∞
r = ∑ aib-i
i=0
where a0 is a non-negative integer, and the ai are integers satisfying 0 ≤ ai < b for i = 1,2,3, ...
r = a0.a1a2a3 ...
An underscored numeral will denote that the numeral is repeated indefinitely. If b=10 for example, then
0.45783 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.
---------------+-----------------------+-----------------------+
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 | | |
---------------+------------+----------+------------+----------+
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?
