Noncanonical number systems in the integers Original Research Article

The well-known binary and decimal representations of the integers, and other similar number systems, admit many generalisations. Here, we investigate whether still every integer could have a finite expansion on a given integer base b, when we choose a digit set that does not contain 0. We prove that such digit sets exist and we provide infinitely many examples for every base b with bgreater-or-equal, slanted4, and for b=−2. For the special case b=−2, we give a full characterisation of all valid digit sets.