Numbers and Proofs


Author: Reg Allenby

Publisher: Elsevier

ISBN: 0080928773

Category: Mathematics

Page: 288

View: 1588

'Numbers and Proofs' presents a gentle introduction to the notion of proof to give the reader an understanding of how to decipher others' proofs as well as construct their own. Useful methods of proof are illustrated in the context of studying problems concerning mainly numbers (real, rational, complex and integers). An indispensable guide to all students of mathematics. Each proof is preceded by a discussion which is intended to show the reader the kind of thoughts they might have before any attempt proof is made. Established proofs which the student is in a better position to follow then follow. Presented in the author's entertaining and informal style, and written to reflect the changing profile of students entering universities, this book will prove essential reading for all seeking an introduction to the notion of proof as well as giving a definitive guide to the more common forms. Stressing the importance of backing up "truths" found through experimentation, with logically sound and watertight arguments, it provides an ideal bridge to more complex undergraduate maths.

Numbers, Sequences and Series


Author: Keith E. Hirst

Publisher: Butterworth-Heinemann

ISBN: 0340610433

Category: Mathematics

Page: 198

View: 9770

Concerned with the logical foundations of number systems from integers to complex numbers.

The Art of Proof

Basic Training for Deeper Mathematics


Author: Matthias Beck,Ross Geoghegan

Publisher: Springer Science & Business Media

ISBN: 9781441970237

Category: Mathematics

Page: 182

View: 8731

The Art of Proof is designed for a one-semester or two-quarter course. A typical student will have studied calculus (perhaps also linear algebra) with reasonable success. With an artful mixture of chatty style and interesting examples, the student's previous intuitive knowledge is placed on solid intellectual ground. The topics covered include: integers, induction, algorithms, real numbers, rational numbers, modular arithmetic, limits, and uncountable sets. Methods, such as axiom, theorem and proof, are taught while discussing the mathematics rather than in abstract isolation. The book ends with short essays on further topics suitable for seminar-style presentation by small teams of students, either in class or in a mathematics club setting. These include: continuity, cryptography, groups, complex numbers, ordinal number, and generating functions.

Groups - Modular Mathematics Series


Author: Camilla Jordan,David Jordan

Publisher: Butterworth-Heinemann

ISBN: 0080571654

Category: Mathematics

Page: 224

View: 8720

This text provides an introduction to group theory with an emphasis on clear examples. The authors present groups as naturally occurring structures arising from symmetry in geometrical figures and other mathematical objects. Written in a 'user-friendly' style, where new ideas are always motivated before being fully introduced, the text will help readers to gain confidence and skill in handling group theory notation before progressing on to applying it in complex situations. An ideal companion to any first or second year course on the topic.

Heinemann Modular Maths Edexcel Further Pure Maths 3


Author: Geoff Mannall,Michael Kenwood

Publisher: Heinemann

ISBN: 9780435511029

Category: A-level examinations

Page: 202

View: 1904

Drawing on over 10 years' experience of publishing for Edexcel maths, Heinemann Modular Maths for Edexcel AS and A Level brings you dedicated textbooks to help you give your students a clear route to success, now with new Core maths titles to match the new 2004 specification. Further Pure 3 replaces Pure 6 in the new specification.

Modular Functions in Analytic Number Theory


Author: Marvin Isadore Knopp

Publisher: American Mathematical Soc.

ISBN: 9780821844885

Category: Mathematics

Page: 154

View: 459

Knopp's engaging book presents an introduction to modular functions in number theory by concentrating on two modular functions, $\eta(\tau)$ and $\vartheta(\tau)$, and their applications to two number-theoretic functions, $p(n)$ and $r_s(n)$. They are well chosen, as at the heart of these particular applications to the treatment of these specific number-theoretic functions lies the general theory of automorphic functions, a theory of far-reaching significance with important connections to a great many fields of mathematics. The book is essentially self-contained, assuming only a good first-year course in analysis. The excellent exposition presents the beautiful interplay between modular forms and number theory, making the book an excellent introduction to analytic number theory for a beginning graduate student.

