A Pythagorean Introduction to Number Theory: Right Triangles, Sums of Squares, and Arithmetic (Undergraduate Texts in Mathematics) - Hardcover

Takloo-Bighash, Ramin

Hardcover

ISBN 10: 3030026035 ISBN 13: 9783030026035

Verlag: Springer-Verlag GmbH, 2018

Inhaltsangabe

<P>RIGHT TRIANGLES ARE AT THE HEART OF THIS TEXTBOOK’S VIBRANT NEW APPROACH TO ELEMENTARY NUMBER THEORY. INSPIRED BY THE FAMILIAR PYTHAGOREAN THEOREM, THE AUTHOR INVITES THE READER TO ASK NATURAL ARITHMETIC QUESTIONS ABOUT RIGHT TRIANGLES, THEN PROCEEDS TO DEVELOP THE THEORY NEEDED TO RESPOND. THROUGHOUT, STUDENTS ARE ENCOURAGED TO ENGAGE WITH THE MATERIAL BY POSING QUESTIONS, WORKING THROUGH EXERCISES, USING TECHNOLOGY, AND LEARNING ABOUT THE BROADER CONTEXT IN WHICH IDEAS DEVELOPED.</P> PROGRESSING FROM THE FUNDAMENTALS OF NUMBER THEORY THROUGH TO GAUSS SUMS AND QUADRATIC RECIPROCITY, THE FIRST PART OF THIS TEXT PRESENTS AN INNOVATIVE FIRST COURSE IN ELEMENTARY NUMBER THEORY. THE ADVANCED TOPICS THAT FOLLOW, SUCH AS COUNTING LATTICE POINTS AND THE FOUR SQUARES THEOREM, OFFER A VARIETY OF OPTIONS FOR EXTENSION, OR A HIGHER-LEVEL COURSE; THE BREADTH AND MODULARITY OF THE LATER MATERIAL IS IDEAL FOR CREATING A SENIOR CAPSTONE COURSE. NUMEROUS EXERCISES ARE INCLUDED THROUGHOUT, MANY OF WHICH ARE DESIGNED FOR SAGEMATH.<P></P><P></P> <P>BY INVOLVING STUDENTS IN THE ACTIVE PROCESS OF INQUIRY AND INVESTIGATION, THIS TEXTBOOK IMBUES THE FOUNDATIONS OF NUMBER THEORY WITH INSIGHTS INTO THE LIVELY MATHEMATICAL PROCESS THAT CONTINUES TO ADVANCE THE FIELD TODAY. EXPERIENCE WRITING PROOFS IS THE ONLY FORMAL PREREQUISITE FOR THE BOOK, WHILE A BACKGROUND IN BASIC REAL ANALYSIS WILL ENRICH THE READER’S APPRECIATION OF THE FINAL CHAPTERS.<BR></P>

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Reseña del editor

Right triangles are at the heart of this textbook’s vibrant new approach to elementary number theory. Inspired by the familiar Pythagorean theorem, the author invites the reader to ask natural arithmetic questions about right triangles, then proceeds to develop the theory needed to respond. Throughout, students are encouraged to engage with the material by posing questions, working through exercises, using technology, and learning about the broader context in which ideas developed.

Progressing from the fundamentals of number theory through to Gauss sums and quadratic reciprocity, the first part of this text presents an innovative first course in elementary number theory. The advanced topics that follow, such as counting lattice points and the four squares theorem, offer a variety of options for extension, or a higher-level course; the breadth and modularity of the later material is ideal for creating a senior capstone course. Numerous exercises are included throughout, many of which are designed for SageMath.

By involving students in the active process of inquiry and investigation, this textbook imbues the foundations of number theory with insights into the lively mathematical process that continues to advance the field today. Experience writing proofs is the only formal prerequisite for the book, while a background in basic real analysis will enrich the reader’s appreciation of the final chapters.

Biografía del autor

Ramin Takloo-Bighash is a Professor of Mathematics at the University of Illinois at Chicago, where his research is centered on Diophantine geometry and automorphic forms. His enthusiasm for research inspires his teaching at all levels.

Details zur Ausgabe

• Verlag: Springer-Verlag GmbH
• Erscheinungsdatum: 2018
• ISBN 10: 3030026035
• ISBN 13: 9783030026035
• Einband: Tapa dura
• Sprache: Englisch
• Auflage: 1
• Anzahl der Seiten: 300
• Kontakt zum Hersteller: Nicht verfügbar