TITLE:
Extensions of the System N of Natural Numbers Assigned to Primary Teachers
AUTHORS:
Milosav M. Marjanović
KEYWORDS:
Properties of Operations and the Order Relation in the System N of Natural Numbers, Non-Negative Rational Numbers, Integers, Permanence of Properties
JOURNAL NAME:
Open Access Library Journal,
Vol.5 No.1,
January
5,
2018
ABSTRACT: This paper has resulted from a discussion conducted about which content of mathematics course should be the most suitable for the students of the institutions where primary teachers are educated and when such a course exists on the curriculum. It is normal to require that a primary teacher knows, in a deeper way, mathematics at proximate levels (preschool and upper classes of elementary school). In particular, this supposes a good acquaintance of the teacher with extensions of number systems. We propose that algebra is used as a tool of expressing properties of operations and the order relation in the system N of natural numbers. The jottings m:n,(n≠0)and m-n have their meaning in N only when m is divisible by n and when m≥n respectively. But these jottings are potentially numbers in an extended sense and we use former of them to extend the system N to the system Q+ of non-negative rational numbers and the latter of them to extend N to the system Z of integers. This natural way of extension is based on the permanence of properties already established in N and they are used to define operations and the order relation in the extended systems (Peacock’s principle). In this paper, we present the extension of N to Q+ in detail and we just sketch the extension of N to Z. Let us note that the fractions are interpreted as the result of equal sharing of quantities and they are represented on the number axis as the sequences of adjacent intervals of equal length whose union is the interval representing the corresponding rational number. We leave open the question up to which degree these extensions should be supplied with all technical details and when some of these details should be just suggested. The answers to such a question certainly depend on many specific factors.