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Problems and Proofs in Numbers and Algebra

Problems and Proofs in Numbers and Algebra Author Richard S. Millman
ISBN-10 9783319144276
Release 2015-02-09
Pages 223
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Focusing on an approach of solving rigorous problems and learning how to prove, this volume is concentrated on two specific content themes, elementary number theory and algebraic polynomials. The benefit to readers who are moving from calculus to more abstract mathematics is to acquire the ability to understand proofs through use of the book and the multitude of proofs and problems that will be covered throughout. This book is meant to be a transitional precursor to more complex topics in analysis, advanced number theory, and abstract algebra. To achieve the goal of conceptual understanding, a large number of problems and examples will be interspersed through every chapter. The problems are always presented in a multi-step and often very challenging, requiring the reader to think about proofs, counter-examples, and conjectures. Beyond the undergraduate mathematics student audience, the text can also offer a rigorous treatment of mathematics content (numbers and algebra) for high-achieving high school students. Furthermore, prospective teachers will add to the breadth of the audience as math education majors, will understand more thoroughly methods of proof, and will add to the depth of their mathematical knowledge. In the past, PNA has been taught in a "problem solving in middle school” course (twice), to a quite advanced high school students course (three semesters), and three times as a secondary resource for a course for future high school teachers. PNA is suitable for secondary math teachers who look for material to encourage and motivate more high achieving students.

Problems and Theorems in Linear Algebra

Problems and Theorems in Linear Algebra Author Viktor Vasil_evich Prasolov
ISBN-10 9780821802366
Release 1994-06-13
Pages 225
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There are a number of very good books available on linear algebra. However, new results in linear algebra appear constantly, as do new, simpler, and better proofs of old results. Many of these results and proofs obtained in the past thirty years are accessible to undergraduate mathematics majors, but are usually ignored by textbooks. In addition, more than a few interesting old results are not covered in many books. In this book, the author provides the basics of linear algebra, with an emphasis on new results and on nonstandard and interesting proofs. The book features about 230 problems with complete solutions. It can serve as a supplementary text for an undergraduate or graduate algebra course.

Approximation by Algebraic Numbers

Approximation by Algebraic Numbers Author Yann Bugeaud
ISBN-10 1139455672
Release 2004-11-08
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Algebraic numbers can approximate and classify any real number. Here, the author gathers together results about such approximations and classifications. Written for a broad audience, the book is accessible and self-contained, with complete and detailed proofs. Starting from continued fractions and Khintchine's theorem, Bugeaud introduces a variety of techniques, ranging from explicit constructions to metric number theory, including the theory of Hausdorff dimension. So armed, the reader is led to such celebrated advanced results as the proof of Mahler's conjecture on S-numbers, the Jarnik–Besicovitch theorem, and the existence of T-numbers. Brief consideration is given both to the p-adic and the formal power series cases. Thus the book can be used for graduate courses on Diophantine approximation (some 40 exercises are supplied), or as an introduction for non-experts. Specialists will appreciate the collection of over 50 open problems and the rich and comprehensive list of more than 600 references.

Equations and Inequalities

Equations and Inequalities Author Jiri Herman
ISBN-10 9781461212706
Release 2012-12-06
Pages 344
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A look at solving problems in three areas of classical elementary mathematics: equations and systems of equations of various kinds, algebraic inequalities, and elementary number theory, in particular divisibility and diophantine equations. In each topic, brief theoretical discussions are followed by carefully worked out examples of increasing difficulty, and by exercises which range from routine to rather more challenging problems. While it emphasizes some methods that are not usually covered in beginning university courses, the book nevertheless teaches techniques and skills which are useful beyond the specific topics covered here. With approximately 330 examples and 760 exercises.

Doing Mathematics

Doing Mathematics Author Steven Galovich
ISBN-10 0495108162
Release 2007
Pages 307
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This book introduces students to the process of doing mathematics and prepares them to succeed in higher-level mathematics courses. By discussing proof techniques, problem solving methods, and the understanding of mathematical ideas, the book provides a solid foundation for students majoring in mathematics, science, and engineering. Students will learn to grasp the underlying concepts of a subject and how to apply these concepts to solving problems. While being able to understand and reproduce proofs of theorems, they will also gain the ability to comprehend the connections among the important concepts and techniques of each subject. This book is intended for a shorter course on proofs and mathematical reasoning, and could also be used as a supplemental text in courses such as algebra, analysis, and linear algebra.

