It's not quite true that we haven't looked at arbitrary real numbers. I've looked at real numbers, and it seems pretty hopeless. I think there's a fundamental incompatibility between using the ordinary (Archimedean) completion of Q and using the number 3/2 as a base. You get too many representations and no criterion for choosing one that's canonical. But I'd love to be wrong about that. Jim Propp On Wed, Dec 19, 2018 at 12:52 PM Lucas, Stephen K - lucassk <lucassk@jmu.edu> wrote:
On Dec 18, 2018, at 11:38 PM, Keith F. Lynch <kfl@KeithLynch.net<mailto: kfl@KeithLynch.net>> wrote:
Any real number except 0, 1, and -1 can be used as the radix (base) in a place value system intended to represent all real numbers. But what if we restrict it to radices such that all integers terminate? Any integer radix greater than 1 will work. So will integer roots of those integers, for instance the square root of 2 or the fifth root of 12. Phi will work. No transcendental radix will work. The reciprocals and the negatives of anything that will work will also work. 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). Jim Propp came up with the idea years ago and has publicized it a few times, including his blog post that I hope everyone who hasn’t read will look at, at
https://mathenchant.wordpress.com/2017/09/17/how-do-you-write-one-hundred-in... Jim proved that every integer has a unique representation in base p/q (relatively prime) with digits 0,1,…,p-1. Recently he has shown that using p-adic notation you can represent any fraction uniquely. We haven’t looked at arbitrary reals…
Is the complete solution set known? Is it dense?
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.) 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.
I'll save complex radices for another day. And I'm pretty sure that quaternion radices won't work, i.e. can't represent all quaternions given nothing but real integer digits.
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