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)
“Since the Greeks, Western man has believed that Being, all Being, is intelligible, that there is a reason for everything ... and that the cosmos is, finally, intelligible. The Oriental, on the other hand, has accepted his existence within a universe that would appear to be meaningless, to the rational Western mind, and has lived with this meaninglessness. Hence the artistic form that seems natural to the Oriental is one that is just as formless or formal, as irrational, as life itself.”
—William Barrett (b. 1913)
“Think of the earth as a living organism that is being attacked by billions of bacteria whose numbers double every forty years. Either the host dies, or the virus dies, or both die.”
—Gore Vidal (b. 1925)
“The Maiden caught me in the Wild,
Where I was dancing merrily;
She put me into her Cabinet
And Lock’d me up with a golden Key.”
—William Blake (1757–1827)
“Official dignity tends to increase in inverse ratio to the importance of the country in which the office is held.”
—Aldous Huxley (1894–1963)
“Shall we now
Contaminate our fingers with base bribes,
And sell the mighty space of our large honors
For so much trash as may be grasped thus?
I had rather be a dog and bay the moon
Than such a Roman.”
—William Shakespeare (1564–1616)