Introduction: In the present-day didactic process of transition from natural numbers to integers, a certain kind of measure is involved which is neither easy nor appropriate to integrate into the familiar numerical systems. We refer to measures and comparisons for which we need a third numerical system--what we will call "relative natural" numbers; this numerical system presents five structural differences from the systems of natural numbers and of integers. In this paper we present part of a study whose purpose was to make evident that such differences are also cognitive. Method: Qualitative meta-analysis and didactic analysis are used for theoretical study and for analysis of responses, for the descriptive study and correspondence analysis of responses to structured questionnaires. Results: The algebraic structures and the epistemology and phenomenology of relative natural numbers and of integers are different. Subjects distinguish between the two types of numbers by responding differently to situations which pertain to one or the other type of numerical structure. Discussion: Results confirm our conjectures, and call for a new organization of the additive numerical field, in which additive relativity and relative natural numbers would be considered, as is the case with proportionality and fractions in the conceptual field of multiplication. (Contains 3 figures, 4 tables, and 6 footnotes.)