Numbers, Sequences and Series
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 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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The Rational Numbers
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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 expansion exponential expressed fact false formula function geometric series given gives graphical graphs idea 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 satisfying number theory number whose square obtain partial sums polar positive integer positive number positive real number power series prime number proof properties Prove quadratic quotient radius of convergence ratio test rational numbers rearranged relationship remainder roots of unity sequence of partial series converges set of numbers solution Solve the inequality square roots subset Suppose tells tends to infinity triangle true TUTORIAL PROBLEM values variable well-ordering principle write