Partitions, q-Series, and Modular Forms


Author: Krishnaswami Alladi,Frank Garvan

Publisher: Springer Science & Business Media

ISBN: 1461400287

Category: Mathematics

Page: 224

View: 7765

Partitions, q-Series, and Modular Forms contains a collection of research and survey papers that grew out of a Conference on Partitions, q-Series and Modular Forms at the University of Florida, Gainesville in March 2008. It will be of interest to researchers and graduate students that would like to learn of recent developments in the theory of q-series and modular and how it relates to number theory, combinatorics and special functions.

Elementary Number Theory


Author: Gareth A. Jones,Josephine M. Jones

Publisher: Springer Science & Business Media

ISBN: 9783540761976

Category: Mathematics

Page: 302

View: 758

An undergraduate-level introduction to number theory, with the emphasis on fully explained proofs and examples. Exercises, together with their solutions are integrated into the text, and the first few chapters assume only basic school algebra. Elementary ideas about groups and rings are then used to study groups of units, quadratic residues and arithmetic functions with applications to enumeration and cryptography. The final part, suitable for third-year students, uses ideas from algebra, analysis, calculus and geometry to study Dirichlet series and sums of squares. In particular, the last chapter gives a concise account of Fermat's Last Theorem, from its origin in the ancient Babylonian and Greek study of Pythagorean triples to its recent proof by Andrew Wiles.

Modular Functions and Dirichlet Series in Number Theory


Author: Tom M. Apostol

Publisher: Springer Science & Business Media

ISBN: 1461209994

Category: Mathematics

Page: 207

View: 2595

A new edition of a classical treatment of elliptic and modular functions with some of their number-theoretic applications, this text offers an updated bibliography and an alternative treatment of the transformation formula for the Dedekind eta function. It covers many topics, such as Hecke’s theory of entire forms with multiplicative Fourier coefficients, and the last chapter recounts Bohr’s theory of equivalence of general Dirichlet series.

Discrete Mathematics

Numbers and Beyond


Author: Stephen Barnett

Publisher: Addison-Wesley


Category: Mathematics

Page: 441

View: 9098

For the increasing number of students who need an understanding of the subject, Discrete Mathematics: Numbers and Beyond provides the perfect introduction. Aimed particularly at non-specialists, its attractive style and practical approach offer easy access to this important subject. With an emphasis on methods and applications rather than rigorous proofs, the book's coverage is based an the essential topics of numbers, counting and numerical processes. Discrete Mathematics: Numbers and Beyond supplies the reader with a thorough grounding in number systems, modular arithmetic, combinatorics, networks and graphs, coding theory and recurrence relations. Throughout the book, learning is aided and reinforced by the following features: a wealth of exercises and problems of varying difficulty a wide range of illustrative applications of general interest numerous worked examples and diagrams team-based student projects in every chapter concise, informal explanations tips for further reading Discrete Mathematics: Numbers and Beyond is an ideal textbook for an introductory discrete mathematics course taken by students of economics, computer science, mathematics, business, finance, engineering and the sciences. 0201342928B04062001

Encyclopaedia of Mathematics

Volume 6: Subject Index — Author Index


Author: Michiel Hazewinkel

Publisher: Springer Science & Business Media

ISBN: 9400903650

Category: Mathematics

Page: 732

View: 2465

This ENCYCLOPAEDIA OF MATHEMATICS aims to be a reference work for all parts of mathe matics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-1985. The annotated translation consists of ten volumes including a special index volume. There are three kinds of articles in this ENCYCLOPAEDIA. First of all there are survey-type articles dealing with the various main directions in mathematics (where a rather fine subdivi sion has been used). The main requirement for these articles has been that they should give a reasonably complete up-to-date account of the current state of affairs in these areas and that they should be maximally accessible. On the whole, these articles should be understandable to mathematics students in their first specialization years, to graduates from other mathematical areas and, depending on the specific subject, to specialists in other domains of science, en gineers and teachers of mathematics. These articles treat their material at a fairly general level and aim to give an idea of the kind of problems, techniques and concepts involved in the area in question. They also contain background and motivation rather than precise statements of precise theorems with detailed definitions and technical details on how to carry out proofs and constructions. The second kind of article, of medium length, contains more detailed concrete problems, results and techniques.

