Nita H. Shah - Journey from Natural Numbers to Complex Numbers

This book is about different types of numbers. Counting process starts with positive whole numbers, known as natural numbers. The system is extended to have the negative whole numbers and zero in it. It is further extended to rational numbers, followed by real numbers, and ultimately by complex numbers. For natural numbers, integers, and rational numbers, before going for the formal study, we have started with real-life problems and introduced the basis of formal definition with real-life problems.

The formal study of the system has references to some topics in mathematics. For the sake of completeness, we have included a brief introduction of these topics in [13]. We have started with set theory. Set theory starts with an informal definition of set, followed by the definition of subset, superset, null set, and universal set. Operations like union, intersection, and product are introduced. The collection of sets and union and intersection operations on the collection of sets are introduced. Power set is introduced, followed by the relation between two sets. A special case of the relation of sets with itself is taken. Properties of relations such as reflexivity, symmetry, transitivity, and antisymmetry are introduced. Based on these properties, equivalence relation and partial order are defined. Partition of the set on the basis of the equivalence relation is defined. The partial order is extended to the total order. On the basis of relation, functions are defined. Properties of functions and composite functions are defined. The concept of invertible functions is introduced. The cardinality of the set is defined using functions. Some interesting facts related to cardinality are proved or stated. Binary operations are introduced as a function from product set of a set with itself to the same set. Properties of binary operations such as commutativity and associativity are defined. Identity element and the inverse of the member operand are defined. The structure of a group is defined, and Abelian groups are defined. A set with two operations (one called addition and another multiplication) is introduced, and the distributivity of multiplication over addition is defined. Ring and fields are defined for the set with operations of addition and multiplication. This brief introduction of concepts can be used as a quick reference for all formal mathematics courses. With this background knowledge, we proceed to the number systems.

We have given an informal introduction of natural numbers. Addition and multiplication are explained with real-life physical activities and structures, keeping the formal definition of the operations in mind. Informal proof of commutativity and associativity is given. After the informal introduction of natural numbers, we move to the formal definition. Axiomatic definition of natural numbers is done using set-theoretic definition and Peano axioms. The formal definitions of addition and multiplication are given. Ordering is defined in natural numbers. The first principle and the second principle of mathematical induction and well-ordering principles are stated. The equivalence of three principles is stated. Prime numbers and composite numbers are defined. A major limitation of the absence of additive inverse and multiplicative inverse is identified, which forms the basis for the extension of natural numbers to integers. Coming back to the physical aspect of natural numbers, the representation of natural numbers is taken up. Position-based value system for writing numbers is explained by two models. This is generalized to any positive integer base greater than 1. Conversions of number from one base to another base are explained. Popular bases 2 (binary), 8 (octal), and 16 (hexadecimal) are taken up. Special cases of converting numbers to base, where the target base is the power of the source base or the source base is the power of the target base, are taken up. Computers use the binary system. The relation between binary systembased arithmetic and Boolean algebra is explained. The rationale of the usage of the octal and hexadecimal systems by computer professionals is explained. Binary Coded Decimal (BCD), another important system, is explained. In the next chapter, building up of integers using natural numbers is done.