Algebraic Number Theory and Fermat s Last Theorem Fourth Edition

Algebraic Number Theory and Fermat s Last Theorem  Fourth Edition Author Ian Stewart
ISBN-10 9781498738408
Release 2015-10-14
Pages 322
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Updated to reflect current research, Algebraic Number Theory and Fermat’s Last Theorem, Fourth Edition introduces fundamental ideas of algebraic numbers and explores one of the most intriguing stories in the history of mathematics—the quest for a proof of Fermat’s Last Theorem. The authors use this celebrated theorem to motivate a general study of the theory of algebraic numbers from a relatively concrete point of view. Students will see how Wiles’s proof of Fermat’s Last Theorem opened many new areas for future work. New to the Fourth Edition Provides up-to-date information on unique prime factorization for real quadratic number fields, especially Harper’s proof that Z(√14) is Euclidean Presents an important new result: Mihăilescu’s proof of the Catalan conjecture of 1844 Revises and expands one chapter into two, covering classical ideas about modular functions and highlighting the new ideas of Frey, Wiles, and others that led to the long-sought proof of Fermat’s Last Theorem Improves and updates the index, figures, bibliography, further reading list, and historical remarks Written by preeminent mathematicians Ian Stewart and David Tall, this text continues to teach students how to extend properties of natural numbers to more general number structures, including algebraic number fields and their rings of algebraic integers. It also explains how basic notions from the theory of algebraic numbers can be used to solve problems in number theory.

Mathematical Problems and Proofs

Mathematical Problems and Proofs Author Branislav Kisacanin
ISBN-10 9780306469633
Release 2007-05-08
Pages 220
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A gentle introduction to the highly sophisticated world of discrete mathematics, Mathematical Problems and Proofs presents topics ranging from elementary definitions and theorems to advanced topics -- such as cardinal numbers, generating functions, properties of Fibonacci numbers, and Euclidean algorithm. This excellent primer illustrates more than 150 solutions and proofs, thoroughly explained in clear language. The generous historical references and anecdotes interspersed throughout the text create interesting intermissions that will fuel readers' eagerness to inquire further about the topics and some of our greatest mathematicians. The author guides readers through the process of solving enigmatic proofs and problems, and assists them in making the transition from problem solving to theorem proving. At once a requisite text and an enjoyable read, Mathematical Problems and Proofs is an excellent entrée to discrete mathematics for advanced students interested in mathematics, engineering, and science.

Elements of Algebra

Elements of Algebra Author John Stillwell
ISBN-10 9781475739763
Release 2013-04-18
Pages 184
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Algebra is abstract mathematics - let us make no bones about it - yet it is also applied mathematics in its best and purest form. It is not abstraction for its own sake, but abstraction for the sake of efficiency, power and insight. Algebra emerged from the struggle to solve concrete, physical problems in geometry, and succeeded after 2000 years of failure by other forms of mathematics. It did this by exposing the mathematical structure of geometry, and by providing the tools to analyse it. This is typical of the way algebra is applied; it is the best and purest form of application because it reveals the simplest and most universal mathematical structures. The present book aims to foster a proper appreciation of algebra by showing abstraction at work on concrete problems, the classical problems of construction by straightedge and compass. These problems originated in the time of Euclid, when geometry and number theory were paramount, and were not solved until th the 19 century, with the advent of abstract algebra. As we now know, alge bra brings about a unification of geometry, number theory and indeed most branches of mathematics. This is not really surprising when one has a historical understanding of the subject, which I also hope to impart.

Problems in Algebraic Number Theory

Problems in Algebraic Number Theory Author M. Ram Murty
ISBN-10 9780387269986
Release 2006-03-30
Pages 352
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The problems are systematically arranged to reveal the evolution of concepts and ideas of the subject Includes various levels of problems - some are easy and straightforward, while others are more challenging All problems are elegantly solved

Transcendental and Algebraic Numbers

Transcendental and Algebraic Numbers Author A. O. Gelfond
ISBN-10 9780486802251
Release 2015-01-05
Pages 208
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Primarily an advanced study of the modern theory of transcendental and algebraic numbers, this treatment by a distinguished Soviet mathematician focuses on the theory's fundamental methods. The text also chronicles the historical development of the theory's methods and explores the connections with other problems in number theory. The problem of approximating algebraic numbers is also studied as a case in the theory of transcendental numbers. Topics include the Thue-Siegel theorem, the Hermite-Lindemann theorem on the transcendency of the exponential function, and the work of C. Siegel on the transcendency of the Bessel functions and of the solutions of other differential equations. The final chapter considers the Gelfond-Schneider theorem on the transcendency of alpha to the power beta. Each proof is prefaced by a brief discussion of its scheme, which provides a helpful guide to understanding the proof's progression.

