Structuring systems of natural, positive rational, and rational numbers

Abstract

In this paper, we are concerned with the extensions of the system N of natural numbers with 0 to the system Q+ of positive rational numbers with 0 and then, the extension of this latter system to the system Q of rational numbers. Structuring these systems, we start with the system of natural numbers forming the list of its basic operative properties (i.e. the properties of operations and the order relation) and using only these postulated properties, we deduce from them a series of properties of proportions and equalities that relate differences. While deducing, we suppose that all involved ratios and differences are defined in N, i.e. they have the value which is a natural number.