And now I probably need to retract some of my “discovery,” because much of it has been done by William Gilbert of Waterloo. He has a whole set of research papers on fractal geometry and complex bases, and if you are interested, please check out his manuscript (unpublished but dated 2002) at https://www.math.uwaterloo.ca/~wgilbert/Research/CxBases/CxBases.html His section 4 shows how to convert to a Gaussian integer to a complex base, and what he calls a “clearing algorithm” is exactly the idea of a carry rule. The only novelty of my approach over Gilbert’s is that I allow for a carry rule with Gaussian integers, which makes base 2+i easier to work with. Unfortunately, Gilbert references: M. DAVIO, J. P. DESCHAMPS AND C. GOSSART, Complex arithmetic, Philips M. B. L. E. Research Lab. Report R369 (May 1978) Brussels. He states that they study base 1+2i and its arithmetic, which may be what I did in base 2+i, and so nothing here is new at all! Unfortunately, I don’t have access to a Phillips Research Lap report to find out… 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 Apr 28, 2018, at 11:52 AM, Lucas, Stephen K - lucassk <lucassk@jmu.edu<mailto:lucassk@jmu.edu>> wrote: Joerg has told us that he found the digits in base 2+i by brute force searching through all the digits. So I have managed to derive a way of finding the digits using a carry rule approach. This is surprisingly building on current research on representing Gaussian integers with an undergraduate student, that we are hoping to submit by the end of the summer. Katai and Szabo showed you can represent a+ib in base -n+i using digits 0 to n^2. My student discovered that base 1+i can be used with digits 0 and 1 except for occasionally a leading digit i, and later I found that digits -1 and 0 work as well. Base n+i with negative digits appears to work. In base b we have the carry rule [+1,-b] saying if we subtract b from one digit we add one to the next most significant, being how positional notation works. In base 2+i, the carry rule is [+1,-2-i], which is great to reduce numbers where the real part is bigger than the imaginary part. But in other cases we will want to reduce the imaginary part more, so times i we get the carry rule [+i,1-2i]. We can then alternate applying these carry rules to reduce a digit to a valid digit, and have a new digit in the next position to simplify. By the way, this is the first time I’ve attempted to use a carry rule with digits that aren’t a sequence of integers, I’m kind of surprised it works. For example, start with 14, are as a list of digits, (14). Applying the first carry rule 7 times gives us digits (most significant to the left) (7,-7i). Second carry rule (-3) times gives (7-3i,-3-i), then first carry rule times (-1) gives (6-3i,-1). Continuing, we get the sequence (3,-6i,-1), (3-3i,-3,-1), (2-3i,-1+i,-1), (2-2i,-i,-1), (1,-3i,-i,-1), (1-i,-1-i,-i,-1), (-i,1,-1,-1). This also gives us an approach to do arithmetic with these digits. Just add the digits then apply carry rules to reduce them back to the required set. Since possible digits after addition are +/- 1 +/- i, one carry rule application might be all you need. Unfortunately, this is a lot of applications of the carry rule. I think I have come up with an approach that works out how many applications of the two carry rules are needed at every step, identifying the next digit immediately and the carry. It uses modular arithmetic to choose the correct remainder to keep carries as Gaussian integers, which identifies {0,1,-1,i,-i} as necessary. It also proves uniqueness, but existence requires a proof that the process converges. But that should work out fine. Joerg also referenced back to the question of representing a+ib in base -1+i using digits 0 and 1. This goes back to Penney (A “binary” system for complex numbers, Journal of the ACM 12(2) 1965) with a horrible construction based on (-1+i)^4=-4 and lots of special cases. Jamil (The complex binary number system, IEEE Potentials 20(5) 2002) did it slightly better, but still horrible. Using the carry approach, this is almost trivial, and directly from my current research paper: start with a+ib = (a+b)+(b)(-1+i), so digits (b,a+b). Then the carry rule for base -1+i over three digits can be proven to be [+1,+2,+2], so apply to the rightmost digit to make it 0 or 1, then keep applying the carry rule right to left and you get the digits. 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