'Contrary to everybody, this self contained paper will show that continued fractions are not only perfectly amenable to arithmetic, they are amenable to perfect arithmetic.' (Bill Gosper, 1972)
For Gosper, the appealing concept of perfect arithmetic may well have involved using continued fractions to perform calculations to any desired degree of accuracy. Continued fractions are ideal tools for successive approximation.
The examples explored here may offer a glimpse into one of the enduring questions regarding continued fractions: they are wonderful at representing numbers, both rational and irrational, but can they also serve as practical tools for computation?
While some of these examples involve simple, finite rational values, it is unlikely that anyone would realistically wheel in continued fraction arithmetic for such questions.
But for those convoluted rationals and especially those infinite irrationals? Well... it was made for such as these!
https://compasstech.com.au/gxwgosper/index.html
0:00 - 2:00 Introduction
2:00 - 15:00 An Overview of Gosper’s Process
15:00 - 31:00 Step-by-Step Example
31:00 - 41:40 A Simpler Matrix Approach
41:40 - 51:58 Some More Interesting Examples
