"Lucas, Stephen K - lucassk" <lucassk@jmu.edu> wrote:

Keith F. Lynch <kfl@KeithLynch.net> wrote:

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I'm pretty sure that rational numbers, other than integers and their reciprocals, won't work.

Rational numbers work as the base of a number system where the integers terminate if you are careful with the digits allowed, not limiting yourself to digits up to floor(R).

Thanks.

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Once you've chosen a radix, what digits should be allowed? The standard is 0 through R-1, where R is the ceiling of the radix. (Assuming the radix is greater than 1.)

I've since realized that for algebraic irrational bases, in the general case I can't limit myself to digits up to floor(R) or even floor(R+1) if I want all integers to terminate. For instance in base sqrt(10) the current year is 2000108 and Columbus sailed in 1040902. I could represent those numbers with no digits greater than 3, but then neither number would terminate.

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But as balanced ternary demonstrates, that's not the only choice. Will any set of R consecutive digits do, so long as one of them is 0? For instance almost-balanced decimal, where the decimal digits are -4 through +5? Will any other set do?

Other weird sets are possible. I have a vague recollection that digit sets in base ten are possible with gaps. But a search through my literature pile has been unsuccessful in tracking down the paper.

This looks like a job for brute force. Base ten is a bit much for a one-hour project, but I checked ternary with all possible set of three digits between -9 and 9 inclusive. The following digit sets seem to be able to represent all integers, and presumably all reals as well: {-7,0,1}, {-5,0,2}, {-4,0,1}, {-2,-1,0}, {-2,0,5}, {-1,0,1}, {-1,0,4}, {-1,0,7}, {0,1,2}. Note the symmetry that if I replace each digit with its negative, that works too. I then did the same with quaternary, also with digits -9 through 9. The following sets of four digits appear to work: {-9,-7,0,2}, {-9,-2,0,1}, {-9,0,1,2}, {-9,0,2,5}, {-7,-5,0,2}, {-7,-1,0,2}, {-7,-1,0,6}, {-7,0,2,3}, {-6,-5,0,1}, {-6,-1,0,1}, {-6,-1,0,5}, {-6,0,1,7}, {-5,-2,0,1}, {-5,-2,0,9}, {-5,0,1,2}, {-5,0,1,6}, {-3,-2,-1,0}, {-3,-2,0,7}, {-3,-1,0,2}, {-2,-1,0,1}, {-2,-1,0,5}, {-2,-1,0,9}, {-2,0,1,3}, {-2,0,1,7}, {-2,0,5,7}, {-2,0,7,9}, {-1,0,1,2}, {-1,0,1,6}, {-1,0,2,5}, {-1,0,2,9}, {-1,0,5,6}, {0,1,2,3}. Again, if I replace each digit with its negative, that works too. The only other patterns I notice immediately are that 0 must always be included and that both negatives and positives must be included unless the digits are consecutive. Surprisingly, neither 1 nor -1 needs to be included. (If base four is called quaternary, why isn't base three called tertiary? Geologists have abandoned both terms, replacing them with neogene and paleogene, so mathematicians might as well use them. Just so long as we don't confuse triadic fractions with triassic fractions.)