I should mention that one of the things I figured out in the past few months is that base 3/2 is best understood not in the context of the real numbers but in the context of the 3-adics. When you use the ordinary “(-1)-adic” notion of convergence, it’s unclear whether there’s a natural way to write a fraction like one-half in base 3/2 using the digits 0, 1, and 2, but switch to the 3-adics and you find that every 3-adic number has a unique representation. Something similar happens for base two-and-three with the digits 0 through 5; every rational number has a canonical representation that comes from viewing it 2-adically and 3-adically. But I’m not sure what the appropriate completion of Q_2 x Q_3 is. Jim Propp On Wednesday, May 16, 2018, Jeffrey Shallit <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:h baker1@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.
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