4 Number Systems

K-12 mathematics is rich in the study of number systems, with particular focus on the Natural Numbers, Integers, Rational Numbers, Real Numbers, and Complex Numbers. In this chapter we use the properties of set theory and equivalence classes discussed in Chapters 2 and 3 to construct these numbers systems. We follow this discussion with a study of some of the properties of number systems, connecting them to the K-12 curriculum. Lastly, we examine different representations of these number systems helpful for developing numerical fluency and conceptual understanding of the numerical operations.

This process of constructing the number systems is very abstract and challenging for those who have not previously studied these constructions. The process is included because it helps teachers to better understand the abstract nature of the number systems and some of the challenges of their students. While working with many of these number systems has become second nature for math teachers, students struggle much more than new teachers expect. So a deeper knowledge of the properties of the number system helps in this teaching situation.

\[\mathbb{N} \subseteq \mathbb{Z} \subseteq \mathbb{Q} \subseteq \mathbb{R} \subseteq \mathbb{C}\] As we are working with the various number systems, we will use varying notation for our operations. With all of the constructions, subtraction and division are not defined. We instead define the additive inverse and multiplicative inverse and so we can think of subtraction \(a-b\) as \(a+(-b)\) and division \(a\div b\) as \(a \times \frac{1}{b}\). We will also use various symbols to represent multiplication so that \(a\times b\), \(a\cdot b\), and \(ab\) all represent the same thing.