This innovative undergraduate textbook approaches number theory through the lens of abstract algebra. Written in an engaging and whimsical style, this text will introduce students to rings, groups, fields, and other algebraic structures as they discover the key concepts of elementary number theory. Inquiry-based learning (IBL) appears throughout the chapters, allowing students to develop insights for upcoming sections while simultaneously strengthening their understanding of previously covered topics. The text is organized around three core themes: the notion of what a “number” is, and the premise that it takes familiarity with a large variety of number systems to fully explore number theory; the use of Diophantine equations as catalysts for introducing and developing structural ideas; and the role of abstract algebra in number theory, in particular the extent to which it provides the Fundamental Theorem of Arithmetic for various new number systems. Other aspects of modern number theory – including the study of elliptic curves, the analogs between integer and polynomial arithmetic, p-adic arithmetic, and relationships between the spectra of primes in various rings – are included in smaller but persistent threads woven through chapters and exercise sets. Each chapter concludes with exercises organized in four categories: Calculations and Informal Proofs, Formal Proofs, Computation and Experimentation, and General Number Theory Awareness. IBL “Exploration” worksheets appear in many sections, some of which involve numerical investigations. To assist students who may not have experience with programming languages, Python worksheets are available on the book’s website. The final chapter provides five additional IBL explorations that reinforce and expand what students have learned, and can be used as starting points for independent projects. The topics covered in these explorations are public key cryptography, Lagrange’s four-square theorem, units and Pell’s Equation, various cases of the solution to Fermat’s Last Theorem, and a peek into other deeper mysteries of algebraic number theory. Students should have a basic familiarity with complex numbers, matrix algebra, vector spaces, and proof techniques, as well as a spirit of adventure to explore the “numberverse.”
Si seulement toutes les langues étaient aussi claires que le lakota... la langue Sioux ! Josie, 16 ans, est surdouée. Elle pense savoir tout sur tout et adore décortiquer, analyser, disséquer les mots... même ceux qui ne font pas vraiment partie de son quotidien, comme "amour", "petit ami", ou "rupture". Le jour où sa sœur Kate présente son fiancé à toute la famille, autour d'un délicieux plat de pâtes dont leur mère a le secret, Josie est persuadée que cette dernière fait fausse route. Comment pourrait-elle " aimer " ce garçon suffisant et insupportable ! Josie s'engage dans une bataille féroce pour briser ce couple... mais lorsqu'elle craque pour son séduisant professeur de linguistique, sa propre vie sentimentale devient bien chaotique...
This innovative undergraduate textbook approaches number theory through the lens of abstract algebra. Written in an engaging and whimsical style, this text will introduce students to rings, groups, fields, and other algebraic structures as they discover the key concepts of elementary number theory. Inquiry-based learning (IBL) appears throughout the chapters, allowing students to develop insights for upcoming sections while simultaneously strengthening their understanding of previously covered topics. The text is organized around three core themes: the notion of what a “number” is, and the premise that it takes familiarity with a large variety of number systems to fully explore number theory; the use of Diophantine equations as catalysts for introducing and developing structural ideas; and the role of abstract algebra in number theory, in particular the extent to which it provides the Fundamental Theorem of Arithmetic for various new number systems. Other aspects of modern number theory – including the study of elliptic curves, the analogs between integer and polynomial arithmetic, p-adic arithmetic, and relationships between the spectra of primes in various rings – are included in smaller but persistent threads woven through chapters and exercise sets. Each chapter concludes with exercises organized in four categories: Calculations and Informal Proofs, Formal Proofs, Computation and Experimentation, and General Number Theory Awareness. IBL “Exploration” worksheets appear in many sections, some of which involve numerical investigations. To assist students who may not have experience with programming languages, Python worksheets are available on the book’s website. The final chapter provides five additional IBL explorations that reinforce and expand what students have learned, and can be used as starting points for independent projects. The topics covered in these explorations are public key cryptography, Lagrange’s four-square theorem, units and Pell’s Equation, various cases of the solution to Fermat’s Last Theorem, and a peek into other deeper mysteries of algebraic number theory. Students should have a basic familiarity with complex numbers, matrix algebra, vector spaces, and proof techniques, as well as a spirit of adventure to explore the “numberverse.”
Shakespeare and Digital Performance in Practice explores the impact of digital technologies on the theatrical performance of Shakespeare in the twenty-first century, both in terms of widening cultural access and developing new forms of artistry. Through close analysis of dozens of productions, both high-profile and lesser known, it examines the rise of live broadcasting and recording in the theatre, the growing use of live video feeds and dynamic projections on the mainstream stage, and experiments in born-digital theatre-making, including social media, virtual reality, and video-conferencing adaptations. In doing so, it argues that technologically adventurous performances of Shakespeare allow performers and audiences to test what they believe theatre to be, as well as to reflect on what it means to be present—with a work of art, with others, with oneself—in an increasingly online world.
Dozens of realistic cases help students make transition from classroom to clinic The Physical Therapy Case Files series gives students realistic cases designed to help them make the transition from classroom to clinical work and is an outstanding review for the specialty topics included on the American Physical Therapy Association certification exams. This evidence-based series can stand alone or is the perfect complement to textbooks for enhanced learning in the context of real patients. Each case includes clinical tips, evidence-based practice recommendations, analysis, and review questions. These cases teach students how to think through evaluation, assessment, and treatment planning. Includes 42 realistic sports medicine cases A great clinical refresher for practitioners looking to brush up on their skills
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