Thanks Jeffrey, clearly I am going to have to get a copy of your book! There does seem to be quite a scattering of material on positional representation, with some streams of references that run in parallel. It would be nice to be able to see all of them in one place at some point. -- Stephen Lucas, Professor Department of Mathematics and Statistics MSC 1911, James Madison University, Harrisonburg, VA 22807 USA Phone 540 568 5104, Fax 540 568 6857, Web http://educ.jmu.edu/~lucassk/ Email lucassk at jmu dot edu (Work) stephen.k.lucas at gmail dot com (Other) Mathematics is like checkers in being suitable for the young, not too difficult, amusing, and without peril to the state. (Plato) On May 16, 2018, at 5:22 AM, Jeffrey Shallit <shallit@uwaterloo.ca<mailto:shallit@uwaterloo.ca>> wrote: Also there is a pretty long discussion about complex bases and automata in my book with Allouche, "Automatic Sequences", in the chapter on representations of integers. And a very large bibliography. Jeffrey Shallit On 5/15/18 9:42 PM, Lucas, Stephen K - lucassk wrote: More possibilities: negative integer bases are done by Gilbert & Green, 1979, Negative Based Number Systems, Mathematics Magazine, 52(4). Their section on arithmetic can be improved, but the representation in negative bases is beautifully presented. Katai & Szabo, Canonical Number Systems for Complex Integers, prove that base -n+i or -n-i with natural number n can be used to represent Gaussian integers using digits from 0 to n^2. Other bases require other digit sets, which include recent discussion with Joerg Arndt on this list on base 2+i using digits {0,1,-1,i,-i}. -- Stephen Lucas, Professor Department of Mathematics and Statistics MSC 1911, James Madison University, Harrisonburg, VA 22807 USA Phone 540 568 5104, Fax 540 568 6857, Web http://educ.jmu.edu/~lucassk/ Email lucassk at jmu dot edu (Work) stephen.k.lucas at gmail dot com (Other) Mathematics is like checkers in being suitable for the young, not too difficult, amusing, and without peril to the state. (Plato) On May 15, 2018, at 8:54 PM, Henry Baker <hbaker1@pipeline.com<mailto:hbaker1@pipeline.com><mailto:hbaker1@pipeline.com>> wrote: Well, I'm not 100% certain, but *someone* must have written a paper *sometime* about positional number systems using an *algebraic* and/or *algebraic integer* radix and integer numerals. Knuth? Knuth? Anyone? Anyone? Several interesting things: If p(r) is the minimal polynomial for r, and deg(p)=n, then we can express r^n in terms of lower powers of r, and thus there is some possible redundancy in the representations. Also, if n>1, then there are multiple r's satisfying p(r)=0, so we have to relate representations using r and r', s.t. p(r)=p(r')=0. Clearly, complex number systems of the 1+i type qualify, but I don't recall any such systems with n>2. Also, cyclotomic polynomials have the same unfortunate property that base-(e^i) numbers have -- namely, it is a lot more difficult to represent large numbers. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com<mailto:math-fun@mailman.xmission.com><mailto:math-fun@mailman.xmission.com> https://urldefense.proofpoint.com/v2/url?u=https-3A__mailman.xmission.com_cg... _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://urldefense.proofpoint.com/v2/url?u=https-3A__mailman.xmission.com_cg... _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com<mailto:math-fun@mailman.xmission.com> https://urldefense.proofpoint.com/v2/url?u=https-3A__mailman.xmission.com_cg...