Joerg, Yes, you have the reference correct to Katai & Szabo. If you read their proof carefully, you discover that they essentially use a carry rule to represent a Gaussian integer in base -n+i, and prove existence and uniqueness. But explicitly writing things in terms of a carry rule makes number conversion and arithmetic much easier to deal with. I must admit, after looking back at your list of representations in base 2+i using {1,-1,i,-1,0}, I’m intrigued at how you came up with the digits. It is far from obvious, and the number of digits going up then down is bizarre! Steve
On Apr 27, 2018, at 3:55 AM, Joerg Arndt <arndt@jjj.de> wrote:
That carry method: neat!
The "classic work of Katai and Szabo" I assume is I.\ K\'{a}tai, J.\ Szab\'{o}: Canonical number systems for complex integers, Acta Sci.\ Math.\ (Szeged), vol.~37, no.~3-4, pp.~255-260, (1975).
Best regards, jj
* Lucas, Stephen K - lucassk <lucassk@jmu.edu> [Apr 27. 2018 09:05]:
New to the list, so I hope this will go in the right place. But replying to Joerg Arndt’s listing of the first few natural numbers in base 2+i using the digits {0,+1,-1,+i,-i}:
The problem with this digit set is that it is not immediately obvious what the rule is to go from number to number adding one. Here is another idea: use the digits {-1,0,1,2,3}. And base 2+i has the carry rule [-1,+4,-5], which says if a digit is 4 or greater, subtract 5 from it, add 4 to the next most significant, subtract one from the next next most significant. One way of verifying this is to see that (-1)(2+i)^{k+2}+4(2+i)^{k+1}-5(2+i)^k=0. For example, with least significant digits on the right, 1 = 1, 2 = 2 3 = 3 4 = (-1)4(-1) = (-1)3(-1)(-1) 5 = (-1)3(-1)0 6 = (-1)3(-1)1 7 = (-1)3(-1)2 8 = (-1)3(-1)3 9 = (-1)3(-1)4 = (-1)23(-1) 10 = (-1)230 11 = (-1)231 12 = (-1)232 13 = (-1)233 14 = (-1)234 = (-1)17(-1) = (-2)52(-1) = (-1)202(-1) and so on.
A couple of notes: these digits are exactly the same representing natural numbers in base 2-i as well. This is current (!) joint work with Jim Propp on digits representing numbers in different bases simultaneously. And in base 2+i, you can extend the classic work of Katai and Szabo and represent Gaussian integers in base 2+i. If you limit yourself to nonnegative digits, they proved only -n+i works, not n+i. But a negative digit does the job!
Steve
_______________________________________________ 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 https://urldefense.proofpoint.com/v2/url?u=https-3A__mailman.xmission.com_cg...