigious, and rational disordering of all the senses. Every form of love, of suffering, of madness; he searches himself, he consumes all the poisons in him, and keeps only their quintessences.”
—Arthur Rimbaud (1854–1891)
“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)
“My mother dandled me and sang,
“How young it is, how young!”
And made a golden cradle
That on a willow swung.
“He went away,” my mother sang,
“When I was brought to bed....””
—William Butler Yeats (1865–1939)
“A magazine or a newspaper is a shop. Each is an experiment and represents a new focus, a new ratio between commerce and intellect.”
—John Jay Chapman (1862–1933)
“It is a base thing for a man among the people not to obey those in command. Never in a state can the laws be well administered when fear does not stand firm.”
—Sophocles (497–406/5 B.C.)