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

TLDR: This book is a masterpiece. I have had a long and slow journey to mathematics and I find it a challenging subject. I find many books obscure and difficult to follow. I firmly believe this is not a necessary feature of the subject, for the most part, but is more often than not a failing in pedagogy. On the contrary Jay has, in my opinion, succeeded in teaching a subject that although difficult is entirely accessible. Almost by definition any book is going to be either too hard or too easy for most people. For myself - someone approximately at the beginning of undergraduate level in maths - who has never “got” proofs and wanted to start to understand them, I have found this the perfect introduction.

Very useful book

This book covers much the same ground as many other textbooks on proofs, particularly the discreet math books like the one by Epp. However, and compared to other dedicated proof books like the very popular Velleman, this one is much more readable in my opinion. It's also cheaper. I definitely recommend to any student who is struggling with some of the explanations of things like proof by strong mathematical induction given in other textbooks.

Always felt that my stdis left much to be desired on the subject of proofs. Managed to cope with what I had until now (20 years later). I'm a bit rusty anyway and I REALLY need to know my stuff in this area right now. An excellent book with excellent explanations and which enables rapid progress but has plenty of detail and help where you need it. NOTE: Cummings self publishes to keep the cost down, which is why this book is cheap in comparision.

It's a really good book which makes proofs accessible and fun. However, I think a way needs to be found to provide self-learners with solutions to the many exercises. Without access to these (or a teacher) a great deal of educational value is lost.

Proofs are scary things, full of complicated notation and 'rigour' ... but it need not be so. Jay Cummings has written a brilliant book on proofs guiding the reader from simple to more complex examples in the most gentle and fun way possible. Finally here is a book that does not overwhelm the reader but focuses on teaching mathematics in a way that builds up understanding step by step. Where the book could do better is by providing solutions to at least some of the numerous exercises in each chapter, currently there are none.

Excellent service! Product arrived on time and was in mint condition. Highly recommended 👌

The book has a wide selection of topics and unlike most maths books it provides clear concise and interesting explanatations. I bought this book to help with my proof writing ability in mathematical olympiads(the Irish Mathematical Olympiad IRMO), it helped a lot. The open questions and brief introductions into higher maths were very interesting and has sparked interest in maths beyond second level olympiads.I can't wait to read more of Jay's books.

The ‘Introduction To’ sections of the book were my favourite and I loved the inclusion of open problems as it builds a curiosity towards mathematics.

Very good Math book

Amazing book to start proofs!

Libro eccezionale che non richiede una conoscenza approfondita per poter essere compreso. Le spiegazioni sono molto chiare e le dimostrazioni piacevoli e divertenti da svolgere. L'approccio è molto informale ed aiuta a mantenere alto il livello di attenzione durante lo studio. Consigliatissimo!

I just finished reading another wonderful book by Jay Cummings. The book is titled "Proofs" and is intended to help the reader learn how to write diverse kinds of proofs from many different areas of mathematics. I especially liked Chapter 4 on Induction because the writing is very clear and to the point. He discusses both weak and strong induction and his examples are extremely well chosen. I especially like his writing style. He also writes about math proofs by induction that contain mistakes that can mislead the reader. His example of how all people have the same name is the same as a similar example not in his book that all horses have the same color. Trying to find the mistakes in these proofs can be a real challenge, but once you do it you will understand math induction even better than you did before. This book has a rather ambitious aim, as proof writing is all anyone does in upper division math courses. Trying to learn how to write proofs in such a wide open field is not easy. However, the author does not try to teach you any advanced math and that is another reason I am so attracted to his writing style. Here is another small but important example. In discussing functions, Jay explains that writing f(x) is standard for a 1-tuple, but writing f((x,y)) with an order pair is not necessary. This a small notational convention that can trip up some students. Jay gives you permission to be confused at times and is aware that even very small things can make your life complicated! Jay has written three books that are all very different. I recently learned that I read his three books in the reverse time order in which he wrote them. Nonetheless, I think most people will find his books very worth while. The "Proofs" book is as good as any and contains a lot of information.