On Wed, Mar 05, 2003 at 12:53:52AM -0500, Gershon Bialer wrote:
So far I have stayed out of the math-fun politics, but it appears some people like to complain about math-fun. I propose a solution. Someone want ...
Quoting wpthurston@mac.com:
On Thursday, February 27, 2003, at 07:54 AM, David Wilson wrote: ... So let it die down!!! Sorry about my post, it was written immediately after David Wilson's, which I thought merited reasonable discussion at the time; unfortunately it arrived a week later because I forgot and sent it from a different email address. ============= I've kept quiet about the discussion of different bases --- there's a whole lot that is known about these kinds of questions. One interesting angle that hasn't been mentioned is that there are interesting alternate choices of digits even for standard integer bases. For instance, digits {-4,-3,-2,-1,0,1,2,3,4,5} for base 10 work in some sense better than the usual choice, since you don't need a sign to describe negative numbers. As another example--- in base 5, the digits {-3, 0, 1, 3, 4} at least uniquely give all positive numbers. ... can you characterize which sets of digits "work" for integer bases?
One motivation is that for complex bases, there is no canonical choice of digits. BTW, a while ago I gave a characterization of those real or complex bases for which there is an almost-always unique representation of any number given by finite-state rules: this can be done if and only if the base is an algebraic integer whose Galois conjugates are not larger (in modulus) than itself. Bill Thurston