Wed, 2 Feb 2005 10:46:55 -0600 (CST) James Propp <propp@math.wisc.edu> Has anyone ever published Gosper's algorithms for adding and multiplying continued fractions? You can find his write-up on the web at http://www.tweedledum.com/rwg/cfup.htm I don't know if it was ever published. But you should know that the version on the web was not the latest version of his paper. The version on tweedledum.com is one that someone retrieved from the SAIL archives from ~1975. There were later versions (lost?), but I know that Gosper did some merging to get the online version close to *one* of the written versions. I just dug through my files and have found a printed copy from ~1977 that seems pretty close to the version on the web. Sadly, the brief and wonderful "Introduction" is not present in the web version. (I typed it in, below. RWG: was the elision of the intro intentional? If not, and I have a later version than you do, I can (physical) mail you a copy.) The one-sentence abstract is a classic: "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." The introduction, too, is a classic: Introduction Continued fractions are hard to like. People who like continued fractions eat pickled okra and drive Citroens. Books on the subject are filled with dull proofs of dull properties, and recent papers relating continued fractions to computers have bordered on libel. But the literature is not the real problem. Let's face it, a continued fraction is a very awkward object for our intuitions to grasp. Just to estimate the size of a purely numerical continued fraction would seem, at first, to require discarding all but the the first few terms, followed by converting to improper fractions in a bottom-to-top repetition. Since it isn't immediately clear how much error we committed by discarding the "tail", we have been penalized for asking even a simple question about size. Do continued fractions suffer from the "observer effect"? Why, if they are so intractable, are we about to attempt arithmetic with them? In fact, modern mathematical writers have denied the feasibility of the idea! Of course, chopsticks are, at first, very awkward objects for our fingers to grasp, and many Chinatown tourists have doubted the feasibility of eating with them. With practicve and the proper technique, however, we eventually learn to pity those poor Europeans who must stab their salad greens with a sour-tasting, bent metal object with no moving parts. Such is the pity I feel for everyone who must crunch his numbers decimally, or cast his points to float among electronic registers. I will admit that continued fraction techniques are not the best way to handle everything, but then neither are chopsticks.