Problems and Proofs in Numbers and Algebra de Richard S. Millman

Problems and Proofs in Numbers and Algebra

Autor Richard S. Millman, Peter J. Shiue, Eric Brendan Kahn

en Limba Engleză Hardback – 9 mar 2015

Structura și metodologia lucrării Problems and Proofs in Numbers and Algebra sunt concepute special pentru a facilita tranziția de la calculul matematic de bază către rigoarea matematicii abstracte. Considerăm că organizarea materialului în două secțiuni tematice distincte — numerele întregi și algebra polinoamelor — permite o progresie logică a dificultății. Prima parte explorează conceptele de bază, algoritmul diviziunii și aritmetica modulară, în timp ce a doua parte extinde aceste principii către sisteme liniare și factorizarea polinoamelor, pregătind terenul pentru cursuri superioare de algebră modernă.

Subliniem faptul că acest volum nu este o simplă colecție de exerciții, ci un instrument pedagogic ce pune accent pe procesul de construcție a unei demonstrații. Fiecare capitol este punctat de probleme complexe, structurate în mai mulți pași, care provoacă cititorul să formuleze conjecturi și să identifice contraexemple. Această abordare practică este similară cu cea din Reading, Writing, and Proving, însă Problems and Proofs in Numbers and Algebra se concentrează mai aplicat pe conexiunea dintre teoria numerelor și structurile algebrice, oferind o bază mai solidă în manipularea polinoamelor.

În contextul operei autorului, lucrarea continuă tradiția rigurozității stabilită de Richard S. Millman în Geometry. Dacă în volumul dedicat geometriei accentul cădea pe abordarea metrică și modele, aici autorul, împreună cu echipa sa, reușește să distileze complexitatea abstractă în pași accesibili pentru studenții de licență. Cartea acoperă o arie comparabilă cu An Elementary Transition to Abstract Mathematics, dar cu o abordare mai concentrată pe profunzimea temelor alese decât pe o panoramă largă a structurilor matematice.

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Specificații

ISBN-13: 9783319144269
ISBN-10: 331914426X
Pagini: 236
Ilustrații: X, 223 p. 9 illus.
Dimensiuni: 160 x 241 x 19 mm
Greutate: 0.52 kg
Ediția:2015
Editura: Springer
Locul publicării:Cham, Switzerland

Public țintă

Lower undergraduate

De ce să citești această carte

Recomandăm această carte studenților care doresc să depășească etapa calculelor mecanice și să înțeleagă fundamentul logic al matematicii. Este un instrument esențial pentru viitorii profesori, oferind rigoarea necesară pentru a preda concepte avansate elevilor performanți. Cititorul câștigă nu doar cunoștințe de algebră, ci și o disciplină a gândirii critice prin exersarea constantă a demonstrațiilor și identificarea erorilor logice.

Despre autor

Richard S. Millman, Peter J. Shiue și Eric Brendan Kahn sunt cadre didactice cu o vastă experiență în pedagogia matematică. Richard S. Millman este recunoscut pentru lucrările sale care aduc rigoarea matematică în curriculumul universitar, precum Geometry: A Metric Approach with Models. Colectivul de autori a testat materialul din acest volum în diverse contexte academice, de la cursuri pentru viitori profesori de liceu până la seminare pentru elevi supradotați, adaptând conținutul pentru a răspunde nevoilor de tranziție către matematica abstractă.

Descriere scurtă

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.

Cuprins

I. The Integers 1. Number Concepts, Prime Numbers, and the Division Algorithm 2. Greatest Common Divisors, Diophantine Equations, and Combinatorics 3. Equivalence Classes with Applications to Clock Arithmetics and Fractions II. The Algebra of Polynomials and Linear Systems 4. Polynomials and the Division Algorithm 5. Factoring Polynomials, Their Roots, and Some Applications 6. Matrices and Systems of Linear Equations

Recenzii

“Aimed at introducing postcalculus students tohigher mathematics by way of solving rigorous problems and learning how toprove. … the content is less focused on basic mathematical concepts seen inupper-division college mathematics coursework and more so on topics thatteachers might present in their classrooms, and on interesting applications … .Teachers of mathematics at the secondary level would be well served by taking acourse based on this text. Summing Up: Recommended. Upper-divisionundergraduates through faculty.” (D. S. Larson, Choice, Vol. 53 (1), September,2015)

Notă biografică

Richard S. Millman,Ph.D., Director, Center for Education Integrating Science, Mathematics, and Computing (CEISMC) Georgia Institute of Technology Atlanta, GA 30332-0282
Peter J. Shiue Department of Mathematical Sciences University of Nevada, Las Vegas 4505 Maryland Pkwy Las Vegas, NV 89154-4020
Eric Brendan Kahn Department of Mathematics, Computer Science, and Statistics Bloomsburg University 400 East Second Street Bloomsburg, PA 17815

Textul de pe ultima copertă

Designed to facilitate the transition from undergraduate calculus and differential equations to learning about proofs, this book helps students develop the rigorous mathematical reasoning needed for advanced courses in analysis, abstract algebra, and more. Students will focus on both how to prove theorems and solve problem sets in-depth; that is, where multiple steps are needed to prove or solve. This proof technique is developed by examining two specific content themes and their applications in-depth: number theory and algebra. This choice of content themes enables students to develop an understanding of proof technique in the context of topics with which they are already familiar, as well as reinforcing natural and conceptual understandings of mathematical methods and styles.
The key to the text is its interesting and intriguing problems, exercises, theorems, and proofs, showing how students will transition from the usual, more routine calculus to abstraction while also learning how to “prove” or “solve” complex problems. This method of instruction is augmented by examining applications of number theory in systems such as RSA cryptography, Universal Product Code (UPC), and International Standard Book Number (ISBN). The numerous problems and examples included in each section reward curiosity and insightfulness over more simplistic approaches. Each problem set begins with a few easy problems, progressing to problems or proofs with multi-step solutions. Exercises in the text stay close to the examples of the section, allowing students the immediate opportunity to practice developing techniques. 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.

Caracteristici

Provides a foundation for solving proofs and problems as a transition to more abstract algebra and mathematics Readers learn problem solving abilities concerning proofs through numbers and algebra Problems and theorems are focused on many diverse areas of number theory and algebra