book

Journey from Natural Numbers to Complex Numbers

by Nita H. Shah, Vishnuprasad D. Thakkar

December 2020

Intermediate to advanced

94 pages

2h 31m

English

CRC Press

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1.1 Prerequisites1.1.1 Set Theory1.1.2 Relation1.1.3 Function1.1.4 Cardinality1.1.5 Algebra1.2 Positive Integers1.2.1 Positive Integers in Real Life1.2.2 Set Theoretic Definition of Natural Numbers1.2.3 Peano Axioms1.2.4 Ordering in Natural Numbers1.2.5 First Principle of Mathematical Induction1.2.6 Second Principle of Mathematical Induction1.2.7 Well-Ordering Principle1.2.8 Limitations of Natural Numbers1.2.9 Representation of Natural Numbers1.2.9.1 Hexadecimal System1.2.10 Number System Used by Computers1.3 Summary
2.1 Informal Introduction of Integers2.2 Integers as Relation in Ordered Pairs of Natural Numbers2.3 Ordering in Ordered Pairs2.4 Operations in Ordered Pairs of Natural Numbers2.5 Properties of Binary Operations2.6 Interpretation of Relation and Operations2.7 Mapping of Ordered Pairs as Extension of Natural Numbers2.8 Representation of Integers2.9 Summary

3.1 Informal Introduction of Rational Numbers3.2 Rational Numbers as Relation in Ordered Pairs of Integers3.3 Ordering in Ordered Pairs3.4 Operations in Ordered Pairs3.5 Properties of Binary Operations3.6 Interpretation of Relation and Operations3.7 Mapping of Ordered Pairs as Extension of Integers3.8 Representation of Rational Numbers3.9 Limitations of Rational Numbers3.10 Summary
4.1 Least Upper Bound Property4.2 Rational Cuts4.3 Dedekind Cuts4.4 Ordering in Cuts4.5 Binary Operations in Cuts4.6 Least Upper Bound Property4.7 Set of Cuts as Extension of Rational Numbers4.8 Cardinality of Set of Real Numbers4.9 Limitations of Real Numbers4.10 Summary
5.1 Complex Numbers as Ordered Pairs of Real Numbers5.2 Binary Operations in Complex Numbers5.3 Introduction of Imaginary Numbers5.4 Representation of Complex Numbers5.5 Ordering in Complex numbers5.6 Cardinality of the Set of Complex Numbers5.7 Algebraic Numbers5.8 Summary

Overview

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.