Cover image for Journey from Natural Numbers to Complex Numbers

Journey from Natural Numbers to Complex Numbers

Author Nita H. Shah , Vishnuprasad D. Thakkar
Release Date: 2020/12/01
ISBN: 9781000299632
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14
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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.

Table of Contents

  1. Cover
  2. Half-Title
  3. Series
  4. Title
  5. Copyright
  6. Contents
  7. Preface
  8. Author biographies
  9. 1 Natural Numbers
    1. 1.1 Prerequisites
    2. 1.1.1 Set Theory
    3. 1.1.2 Relation
    4. 1.1.3 Function
    5. 1.1.4 Cardinality
    6. 1.1.5 Algebra
    7. 1.2 Positive Integers
    8. 1.2.1 Positive Integers in Real Life
    9. 1.2.2 Set Theoretic Definition of Natural Numbers
    10. 1.2.3 Peano Axioms
    11. 1.2.4 Ordering in Natural Numbers
    12. 1.2.5 First Principle of Mathematical Induction
    13. 1.2.6 Second Principle of Mathematical Induction
    14. 1.2.7 Well-Ordering Principle
    15. 1.2.8 Limitations of Natural Numbers
    16. 1.2.9 Representation of Natural Numbers
    17. 1.2.9.1 Hexadecimal System
    18. 1.2.10 Number System Used by Computers
    19. 1.3 Summary
  10. 2 Integers
    1. 2.1 Informal Introduction of Integers
    2. 2.2 Integers as Relation in Ordered Pairs of Natural Numbers
    3. 2.3 Ordering in Ordered Pairs
    4. 2.4 Operations in Ordered Pairs of Natural Numbers
    5. 2.5 Properties of Binary Operations
    6. 2.6 Interpretation of Relation and Operations
    7. 2.7 Mapping of Ordered Pairs as Extension of Natural Numbers
    8. 2.8 Representation of Integers
    9. 2.9 Summary
  11. 3 Rational Numbers
    1. 3.1 Informal Introduction of Rational Numbers
    2. 3.2 Rational Numbers as Relation in Ordered Pairs of Integers
    3. 3.3 Ordering in Ordered Pairs
    4. 3.4 Operations in Ordered Pairs
    5. 3.5 Properties of Binary Operations
    6. 3.6 Interpretation of Relation and Operations
    7. 3.7 Mapping of Ordered Pairs as Extension of Integers
    8. 3.8 Representation of Rational Numbers
    9. 3.9 Limitations of Rational Numbers
    10. 3.10 Summary
  12. 4 Real Numbers
    1. 4.1 Least Upper Bound Property
    2. 4.2 Rational Cuts
    3. 4.3 Dedekind Cuts
    4. 4.4 Ordering in Cuts
    5. 4.5 Binary Operations in Cuts
    6. 4.6 Least Upper Bound Property
    7. 4.7 Set of Cuts as Extension of Rational Numbers
    8. 4.8 Cardinality of Set of Real Numbers
    9. 4.9 Limitations of Real Numbers
    10. 4.10 Summary
  13. 5 Complex Numbers
    1. 5.1 Complex Numbers as Ordered Pairs of Real Numbers
    2. 5.2 Binary Operations in Complex Numbers
    3. 5.3 Introduction of Imaginary Numbers
    4. 5.4 Representation of Complex Numbers
    5. 5.5 Ordering in Complex numbers
    6. 5.6 Cardinality of the Set of Complex Numbers
    7. 5.7 Algebraic Numbers
    8. 5.8 Summary
  14. Index
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