Theo Todman's Book Collection (Book-Paper Abstracts)

Back Cover Blurb

• Proofs: A Long-Form Mathematics Textbook is designed for students. Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. The proofs are not terse, and aim for understanding over economy. Furthermore, dozens of proofs are preceded by "scratch work" or a proof sketch to give students a big-picture view and an explanation of how they would come up with it on their own.
• This book covers intuitive proofs, direct proofs, sets, induction, logic, the contrapositive, contradiction, functions and relations. The text aims to make the ideas visible, and contains over 200 illustrations. The writing is relaxed and conversational, and includes periodic attempts at humor.
• This text is also an introduction to higher mathematics. This is done in-part through the chosen examples and theorems. Furthermore, following every chapter is an introduction to an area of math. These include Ramsey theory, number theory, topology, sequences, real analysis, big data, game theory, cardinality and group theory.
• After every chapter are "pro-tips," which are short thoughts on things I wish I had known when I took my intro-to-proofs class. They include finer comments on the material, study tips, historical notes, comments on mathematical culture, and more. Also, after each chapter's exercises is an introduction to an unsolved problem in mathematics.
• In the first appendix we discuss some further proof methods, the second appendix is a collection of particularly beautiful proofs, and the third is some writing advice. More content, including hints and solutions to select exercises, may be found at Long(er)-Form Mathematics.
• Jay Cummings is a professor in the Department of Mathematics and Statistics at California State University. Sacramento. He received his Ph.D. from UC San Diego under Ron Graham. Most of his published work is in various areas of combinatorics. He believes that learning math has become far too expensive, and is striving to write textbooks which are enjoyable to read, highlight the beauty in mathematics, and are significantly more affordable than the others on the market. This is Jay's second book. His first is Real Analysis: A Long-Form Mathematics Textbook, which is currently in its second edition.