Women in Numbers 2: Research Directions in Number Theory


Author: Chantal David,Matilde Lalín, Michelle Manes

Publisher: American Mathematical Soc.

ISBN: 1470410222

Category: Mathematics

Page: 206

View: 9295

The second Women in Numbers workshop (WIN2) was held November 6-11, 2011, at the Banff International Research Station (BIRS) in Banff, Alberta, Canada. During the workshop, group leaders presented open problems in various areas of number theory, and working groups tackled those problems in collaborations begun at the workshop and continuing long after. This volume collects articles written by participants of WIN2. Survey papers written by project leaders are designed to introduce areas of active research in number theory to advanced graduate students and recent PhDs. Original research articles by the project groups detail their work on the open problems tackled during and after WIN2. Other articles in this volume contain new research on related topics by women number theorists. The articles collected here encompass a wide range of topics in number theory including Galois representations, the Tamagawa number conjecture, arithmetic intersection formulas, Mahler measures, Newton polygons, the Dwork family, elliptic curves, cryptography, and supercongruences. WIN2 and this Proceedings volume are part of the Women in Numbers network, aimed at increasing the visibility of women researchers' contributions to number theory and at increasing the participation of women mathematicians in number theory and related fields. This book is co-published with the Centre de Recherches Mathématiques.

Number Theory in the Spirit of Ramanujan


Author: Bruce C. Berndt

Publisher: American Mathematical Soc.

ISBN: 0821841785

Category: Mathematics

Page: 187

View: 1836

Ramanujan is recognized as one of the great number theorists of the twentieth century. Here now is the first book to provide an introduction to his work in number theory. Most of Ramanujan's work in number theory arose out of $q$-series and theta functions. This book provides an introduction to these two important subjects and to some of the topics in number theory that are inextricably intertwined with them, including the theory of partitions, sums of squares and triangular numbers, and the Ramanujan tau function. The majority of the results discussed here are originally due to Ramanujan or were rediscovered by him. Ramanujan did not leave us proofs of the thousands of theorems he recorded in his notebooks, and so it cannot be claimed that many of the proofs given in this book are those found by Ramanujan. However, they are all in the spirit of his mathematics.The subjects examined in this book have a rich history dating back to Euler and Jacobi, and they continue to be focal points of contemporary mathematical research. Therefore, at the end of each of the seven chapters, Berndt discusses the results established in the chapter and places them in both historical and contemporary contexts. The book is suitable for advanced undergraduates and beginning graduate students interested in number theory.

What's Happening in the Mathematical Sciences


Author: Barry Cipra

Publisher: American Mathematical Soc.

ISBN: 9780821803554

Category: Science

Page: 111

View: 5962

Beautifully produced and marvelously written, ""What's Happening in the Mathematical Sciences, Volume 3"", contains 10 articles on recent developments in the field. In an engaging, reader-friendly style, Barry Cipra explores topics ranging from Fermat's Last Theorem to Computational Fluid Dynamics. The volumes in this series highlight the many roles mathematics plays in the modern world. This volume includes articles on: a new mathematical method that's taking Wall Street by storm 'Ultra-parallel' supercomputing with DNA, and how a mathematician found the famous flaw in the Pentium chip. Unique in kind, and lively in style, ""What's Happening in the Mathematical Sciences, Volume 3"" is a delight to read and a valuable source of information.

