New Website
Up-to-date information about this book has moved to the website LongFormMath.com.
Proofs Textbook
This book is the second of a (planned) series of textbooks, which I am calling “long-form textbooks.” 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.
I also believe that learning mathematics has become too expensive and commercialized, and textbooks in particular are far overpriced.
I am endeavoring to help change this. This 500-page book is currently available on
for just $16, and in my own classes I offer a free alternative. Click
here
to find the book on Amazon.
Real Analysis Textbook, Second Edition
This book was the first long-form textbook. Just like with the proofs book, the aim of this book was for understanding and enjoyment. This proofs has even more “scratch work” and proof sketches. Examples often drive the narrative and challenge the intuition of the reader, and it has just as many pictures.
For this book and the proofs book, the writing is far smore relaxed than most textbooks, and I have included periodic historical notes, poor attempts at humor, and occasional diversions into other interesting areas of mathematics.
This text covers the real numbers, cardinality, sequences, series, the topology of the reals, continuity, differentiation, integration, and sequences and series of functions. Each chapter ends with exercises, and nearly all include some open questions. In the first appendix we construct the reals, and the second is a collection of additional peculiar and pathological examples from analysis.
This book was my first attempt to push back against expensive and commercialized textbooks.
This 430-page book is currently available on
for just $16, and in my own classes I offer a free alternative. Click
here
to find the book on Amazon.
Future Long-Form Textbooks
I plan to keep writing, and have some work underway on other long-form textbooks. When I know which subject will be next, I will announce it here and on my Twitter feed.
Other Books
I plan to write other books as well, such a book on math in the real world and some math-themed kids books. Stay tuned!