Journey from Natural Numbers to Complex Numbers

Nita H. Shah, Vishnuprasad D. Thakkar

• Publisher: CRC Press
• Year: 2020
• Print ISBN: 9780367613327
• eBook ISBN: 9781000299632
• Edition: 1
• Topic: Business
• Subtopic: Operations
• Index: Business

This book is for those interested in number systems, abstract algebra, and analysis. It provides an understanding of negative and fractional numbers with theoretical background and explains rationale of irrational and complex numbers in an easy to understand format.

This book covers the fundamentals, proof of theorems, examples, definitions, and concepts. It explains the theory in an easy and understandable manner and offers problems for understanding and extensions of concept are included. The book provides concepts in other fields and includes an understanding of handling of numbers by computers.

Research scholars and students working in the fields of engineering, science, and different branches of mathematics will find this book of interest, as it provides the subject in a clear and concise way.

1
Natural Numbers

Our introduction to mathematics starts with positive whole numbers, also known as natural numbers. We will be going through informal and formal aspects of all types of numbers. We know that natural numbers are not enough for mathematics. For formal aspects of the development and extension of number types, we need background knowledge of some topics in mathematics. Hence, we start with the introduction of these concepts.

1.1 Prerequisites

1.1.1 Set Theory

We start with definitions related to sets. First of all, we define set. The definition given here is intuitive and not a formal one. The formal definition may involve some other terms, which may in turn require the definition of those terms. So we start with set as the initial term with an intuitive definition.

Definition 1.1: Set is an unordered collection of distinct objects.

For a collection being considered to be qualified as a set, one should be able to check unambiguously with certainty whether any object is in the collection or not. Generally, sets have the same type of members, but it is not necessary to be so. It may contain different kinds of objects. A common convention is to represent a set by an uppercase letter, and the member is represented by a lowercase letter. Lowercase letters are used as a member variable symbol, and the actual member can be anything. Using a lowercase letter to represent the member does not mean that members are lowercase letters. To indicate that an object a is in set P, we write a ∈ P, read as a belongs to P or a is in P. Similarly, if b is not in P then we write b ∉ P, read as b does not belong to P or b is not in P.

One of the methods to describe a set is listing all its members. If the number of members is large, then … is used to indicate terms similar to listed near it. The method of describing a set using a member list is called roster method. Another method of describing set is by set builder. In this method, a representative member is written on the left side of the vertical bar |, and related details are written on the right side of the vertical bar. In both the methods, set details are enclosed in curly brackets {}.

Some popular sets are: set of natural numbers denoted by N or ℕ, integers denoted by Z or ℤ, rational numbers denoted by Q or ℚ, real numbers denoted by R or ℝ, and complex numbers by C or ℂ .

Definition 1.2: Set A is a subset of set B if and only if all elements of A are elements of B.

Set A is a subset of itself as all members of A are in it. The symbol ⊆ used to indicate that set is subset of another set. To indicate that the subset is not the set itself, symbol ⊂ is used. If the subset being set itself is not significant, the symbol ⊂ is used as a common symbol.

For a set to not become a subset of another set, it should have an element, which is not in the other set. For set A to not become a subset of B, there should be an element a ∈ A ∍ a ∉ B .

An alternate representation of ‘A is subset of B’ ( A ⊆ B ) is ‘B is superset of A’, denoted as B ⊇ A .

An important point to be noted here is two sets A and B are equal if and only if A ⊆ B and B ⊆ A .

Definition 1.3: The set containing no elements is called an empty set or null set.

The null set is well-defined. Any candidate object can be checked for being a member of this set. Given any candidate, the result for membership of the set is that it is not a member. The symbol used for the null set is ϕ.

The null set does not have any element. So, for another set A, null set having an element which is not in se...

Table of contents

• Cover
• Half-Title
• Series
• Title
• Copyright
• Contents
• Preface
• Author biographies
• 1 Natural Numbers
• 2 Integers
• 3 Rational Numbers
• 4 Real Numbers
• 5 Complex Numbers
• Index