Journey from Natural Numbers to Complex Numbers

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
      1. 1.1.1 Set Theory
      2. 1.1.2 Relation
      3. 1.1.3 Function
      4. 1.1.4 Cardinality
      5. 1.1.5 Algebra
   2. 1.2 Positive Integers
      1. 1.2.1 Positive Integers in Real Life
      2. 1.2.2 Set Theoretic Definition of Natural Numbers
      3. 1.2.3 Peano Axioms
      4. 1.2.4 Ordering in Natural Numbers
      5. 1.2.5 First Principle of Mathematical Induction
      6. 1.2.6 Second Principle of Mathematical Induction
      7. 1.2.7 Well-Ordering Principle
      8. 1.2.8 Limitations of Natural Numbers
      9. 1.2.9 Representation of Natural Numbers
         1. 1.2.9.1 Hexadecimal System
      10. 1.2.10 Number System Used by Computers
   3. 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