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This text offers a crucial primer on proofs and the language of mathematics. Brief and to the point, it lays out the fundamental ideas of abstract mathematics and proof techniques that students will need to master for other math courses. Campbell presents these concepts in plain English, with a focus on basic terminology and a conversational tone that draws natural parallels between the language of mathematics and the language students communicate in every day. The discussion highlights how symbols and expressions are the building blocks of statements and arguments, the meanings they convey, and why they are meaningful to mathematicians. In-class activities provide opportunities to practice mathematical reasoning in a live setting, and an ample number of homework exercises are included for self-study. This text is appropriate for a course in Foundations of Advanced Mathematics taken by students who've had a semester of calculus, and is designed to be accessible to students with a wide range of mathematical proficiency. It can also be used as a self-study reference, or as a supplement in other math courses where additional proofs practice is needed. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.

Bridges the gap between calculus and advanced mathematics - improving the student's ability to think and write in a mature mathematical fashion and providing a solid understanding of the material most useful for advanced courses.

An easy-to-use guide that shows how to read, understand, and do proofs. Shows how any proof can be understood as a sequence of techniques. Covers the full range of techniques used in proofs, such as the contrapositive, induction, and proof by contradiction. Explains how to identify which techniques are used and how they are applied in the specific problem. Illustrates how to read written proofs with many step-by-step examples. Includes new, expanded appendices related to discrete mathematics, linear algebra, modern algebra and real analysis.

'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.

The present volume of Research in Collegiate Mathematics Education, like previous volumes in this series, reflects the importance of research in mathematics education at the collegiate level. The editors in this series encourage communication between mathematicians and mathematics educators, and as pointed out by the International Commission of Mathematics Instruction (ICMI), much more work is needed in concert with these two groups. Indeed, editors of RCME are aware of this need and the articles published in this series are in line with that goal. Nine papers constitute this volume. The first two examine problems students experience when converting a representation from one particular system of representations to another. The next three papers investigate students learning about proofs. In the next two papers, the focus is instructor knowledge for teaching calculus. The final two papers in the volume address the nature of ``conception'' in mathematics. Whether they are specialists in education or mathematicians interested in finding out about the field, readers will obtain new insights about teaching and learning and will take away ideas that they can use.

Expanding on the first edition of An Introduction to Continuous-Time Stochastic Processes, this concisely written book is a rigorous and self-contained introduction to the theory of continuous-time stochastic processes. A balance of theory and applications, the work features concrete examples of modeling real-world problems from biology, medicine, industrial applications, finance, and insurance using stochastic methods. No previous knowledge of stochastic processes is required.

Designing interesting problems and writing assignments is one ofthe chief tasks of all teachers, but it can be especiallychallenging to translate and apply learning theory, good teachingtechniques, and writing assignments into STEM and otherquantitative disciplines. Student Writing in the QuantitativeDisciplines offers instructors in math-based disciplines meaningfulapproaches to making their coursework richer and more relevant fortheir students, as well as satisfying institutional imperatives forwriting curricula. This important resource provides instructorswith the hands-on skills needed to guide their students in writingwell in quantitative courses at all levels of the collegecurriculum and to promote students' general cognitive andintellectual growth. Comprehensive in scope, the book includes: Ideas for using writing as a means of learning mathematicalconcepts Illustrative examples of effective writing activities andassignments in a number of different genres Assessment criteria and effective strategies for responding tostudents' writing Examples of ways to help students engage in peer review,revision, and resubmission of their written work "Those of us who spend our lives urging faculty in alldisciplines to integrate more writing into their courses havewished for the day when someone like Patrick Bahls would stepforward with a book like this one."—Chris M. Anson, UniversityDistinguished Professor and director, Campus Writing and SpeakingProgram, North Carolina State University "Written by a mathematician, this readable, theoretically soundbook describes practical strategies for teachers in thequantitative sciences to assign and respond to students' writing.It also describes numerous approaches to writing that engagestudents in disciplinary learning, collaborative discovery, andeffective communication."—Art Young, Campbell Professor ofEnglish emeritus, Clemson University "Loaded with practical advice, this timely, important, andengaging book will be an invaluable resource for instructorswishing to bring the benefits of writing-to-learn to thequantitative disciplines. As a mathematician thoroughly grounded inwriting-across-the-curriculum scholarship, Bahls brings humor,classroom experience, and pedagogical savvy to a mission he clearlyloves—improving the quality of student learning in math andscience."—John C. Bean, professor, Seattle University, andauthor, Engaging Ideas

Ideal for mathematics majors and prospective secondary school teachers, Euclidean and Transformational Geometry provides a complete and solid presentation of Euclidean geometry with an emphasis on solving challenging problems. The author examines various strategies and heuristics for approaching proofs and discusses the process students should follow to determine how to proceed from one step to the next through numerous problem solving techniques. A large collection of problems, varying in level of difficulty, are integrated throughout the text and suggested hints for the more challenging problems appear in the instructor's solutions manual and can be used at the instructor's discretion.

This is an informal and accessible introduction to plane algebraic curves that also serves as a natural entry point to algebraic geometry. There is a unifying theme to the book: give curves enough living space and beautiful theorems will follow. This book provides the reader with a solid intuition for the subject, while at the same time keeping the exposition simple and understandable, by introducing abstract concepts with concrete examples and pictures. It can be used as the text for an undergraduate course on plane algebraic curves, or as a companion to algebraic geometry at graduate level. This book is accessible to those with a limited mathematical background. This is because for those outside mathematics there is a growing need for an entre to algebraic geometry, a need created by the ever-expanding role algebraic geometry is playing in areas ranging from biology to chemistry and robotics to cryptology.

The heart of mathematics is its elegance; the way it all fits together. Unfortunately, its beauty often eludes the vast majority of people who are intimidated by fear of the difficulty of numbers. Mathematical Elegance remedies this. Using hundreds of examples, the author presents a view of the mathematical landscape that is both accessible and fascinating. At a time of concern that American youth are bored by math, there is renewed interest in improving math skills. Mathematical Elegance stimulates students, along with those already experienced in the discipline, to explore some of the unexpected pleasures of quantitative thinking. Invoking mathematical proofs famous for their simplicity and brainteasers that are fun and illuminating, the author leaves readers feeling exuberant-as well as convinced that their IQs have been raised by ten points. A host of anecdotes about well-known mathematicians humanize and provide new insights into their lofty subjects. Recalling such classic works as Lewis Carroll's Introduction to Logic and A Mathematician Reads the Newspaper by John Allen Paulos, Mathematical Elegance will energize and delight a wide audience, ranging from intellectually curious students to the enthusiastic general reader.