Liebe und Mathematik

Im Herzen einer verborgenen Wirklichkeit


Author: Edward Frenkel

Publisher: Springer-Verlag

ISBN: 3662434210

Category: Mathematics

Page: 317

View: 6022

Mathematical Thinking

Problem-solving and Proofs


Author: John P. D'Angelo,Douglas Brent West

Publisher: Pearson College Division

ISBN: 9780130144126

Category: Mathematics

Page: 412

View: 1642

This survey of both discrete and continuous mathematics focuses on the logical thinking skills necessary to understand and communicate fundamental ideas and proofs in mathematics, rather than on rote symbolic manipulation. Coverage begins with the fundamentals of mathematical language and proof techniques (such as induction); then applies them to easily-understood questions in elementary number theory and counting; then develops additional techniques of proofs via fundamental topics in discrete and continuous mathematics. Topics are addressed in the context of familiar objects; easily-understood, engaging examples; and over 700 stimulating exercises and problems, ranging from simple applications to subtle problems requiring ingenuity. ELEMENTARY CONCEPTS. Numbers, Sets and Functions. Language and Proofs. Properties of Functions. Induction. PROPERTIES OF NUMBERS. Counting and Cardinality. Divisibility. Modular Arithmetic. The Rational Numbers. DISCRETE MATHEMATICS. Combinatorial Reasoning. Two Principles of Counting. Graph Theory. Recurrence Relations. CONTINUOUS MATHEMATICS. The Real Numbers. Sequences and Series. Continuity. Differentiation. Integration. The Complex Numbers. For anyone interested in learning how to understand and write mathematical proofs, or a reference for college professors and high school teachers of mathematics.

Extending the Frontiers of Mathematics

Inquiries Into Proof and Argumentation


Author: Edward B. Burger

Publisher: Key College Pub


Category: Computers

Page: 171

View: 9730

This book leads students to develop a mature process that will serve them throughout their professional careers, whether inside or outside of mathematics. Its inquiry-based approach to the foundations of mathematics promotes exploration of proofs and other advanced mathematical ideas. The book uses puzzles and patterns, teaches the "prove and extend or disprove and salvage" approach to problems and proofs, and builds mathematical challenge on challenge to boost skills.

Number Theory Through Inquiry


Author: David C. Marshall,Edward Odell,Michael Starbird

Publisher: MAA

ISBN: 0883857510

Category: Mathematics

Page: 140

View: 3047

This innovative textbook leads students on a carefully guided discovery of introductory number theory. The book has two equally significant goals. The first is to help students develop mathematical thinking skills, particularly theorem-proving skills. The other goal is to help students understand some of the wonderfully rich ideas in the mathematical study of numbers. This book is appropriate for a proof transitions course, for independent study, or for a course designed as an introduction to abstract mathematics. It is designed to be used with an instructional technique variously called guided discovery or Modified Moore Method or Inquiry Based Learning (IBL). Instructors' materials explain the instructional method, which gives students a totally different experience compared to a standard lecture course. Students develop an attitude of personal reliance and a sense that they can think effectively about difficult problems; goals that are fundamental to the educational enterprise within and beyond mathematics.

Analytic and Elementary Number Theory

A Tribute to Mathematical Legend Paul Erdos


Author: Krishnaswami Alladi,P.D.T.A. Elliott,A. Granville,G. Tenenbaum

Publisher: Springer

ISBN: 1475745079

Category: Mathematics

Page: 300

View: 2343

This volume contains a collection of papers in Analytic and Elementary Number Theory in memory of Professor Paul Erdös, one of the greatest mathematicians of this century. Written by many leading researchers, the papers deal with the most recent advances in a wide variety of topics, including arithmetical functions, prime numbers, the Riemann zeta function, probabilistic number theory, properties of integer sequences, modular forms, partitions, and q-series. Audience: Researchers and students of number theory, analysis, combinatorics and modular forms will find this volume to be stimulating.

Topics in Number Theory


Author: Minking Eie

Publisher: World Scientific

ISBN: 9812835180

Category: Mathematics

Page: 278

View: 3321

This is a first-ever textbook written in English about the theory of modular forms and Jacobi forms of several variables. It contains the classical theory as well as a new theory on Jacobi forms over Cayley numbers developed by the author from 1990 to 2000. Applications to the classical Euler sums are of special interest to those who are eager to evaluate double Euler sums or more general multiple zeta values. The celebrated sum formula proved by Granville in 1997 is generalized to a more general form here.