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. 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? 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.