GXWeb Continued Logarithm Model

There is a mutation of continued fractions, which I call continued logarithms, which have several advantages over regular continued fractions, especially for computational hardware. As with ordinary continued fractions, each number and subexpression will be a microprocess which describes itself a little at a time, but instead of continued fraction terms, our description language will have only two words, "0" and "1". A 1 means "I was at least 2, so I halved myself". A 0 means "I was between 1 and 2, so I subtracted 1 and reciprocated myself (so now I am > 1)".

So 100/2.54 = 1,1,1,1,1,0,1,1,0,1,1,1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,1,0,1,0. Alternatively, we could write it as the sequence of lengths of bursts of 1s: 5,2,3,1,3,0,0,2,1,1,1. In the latter representation, each term is the integer part of the log base 2 of the number remaining to be described. As with regular continued fractions, oo is the empty sequence, rationals are the finite sequences, and many (but not all!) quadratic surds have periodic sequences.

The primary advantage is the conveniently small information parcel. The restriction to integers of regular continued fractions makes them unsuitable for very large and very small numbers.

(Gosper, 1972)

While continued fractions may formally trace their origins to the time of Euclid, continued (binary) logarithms owe their inception to Bill Gosper in his 1972 unpublished paper on continued fraction arithmetic.

In the newly dawning age of personal computers, Gosper recognised both the limitations of continued fractions in dealing with very large and very small numbers, and the possibilities for a new type of binary representation, reducing such numbers to highly accurate arrays of zeros and ones. Gosper described it as a "sort of recursive version of scientific notation" (1972).

This exploration introduces just a taste of the wonderful research of the past decade, exploring these unusual mathematical creatures. The particular focus here is on the multiple continued fraction-like representations for real numbers made explicit through continued logarithms.

While we count ourselves lucky to know of one or two generalized continued fraction forms for most reals, this exploration reveals the relatively simple processes which deliver up to five distinct forms for most numbers!

Visualising Continued Logarithms

Discovering Continued Logarithms

Defining Continued Logarithms