Key Maths

Key Maths Author David Baker
ISBN-10 9780748759958
Release 2001
Pages 572
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Planned, developed and written by practising classroom teachers with a wide variety of experience in schools, this maths course has been designed to be enjoyable and motivating for pupils and teachers. The course is open and accessible to pupils of all abilities and backgrounds, and is differentiated to provide material which is appropriate for all pupils. It provides spiral coverage of the curriculum which involves regular revisiting of key concepts to promote familiarity through practice. This teacher's file is designed for stage three of Year 9.

Boolean Algebra and Its Applications

Boolean Algebra and Its Applications Author J. Eldon Whitesitt
ISBN-10 9780486158167
Release 2012-05-24
Pages 192
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Introductory treatment begins with set theory and fundamentals of Boolean algebra, proceeding to concise accounts of applications to symbolic logic, switching circuits, relay circuits, binary arithmetic, and probability theory. 1961 edition.

Algebra from A to Z

Algebra from A to Z Author A. W. Goodman
ISBN-10 9810249802
Release 2001
Pages 164
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Explains algebra from basic concepts to college-level skills.

A Mathematical Mosaic

A Mathematical Mosaic Author Ravi Vakil
ISBN-10 1895997046
Release 1996-01
Pages 254
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Excerpt from a review in the "Mathematics Teacher." A Mathematical Mosaic is a collection of wonderful topics from nmber theory through combinatorics to game theory, presented in a fashion that seventh- and eighth- grade students can handle yet high school students will find challenging." John Cocharo, Saint Mark's School of Texas, Dallas, TX

Elements of the Theory of Numbers

Elements of the Theory of Numbers Author Joseph B. Dence
ISBN-10 0122091302
Release 1999-01-01
Pages 517
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Elements of the Theory of Numbers teaches students how to develop, implement, and test numerical methods for standard mathematical problems. The authors have created a two-pronged pedagogical approach that integrates analysis and algebra with classical number theory. Making greater use of the language and concepts in algebra and analysis than is traditionally encountered in introductory courses, this pedagogical approach helps to instill in the minds of the students the idea of the unity of mathematics. Elements of the Theory of Numbers is a superb summary of classical material as well as allowing the reader to take a look at the exciting role of analysis and algebra in number theory. * In-depth coverage of classical number theory * Thorough discussion of the theory of groups and rings * Includes application of Taylor polynomials * Contains more advanced material than other texts * Illustrates the results of a theorem with an example * Excellent presentation of the standard computational exercises * Nearly 1000 problems--many are proof-oriented, several others require the writing of computer programs to complete the computations * Clear and well-motivated presentation * Provides historical references noting distinguished number theory luminaries such as Euclid, de Fermat, Hilbert, Brun, and Lehmer, to name a few * Annotated bibliographies appear at the end of all of the chapters * Instructor's Solution Manual is free to adopters

Famous Problems of Geometry and How to Solve Them

Famous Problems of Geometry and How to Solve Them Author Benjamin Bold
ISBN-10 9780486137636
Release 2012-05-11
Pages 144
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Delve into the development of modern mathematics and match wits with Euclid, Newton, Descartes, and others. Each chapter explores an individual type of challenge, with commentary and practice problems. Solutions.

Algebra and Number Theory

Algebra and Number Theory Author Benjamin Fine
ISBN-10 9783110516142
Release 2017-09-11
Pages 342
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This two-volume set collects and presents some fundamentals of mathematics in an entertaining and performing manner. The present volume examines many of the most important basic results in algebra and number theory, along with their proofs, and also their history. Contents The natural, integral and rational numbers Division and factorization in the integers Modular arithmetic Exceptional numbers Pythagorean triples and sums of squares Polynomials and unique factorization Field extensions and splitting fields Permutations and symmetric polynomials Real numbers The complex numbers, the Fundamental Theorem of Algebra and polynomial equations Quadratic number fields and Pell’s equation Transcendental numbers and the numbers e and π Compass and straightedge constructions and the classical problems Euclidean vector spaces