In this article we study a numeration system previously used to prove combinatorial properties in discrete geometry: the ∆-numeration. Since this system, introduced via the fully subtractive algorithm, has been seen mainly as a tool, we propose here to study it from the point of view of numeration systems. In particular, we make the link with βnumeration and Cantor real bases. We reintroduce the rewriting system introduced to calculate in ∆-numeration. This systems is based on the properties of the fully subtractive algorithm and is normalising. Finally, we study the ultimately periodic case, a special case of alternate bases, and show that the ultimately periodic words represent exactly the elements of Q[β] where β is the inverse of a Pisot number.