_{∞}

r = ∑ a

_{i}b

^{-i}

^{i=0}

where a

_{0}is a non-negative integer, and the a

_{i}are integers satisfying 0 ≤ a

_{i}< b for i = 1,2,3, ...

r = a

_{0}.a

_{1}a

_{2}a

_{3}...

An underscored numeral will denote that the numeral is repeated indefinitely. If b=10 for example, then

0.45

__783__denotes 0.45783783783783....

No matter what the base b, an irrational number has only one b-nary representation.

Looking at the table below we see that (depending on the base) a rational number can have one or two b-nary representations.

If p, q, and b are integers with p ≥ 0, q > 0, and b > 1, then what is the relation between p, q, and b such that the rational number p/q has only one b-nary representation?---------------+-----------------------+-----------------------+ Base | Ten | Three | ---------------+------------+----------+------------+----------+ Representation | fraction | b-nary | fraction | b-nary | ---------------+------------+----------+------------+----------+ | 1/3 | 0.3| 1/10 | 0.10| | | | | or | | | | | 0.02| ---------------+------------+----------+------------+----------+ | 1/5 | 0.20| 1/12 | 0.0121| | | or | | | | | 0.19| | | ---------------+------------+----------+------------+----------+