Unpacking Continued Logarithms
Channel: Compass Learning Technologies
Date: 2022-05-10

In how many different ways can you express a number like 22/9?
As a decimal? Sure…
As a mixed numeral? Perhaps…
What about as a continued fraction?
More than you might think using Continued Logarithms (especially once you start using different base numbers!)

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trace their origins to the time of Euclid, (binary) continued logarithms owe their inception to Bill Gosper in his 1972 unpublished paper on continued fraction arithmetic.

In the newly dawning computer age, 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).

Building upon the wonderful research of the past two decades coming from the University of Newcastle, Australia (led and inspired by the late Jonathan Borwein), this current exploration - this mathematical playground -  introduces just a taste of the wonderful research of the past two decades, 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 might generally count ourselves lucky to know of one or two generalized continued fraction forms for most reals, this exploration reveals the relatively simple processes which can deliver up to five distinct forms for most numbers - potentially in every different base number!

Use the tools provided on this web page to explore, visualise and dig more deeply into these wonderful numbers.

https://compasstech.com.au/gxwclog/index.html

Dr Scott Lindstrom presents a much more complete description of the CARMA research on continued logarithms:

https://youtu.be/pmmDlD9K4o0

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0:00-00:48 Introduction
00:49-5:17 Gosper’s Continued Logarithms
5:18-7:25  CARMA: Types I, II and III
7:26-21:58 Visualising Continued Logarithms
22:00-30:55 Types II and III
30:58-34:51 Overview of the Continued Logarithm Playground
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