## Numbers, Sequences and SeriesNumber 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 processes involving sequences and series of numbers, including power series. The book also has worked examples throughout and includes some suggestions for self-study projects. In addition there are tutorial problems aimed at stimulating group work and discussion. |

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### Contents

Chapter 1 Sets and Logic | 1 |

Chapter 2 The Integers | 19 |

Chapter 3 The Rational Numbers | 38 |

Chapter 4 Inequalities | 55 |

Chapter 5 The Real Numbers | 68 |

Chapter 6 Complex Numbers | 83 |

Chapter 7 Sequences | 106 |

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absolutely convergent algebraic approximations arithmetic base calculator Chapter complex numbers complex plane consider continued fraction decimal deduce deﬁned deﬁnition denominator digits discussed diverges division divisor equal equation equivalence classes equivalence relation Euclidean Algorithm Euler Example EXERCISES exponential expressed fact false ﬁnd ﬁnite ﬁrst formula function geometric series gives graphical graphs illustrate inﬁnite sequence inﬁnite series integer investigate iteration least upper bound lim(a limit logical mathematical induction mathematicians method natural numbers notation nth term number whose square obtain partial sums polar positive integer positive number positive real number power series prime number proof properties Prove quantiﬁers quotient radius of convergence ratio test rational numbers real numbers rearranged reﬂects relationship remainder roots of unity satisﬁed sequence of partial series converges set of numbers solution Solve the inequality speciﬁed square root subset Suppose tells tends to inﬁnity triangle true TUTORIAL PROBLEM values variable well-ordering principle write