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  • Proofs: A Long-Form Mathematics Textbook (The Long-Form Math Textbook Series)

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Proofs: A Long-Form Mathematics Textbook (The Long-Form Math Textbook Series)


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This 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.

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About the author

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Jay Cummings
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Jay Cummings 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. After receiving his PhD from UC San Diego under Ron Graham, he joined the faculty at California State University, Sacramento. His combinatorics research continues and he particularly enjoys involving students in his work. He is passionate about his teaching and enjoys designing new courses.

Customer reviews

4.8 out of 5 stars
1,044 global ratings

Customers say

Customers find the textbook well-suited for self-study, taking them through higher-level mathematics concepts in an active learning setup. The book is easy to comprehend, written in a conversational tone, and keeps readers engaged with its entertaining puzzle-book style. Moreover, it features wildly amusing footnotes and is worth more than its price suggests. However, the exercise quality receives mixed feedback, with some customers appreciating the good exercises while others note the lack of solutions.
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73 customers mention content, 66 positive, 7 negative
Customers find the book's content well-suited for self-study and active learning, taking readers through higher-level mathematics concepts.
“Proofs” is a great book for those of you who wish to improve your advanced math skills and get better doing proofs....Read more
Excellent. All math books should be written this way, it's just a more fun and efficient way to learn.Read more
Excellent book ruined by page after page of substandard printing. Appears there was no QC in this print shop.Read more
...pursuing self-study, and Cummings brings the reader along with clear explanations and very insightful exercises....Read more
25 customers mention readability, 23 positive, 2 negative
Customers find the textbook easy to comprehend and wonderfully clear, with one customer noting that it clearly walks through each step.
...The author has kept it engaging and understandable. The information isn’t just presented, but taught and taught well....Read more
Easy to read. I wasn’t scratching my head trying to figure out notation or what a variable represented....Read more
...It’s a fun and challenging read and really illuminates the problem solving strategies well. I can’t wait for more books from this author!Read more
Exceptionally good textbook for self-learning and digestible explanations.Read more
24 customers mention engaging, 24 positive, 0 negative
Customers find the book engaging, describing it as an entertaining puzzle book that keeps readers interested. One customer mentions being excited after reading the first 20 pages.
...This book has been great for him because it is engaging and he's enjoying learning about different areas of math. I highly recommend it!Read more
...It will make math feel fuzzy and warm and exciting and delightful and fun....Read more
I thoroughly enjoyed reading this book. Most books don’t dive right into doing proofs in the beginning, which is what this book does....Read more
...Makes it a joy to read and think about the material. Really appreciate it!...Read more
19 customers mention writing style, 19 positive, 0 negative
Customers appreciate the writing style of the textbook, noting it is well written and conversational in tone.
Really appreciate Jay Cummings publishing this himself. It is well written and even funny at times....Read more
...I especially like his writing style. He also writes about math proofs by induction that contain mistakes that can mislead the reader....Read more
...The approach is interesting and engaging and the book is very well written....Read more
...As someone with a degree in mathematics, I find the book well written, although a little bit wordy at times. My friend, though, absolutely loved it!Read more
11 customers mention humor, 11 positive, 0 negative
Customers enjoy the book's humor, particularly its wildly amusing footnotes, with one customer noting how it alternates between hardcore sections and "fun" sections.
...There are humorous footnotes, quirky diagrams/pictures, and insightful intuitions - all in an effort to acclimate the reader to a thinkspace where...Read more
...The author writes with a kind of clarity and humor that is far more engaging than almost any other serious math book I've read....Read more
...It is well written and even funny at times. Still, though, this book serves as an extremely informative text.Read more
...This book have given me more than I expected. It is like having a witty and knowledgeable tutor built into the book guiding me through the material....Read more
11 customers mention value for money, 9 positive, 2 negative
Customers find the textbook worth more than its price would suggest.
...Nonetheless, I think most people will find his books very worth while. The "Proofs" book is...Read more
...the author self-published the book - our benefit is the fair price (thanks Jay), the downside might be the rather poor paper quality...Read more
...I picked this one due to the low price, but, wow, is it worth more than it's price would suggest....Read more
...All this for an affordable price. Do yourself a favor and keep this and the author's other book, Real Analysis, lying around.Read more
9 customers mention conversational, 9 positive, 0 negative
Customers appreciate the textbook's conversational and approachable style.
...It is well written and approachable. The concepts are introduced in a non intimidating manner and understandable. Very well written. Thank you!Read more
I love the conversational style and the humourous approach taken by the author while teaching the material....Read more
...It’s excellent. It’s rigorous and conversational, nothing like a typical text that might cost 10x what this one does.Read more
...The choice of topics is quite good and the tone is conversational and times but never chatty. Always entertaining and encouraging....Read more
12 customers mention exercise quality, 8 positive, 4 negative
Customers have mixed opinions about the exercises in the textbook, with some finding lots of good ones while others note there are no solutions provided.
...The exercises are great. I already bought the author’s Real Analysis book and hope to get to it after working through every chapter in this book....Read more
Seems very well written, but I was disappointed that no solutions to exercises can be found in the book....Read more
...Even so, you'll find many, many new insights. Lots of good exercises as well....Read more
...The book contains many exercises which are certainly valuable to help student understanding....Read more