2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/170470The 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 |b|\ge 4, and for b=-2. For the special case b=-2, we give a full characterisation of all valid digit sets.30 pages, to appear in Journal of Number TheoryNumber Theory11A63text