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 (18191900)
“So far as discipline is concerned, freedom means not its absence but the use of higher and more rational forms as contrasted with those that are lower or less rational.”
—Charles Horton Cooley (18641929)
“Green grow the rushes-O
What is your one-O?”
—Unknown. Carol of the Numbers (l. 23)
“In books one finds golden mansions and women as beautiful as jewels.”
—Chinese proverb.
“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 (18621933)
“What I often forget about students, especially undergraduates, is that surface appearances are misleading. Most of them are at base as conventional as Presbyterian deacons.”
—Muriel Beadle (b. 1915)