Henry, There are several papers on the topic, and at least one more currently in preparation. Knuth certainly has his famous base 2i paper, but the most famous algebraic positional system is base golden ratio using digits 0 and 1: Bergman, Mathematics Magazine, 1957. Rousseau (1995 Mathematics Magazine) also spends some time with base ph, and the website http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/phigits.html has more on the topic gathered in one place than any other reference I know. There is a lot of work on so-called “beta expansions”, which is all about representing numbers in irrational bases with a finite number of digits. This goes back to Renyi, 1957, Representations for real numbers and their ergodic properties, but there are lots of recent references, including Ambroz et al, Arithmetics on number systems with irrational bases, Bull. Belg. Math. Soc 2003. You will be pleased (?) to know that the theory of natural numbers and their representation in bases that are the solutions of quadratic equations is the topic of current research by Jim Propp and myself. At the most recent gathering for Gardner, Jim gave a presentation on how to represent natural numbers using digits that are simultaneously correct in base 2 AND base 3. This is equivalent to the carry rule (how to manipulate digits without changing the number) [+1,-5,+6] using the digits from zero to four. The good news is that the same approach works for any pair of bases that satisfy the same quadratic equation. Including irrationals and complex conjugate pairs. The bad news is that you will have to wait a while for this particular paper, because we are still writing it and trying to understand various special cases. But as a fun teaser, everything you read about base golden ration (1+sqrt(5))/2, it turns out for natural numbers the digits also represent the number in base (1-sqrt(5))/2, which is not something yet discussed in the literature. All of the “metallic means” (see the Wikipedia article with the same name) have nice carry rules [+1,-n,+1] using digits from 0 to n-1, so any natural number has a representation with a finite number of digits these irrational bases, and a simple finite construction exists — and arithmetic can be done on numbers in these forms automatically. But of course, we need to finish writing the paper :-) Steve -- 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>> 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> https://urldefense.proofpoint.com/v2/url?u=https-3A__mailman.xmission.com_cg...