Numbers, sequences and series / Keith E. Hirst.

Includes index.

Front Cover; Numbers, Sequences and Series; Copyright Page; Table of Contents; Series Preface; Preface; Chapter 1. Sets and Logic; 1.1 Symbolism; 1.2 Sets; 1.3 The logic of mathematical discourse; 1.4 Statements involving variables; 1.5 Statements involving more than one variable; 1.6 Equivalence relations; Summary; Exercises on Chapter 1; Chapter 2. The Integers; 2.1 Peano's Axioms; 2.2 Proof by mathematical induction; 2.3 Negative integers; 2.4 Division and the highest common factor; 2.5 The Euclidean Algorithm; 2.6 Digital representation; Summary; Exercises on Chapter 2

Chapter 3. The Rational Numbers3.1 Solving equations; 3.2 Constructing the rational numbers; 3.3 Continued fractions; Summary; Exercises on Chapter 3; Chapter 4. Inequalities; 4.1 The basic rules for inequalities; 4.2 Solving inequalities graphically; 4.3 Solving inequalities algebraically; 4.4 A tabular approach to inequalities; 4.5 Increasing and decreasing functions; Summary; Exercises on Chapter 4; Chapter 5. The Real Numbers; 5.1 Gaps in the rational number system; 5.2 An historical interlude; 5.3 Bounded sets; 5.4 Arithmetic and algebra with real numbers; Summary; Exercises on Chapter 5

Chapter 6. Complex Numbers6.1 Hamilton's definition; 6.2 The algebra of complex numbers; 6.3 The geometry of complex numbers; 6.4 Polar representation; 6.5 Euler's Formula; 6.6 The roots of unity; Summary; Exercises on Chapter 6; Chapter 7. Sequences; 7.1 Defining an infinite sequence; 7.2 Solving equations; 7.3 Limits of sequences; 7.4 Increasing and decreasing sequences; 7.5 Iteration; 7.6 Complex sequences; Summary; Exercises on Chapter 7; Chapter 8. Infinite Series; 8.1 Convergence; 8.2 Tests for convergence; 8.3 Series and integrals; 8.4 Complex series and absolute convergence

8.5 Power seriesSummary; Exercises on Chapter 8; Chapter 9. Decimals; 9.1 Infinite decimal expansions; 9.2 Periodic decimals; 9.3 Point nine recurring; Summary; Exercises on Chapter 9; Chapter 10. Further Developments; 10.1 Sets, logic and Boolean algebra; 10.2 Number theory and continued fractions; 10.3 Real numbers and more; 10.4 Complex numbers and beyond; 10.5 Sequences and series; 10.6 Decimals; Answers to Exercises; Index

Number and geometry are the foundations upon which mathematics has been built over some 3000 years. This book is concerned with the logical foundations of number systems from integers to complex numbers. The author has chosen to develop the ideas by illustrating the techniques used throughout mathematics rather than using a self-contained logical treatise. The idea of proof has been emphasised, as has the illustration of concepts from a graphical, numerical and algebraic point of view. Having laid the foundations of the number system, the author has then turned to the analysis of infinite proc.

English.

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