Golden Ratio Base - Representing Rational Numbers As Golden Ratio Base Numbers

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 = 10.1_φ
• 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 = 1/10.01_φ = 100_φ/1001_φ 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.