The Stern-Brocot tree is _incredible_; I've been trying to grasp the implications for a while. It entirely includes the ability to programmatically search all of the rationals, to automatically find the best rational approximation to any testable real (for several excellent and useful definitions of "best"), and so on. It's been expanded to include the n-tuples of rationals as well, including limited support for complex numbers (although that theory isn't _quite_ as simple).
You don't need to "know" your target (and think about it, an irrational real number cannot possibly be _known_ without computing it); you only need to be able to test whether you're below, above, or at your target (for each dimension you're considering). So you have to know something _about_ your target. I suspect this means your target must be computable -- Chaitin's Omega is one of aleph-1 examples of such uncomputable reals. (Wikipedia: "a Chaitin constant (Chaitin omega number)[1] or halting probability is a real number that, informally speaking, represents the probability that a randomly constructed program will halt.") It's an incredible way to look at the deep structure of the rationals, which we're often tempted to mistake as being a simple, flat, and boring continuum. The following is a good video introduction -- not multimedia, just a good clear lecture. The next few episodes add some more recent research in the tree as well. https://www.youtube.com/watch?v=CiO8iAYC6xI -Wm On Wed, Jan 31, 2018 at 3:56 AM Martin Kreuzer <[email protected]> wrote: > After some reading I now accept that going through the tree I could > approximate any real number with fractions (taking mediants on the > way). But doesn't that mean I have to know the target in the first place..? > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
