This paper presents recent results of this author and Z. Kadelburg and also contains additional comments and remarks. These results complete the study of number systems which this author has published in OALib Journal in search of a ground for designing a simplified content assigned to primary teachers. This ground consisted of derivation of properties of the system N of natural numbers with 0 and of the use of these properties as a basis for the extension of N to the systems of positive rational numbers and the system of integers. The crucial step in our final research paper has been the selection of basic operative properties of the system N (properties of operations and the order relation). Using these properties and those deduced from them, the system Q of positive rational numbers with 0 has been constructed and its basic operative properties verified (being those of N, plus the property of existence of multiplicative inverse). Then, using properties of Q , the system Q of rational numbers is constructed and its properties verified (being those of Q , plus the existence of additive inverse). Taking the basic operative properties of N for the axioms, we define the algebraic structure {S, , ×, <}, where S is a non-empty set, “ ” and “×” two binary operations and “<” the order relation, which we call N-structure, The number systems N, Q and Q, as well as the system R of real numbers are examples of N-structure. Thus, they share all properties that the axioms express and all those deduced from them. Hence all these properties when transcribed with the letters denoting the corresponding variables are transferable from N to the systems Q , Q and R. This is a precise formulation of the Peacock’s principle of permanence and a firm logical basis for its validity. Finally, we prove that each system satisfying the axioms of N-structure contains an isomorphicb copy of N. Thus, the natural numbers with 0 are the smallest system satisfying the axioms of N-structure, This is a characterization of the system N which is analogous to that of Q which is the smallest ordered field.