Proofs; A Long Form Mathematics Textbook

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Book Description

“Proofs: A Long-Form Mathematics Textbook” by Jay Cummings is a unique and student-friendly resource designed to transform the way students approach and understand the art of mathematical proofs. Departing from the conventional definition-theorem-proof format, this textbook provides a refreshing approach with an emphasis on commentary, motivation, and thorough explanations, making it accessible and engaging for students.

The distinguishing features of this textbook are its in-depth proofs that prioritize comprehension over brevity. To aid in understanding, numerous proofs are accompanied by “scratch work” or proof sketches, offering students a holistic perspective on how to develop these proofs independently. This approach not only demystifies the world of mathematical proofs but also encourages students to actively engage with the material.

Cummings covers a wide range of mathematical concepts, including intuitive proofs, direct proofs, sets, induction, logic, contrapositive, contradiction, functions, and relations. The text doesn't just introduce mathematical concepts; it strives to make them visually tangible with over 200 illustrations. The writing style is casual and conversational, even incorporating moments of humor to keep students engaged.

Moreover, “Proofs: A Long-Form Mathematics Textbook” serves as an introductory guide to higher mathematics. Through carefully chosen examples and theorems, students are introduced to various mathematical domains. Each chapter concludes with a foray into a new area of mathematics, such as Ramsey theory, number theory, topology, sequences, real analysis, big data, game theory, cardinality, and group theory.

As an added bonus, the textbook features “pro-tips” at the end of each chapter, offering valuable insights and advice, including finer points about the material, study tips, historical notes, and cultural context in mathematics. Furthermore, after each chapter's exercises, students are introduced to intriguing unsolved problems in mathematics, sparking curiosity and exploration.

Jay Cummings, the author, is not only passionate about mathematics but is also dedicated to making math education more accessible and affordable. His experience and expertise, including a PhD from UC San Diego under the mentorship of Ron Graham and his position on the faculty at California State University, Sacramento, make him a credible and engaging source of mathematical instruction.

In summary, “Proofs: A Long-Form Mathematics Textbook” offers a fresh and engaging approach to understanding mathematical proofs, making it an invaluable resource for students and educators alike. It stands out for its emphasis on clarity, motivation, and the beauty of mathematics, all while striving to keep education accessible and affordable.