Problems and Proofs in Numbers and Algebra

Author: Richard S. Millman
Publisher: Springer
ISBN: 3319144278
Category : Mathematics
Languages : en
Pages : 230

Book Description
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.

Proofs from THE BOOK

Author: Martin Aigner
Publisher: Springer Science & Business Media
ISBN: 3662223430
Category : Mathematics
Languages : en
Pages : 194

Book Description
According to the great mathematician Paul Erdös, God maintains perfect mathematical proofs in The Book. This book presents the authors candidates for such "perfect proofs," those which contain brilliant ideas, clever connections, and wonderful observations, bringing new insight and surprising perspectives to problems from number theory, geometry, analysis, combinatorics, and graph theory. As a result, this book will be fun reading for anyone with an interest in mathematics.

Numbers and Proofs

Author: Reg Allenby
Publisher: Elsevier
ISBN: 0080928773
Category : Mathematics
Languages : en
Pages : 289

Book Description
'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.

Book of Proof

Author: Richard H. Hammack
Publisher:
ISBN: 9780989472111
Category : Mathematics
Languages : en
Pages : 314

Book Description
This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra. Although it may be more meaningful to the student who has had some calculus, there is really no prerequisite other than a measure of mathematical maturity.

Proofs and Fundamentals

Author: Ethan D. Bloch
Publisher: Springer Science & Business Media
ISBN: 1461221307
Category : Mathematics
Languages : en
Pages : 434

Book Description
The aim of this book is to help students write mathematics better. Throughout it are large exercise sets well-integrated with the text and varying appropriately from easy to hard. Basic issues are treated, and attention is given to small issues like not placing a mathematical symbol directly after a punctuation mark. And it provides many examples of what students should think and what they should write and how these two are often not the same.

Mathematical Problems and Proofs

Author: Branislav Kisacanin
Publisher: Springer Science & Business Media
ISBN: 0306469634
Category : Mathematics
Languages : en
Pages : 219

Book Description
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.

Journey into Mathematics

Author: Joseph J. Rotman
Publisher: Courier Corporation
ISBN: 0486151689
Category : Mathematics
Languages : en
Pages : 323

Book Description
This treatment covers the mechanics of writing proofs, the area and circumference of circles, and complex numbers and their application to real numbers. 1998 edition.

Proofs that Really Count

Author: Arthur T. Benjamin
Publisher: American Mathematical Society
ISBN: 1470472597
Category : Mathematics
Languages : en
Pages : 210

Book Description
Mathematics is the science of patterns, and mathematicians attempt to understand these patterns and discover new ones using a variety of tools. In Proofs That Really Count, award-winning math professors Arthur Benjamin and Jennifer Quinn demonstrate that many number patterns, even very complex ones, can be understood by simple counting arguments. The book emphasizes numbers that are often not thought of as numbers that count: Fibonacci Numbers, Lucas Numbers, Continued Fractions, and Harmonic Numbers, to name a few. Numerous hints and references are given for all chapter exercises and many chapters end with a list of identities in need of combinatorial proof. The extensive appendix of identities will be a valuable resource. This book should appeal to readers of all levels, from high school math students to professional mathematicians.

How to Prove It

Author: Daniel J. Velleman
Publisher: Cambridge University Press
ISBN: 0521861241
Category : Mathematics
Languages : en
Pages : 401

Book Description
Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This new edition of Velleman's successful text will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. The author shows how complex proofs are built up from these smaller steps, using detailed 'scratch work' sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. To give students the opportunity to construct their own proofs, this new edition contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software. No background beyond standard high school mathematics is assumed. This book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and of course mathematicians.

An Illustrated Theory of Numbers

Author: Martin H. Weissman
Publisher: American Mathematical Soc.
ISBN: 1470463717
Category : Education
Languages : en
Pages : 341

Book Description
News about this title: — Author Marty Weissman has been awarded a Guggenheim Fellowship for 2020. (Learn more here.) — Selected as a 2018 CHOICE Outstanding Academic Title — 2018 PROSE Awards Honorable Mention An Illustrated Theory of Numbers gives a comprehensive introduction to number theory, with complete proofs, worked examples, and exercises. Its exposition reflects the most recent scholarship in mathematics and its history. Almost 500 sharp illustrations accompany elegant proofs, from prime decomposition through quadratic reciprocity. Geometric and dynamical arguments provide new insights, and allow for a rigorous approach with less algebraic manipulation. The final chapters contain an extended treatment of binary quadratic forms, using Conway's topograph to solve quadratic Diophantine equations (e.g., Pell's equation) and to study reduction and the finiteness of class numbers. Data visualizations introduce the reader to open questions and cutting-edge results in analytic number theory such as the Riemann hypothesis, boundedness of prime gaps, and the class number 1 problem. Accompanying each chapter, historical notes curate primary sources and secondary scholarship to trace the development of number theory within and outside the Western tradition. Requiring only high school algebra and geometry, this text is recommended for a first course in elementary number theory. It is also suitable for mathematicians seeking a fresh perspective on an ancient subject.