[math-fun] radix r numbers, |r|<1

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.