A question insipired by my previous post, in which I mentioned seventeen in base pi (120.2200211010202300200030...). In what non-integer real-number bases greater than 1 can the representations of all integers terminate? (The representations may need to be non-canonical, i.e. don't necessarily use a greedy algorithm.) Examples are base sqrt(2) and base phi. I believe all such bases are irrational but not transcendental. (Are they all quadratic?) Can anyone prove this? Can anyone list all such bases? I mention "greater than 1" since there's a symmetry around 1. If all integers terminate in base b, they also terminate in base 1/b. Just reverse the order of the digits and the radix point. For instance 2013 in base 1/10 is 0.3102. They also terminate in base -b and base -1/b. Also, what can be said about bases in which the representations of some integers repeat rather than terminate? I know that N can be non-canonically represented as 1.111111111111111... in base N/(N-1), but I don't think any other integers repeat in that same base except powers (and roots?) of N.