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The fundamental theorem of algebra states that any complex polynomial must have a complex root. This book examines three pairs of proofs of the theorem from three different areas of mathematics: abstract algebra, complex analysis and topology. The first proof in each pair is fairly straightforward and depends only on what could be considered elementary mathematics. However, each of these first proofs leads to more general results from which the fundamental theorem can be deduced as a direct consequence. These general results constitute the second proof in each pair. To arrive at each of the proofs, enough of the general theory of each relevant area is developed to understand the proof. In addition to the proofs and techniques themselves, many applications such as the insolvability of the quintic and the transcendence of e and pi are presented. Finally, a series of appendices give six additional proofs including a version of Gauss'original first proof. The book is intended for junior/senior level undergraduate mathematics students or first year graduate students, and would make an ideal "capstone" course in mathematics.
Suitable for advanced courses in applied mathematics, this text covers analysis of lumped parameter systems, distributed parameter systems, and important areas of applied mathematics. Answers to selected problems. 1970 edition.
Publisher description -- This book will save you time as you master the basics taught in first-year, calculus-based college physics courses. You'll firmly grasp the all-important building blocks needed for every physical science and all branches of engineering. The many problems included with guided solutions make this potentially daunting subject much easier. Additional problems with answers give you a chance to reinforce what you've learned and gauge your progress as you go. This next-best thing to a private tutor makes especially clear the topics most students find most difficult. It's ideal for independent study, brushup before an exam, or preparation for the MED-CAT and GRE.
Martin Gardner's Mathematical Games columns in Scientific American inspired and entertained several generations of mathematicians and scientists. Gardner in his crystal-clear prose illuminated corners of mathematics, especially recreational mathematics, that most people had no idea existed. His playful spirit and inquisitive nature invite the reader into an exploration of beautiful mathematical ideas along with him. These columns were both a revelation and a gift when he wrote them; no one--before Gardner--had written about mathematics like this. They continue to be a marvel. This volume, originally published in 1959, contains the first sixteen columns published in the magazine from 1956-1958. They were reviewed and briefly updated by Gardner for this 1988 edition.