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:
“He who has learned what is commonly considered the whole art of painting, that is, the art of representing any natural object faithfully, has as yet only learned the language by which his thoughts are to be expressed.”
—John Ruskin (1819–1900)
“I often wish for the end of the wretched remnant of my life; and that wish is a rational one; but then the innate principle of self-preservation, wisely implanted in our natures, for obvious purposes, opposes that wish, and makes us endeavour to spin out our thread as long as we can, however decayed and rotten it may be.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“I had but three chairs in my house; one for solitude, two for friendship; three for society. When visitors came in larger and unexpected numbers there was but the third chair for them all, but they generally economized the room by standing up.”
—Henry David Thoreau (1817–1862)
“He, that holds fast the golden mean,
And lives contentedly between
The little and the great,
Feels not the wants that pinch the poor,
Nor plagues that haunt the rich man’s door,
Imbitt’ring all his state.”
—Horace [Quintus Horatius Flaccus] (65–8)
“Official dignity tends to increase in inverse ratio to the importance of the country in which the office is held.”
—Aldous Huxley (1894–1963)
“Were I as base as is the lowly plain,
And you, my Love, as high as heaven above,
Yet should the thoughts of me, your humble swain,
Ascend to heaven in honour of my love.”
—Joshua Sylvester (1561–1618)