We introduce a number system with a fixed rational base. For instance, we consider a number system based on powers of 3/2 with digits 0,1,2, with some restrictions. This system is also produced by an accelerated adding machine. The system gives a way to expand positive integers. Though the associated language is not even context-free, the odometer is given by an automaton.
The system can be "compactified" by extending to the right. Each real number is aperiodic in this expansion. Up to countable many exceptions, the expansion is unique. We characterized this countable set of exceptions and found an interesting connection with Mahler's problem on the distribution of fractional parts of (3/2)^n.