Oops! Posted too soon. What I was really trying to get at was irrational/transcendental bases r, |r| close to 1, and what "numerals" to use. It appears that when 0.5<|r|<1.5, we have to leave digit-0 "holes". E.g., suppose for the moment that r=1.1. It will take (1.1)^8 = ~2.14 before we will need another "1" digit. So we get representations with subsequences of 0's in them. Ditto for r=0.9, where it will take 0.9^(-7) = ~2.09. At 09:16 AM 4/28/2018, Henry Baker wrote:

Positional number systems that can express *every* number rely on the fact that the geometric series

2 3 4 5 6 7 8 9 10 (%o7)/T/ 1 + r + r + r + r + r + r + r + r + r + r + . . .

*diverges* for |r|>=1.

But what if we don't care about representing *every* number?

What if we only care about the interval |x|<2 ?

In this case, r=1/2 should work, right?

Then 1.5 = 1+1/2 = "1.1"; 1.25 = 1+1/4 = "1.01"; etc.

A greedy algorithm should produce an acceptable representation.