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

Description:

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.


Reviews:

5.0 out of 5 stars Masterpiece

A.M. · 27 September 2024

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.

5.0 out of 5 stars A very readable introduction to proof reading and writing

Q. · 22 February 2025

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.

5.0 out of 5 stars Wish I'd had this during my undergrad studies.

R.S. · 30 March 2023

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.

5.0 out of 5 stars This is an amazing book! I would also recommend to participants of mathematical olympiads.

R. · 13 May 2023

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.

4.0 out of 5 stars Great but needs solutions to exercises for self-learners

P.R.A. · 21 July 2021

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.

5.0 out of 5 stars Useful

N. · 30 July 2025

Very useful book

5.0 out of 5 stars Proofs, Jay Cummings

A.S. · 16 June 2025

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

5.0 out of 5 stars Brilliant Book

A.C. · 3 June 2021

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.

Simplesmente fantástico

G.G. · 3 June 2025

Ainda estou no começo, capítulo 2, mas já pude perceber a qualidade do produto. O autor conversa com você, comete erros junto com você, tem um tom mais informal e engraçado, começando pela dedicatória a sua esposa, que Léo I livro inteiro com exceção das partes de matemática, rsrsrs.Simplesmente sensacional, um curso de matemática dado por aquele tipo de professor que transforma sua visão de um determinado assunto e que cruzamos raramente na nossa carreira de estudos.

Tolles Buch über mathematische Beweise

D. · 17 May 2025

Das Buch liest sich großartig. Der Author hat sich sehr viel Mühe gegeben die Themen interessant und verständlich zu erklären und zu veranschaulichen. Dazwischen immer wieder humorvolle Spitzen, sehr erfrischend!

Finally, a book that does its job, and does it exceptionally at that.

p. · 16 November 2024

I find that this book is sooo good all around the board. It has exercises which is always very important, and also the quality is perfect. Its big, wonderful, and cheap. What else can one ask for?

Excellent Introduction to Wider Mathematics

L.D.B. · 7 April 2024

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.

Good and Funny Math book

D. · 26 March 2025

Very good Math book

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

Product ID: KO08G8WPIS1
Condition: New

4.8

AED16876

Price includes VAT & Import Duties
Type: Paperback
Availability: In Stock

Quantity:

|

Order today to get by 7-14 business days

Delivery fee of AED 20. Free for orders above AED 200.

Returns & Warranty policies

Imported From: United Kingdom

At BOLO, we work hard to ensure the products you receive are new, genuine, and sourced from reputable suppliers.

BOLO is not an authorized or official retailer for most brands, nor are we affiliated with manufacturers unless specifically stated on a product page. Instead, we source verified sellers, authorized distributors or directly from the manufacturer.

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Unless otherwise stated, all prices displayed on the product page include applicable taxes and import duties.

BOLO operates in accordance with the laws and regulations of United Arab Emirates. Any items found to be restricted or prohibited for sale within the UAE will be cancelled prior to shipment. We take proactive measures to ensure that only products permitted for sale in United Arab Emirates are listed on our website.

All items are shipped by air, and any products classified as “Dangerous Goods (DG)” under IATA regulations will be removed from the order and cancelled.

All orders are processed manually, and we make every effort to process them promptly once confirmed. Products cancelled due to the above reasons will be permanently removed from listings across the website.

Similar suggestions by Bolo

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Similar items from “Teaching Aids”

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

Product ID: KO08G8WPIS1
Condition: New

4.8

Proofs: A Long-Form Mathematics Textbook (The Long-Form Math Textbook Series)-0
Type: Paperback

AED16876

Price includes VAT & Import Duties
Availability: In Stock

Quantity:

|

Order today to get by 7-14 business days

Delivery fee of AED 20. Free for orders above AED 200.

Returns & Warranty policies

Imported From: United Kingdom

At BOLO, we work hard to ensure the products you receive are new, genuine, and sourced from reputable suppliers.

BOLO is not an authorized or official retailer for most brands, nor are we affiliated with manufacturers unless specifically stated on a product page. Instead, we source verified sellers, authorized distributors or directly from the manufacturer.

Each product undergoes thorough inspection and verification at our consolidation and fulfilment centers to ensure it meets our strict authenticity and quality standards before being shipped and delivered to you.

If you ever have concerns regarding the authenticity of a product purchased from us, please contact Bolo Support. We will review your inquiry promptly and, if necessary, provide documentation verifying authenticity or offer a suitable resolution.

Your trust is our top priority, and we are committed to maintaining transparency and integrity in every transaction.

All product information, images, descriptions, and reviews originate from the manufacturer or from trusted sellers overseas. BOLO is not affiliated with, endorsed by, or an authorized retailer for most brands listed on our website unless stated otherwise.

While we strive to display accurate information, variations in packaging, labeling, instructions, or formulation may occasionally occur due to regional differences or supplier updates. For detailed or manufacturer-specific information, please contact the brand directly or reach out to BOLO Support for assistance.

Unless otherwise stated, all prices displayed on the product page include applicable taxes and import duties.

BOLO operates in accordance with the laws and regulations of United Arab Emirates. Any items found to be restricted or prohibited for sale within the UAE will be cancelled prior to shipment. We take proactive measures to ensure that only products permitted for sale in United Arab Emirates are listed on our website.

All items are shipped by air, and any products classified as “Dangerous Goods (DG)” under IATA regulations will be removed from the order and cancelled.

All orders are processed manually, and we make every effort to process them promptly once confirmed. Products cancelled due to the above reasons will be permanently removed from listings across the website.

Description:

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.


Reviews:

5.0 out of 5 stars Masterpiece

A.M. · 27 September 2024

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.

5.0 out of 5 stars A very readable introduction to proof reading and writing

Q. · 22 February 2025

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.

5.0 out of 5 stars Wish I'd had this during my undergrad studies.

R.S. · 30 March 2023

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.

5.0 out of 5 stars This is an amazing book! I would also recommend to participants of mathematical olympiads.

R. · 13 May 2023

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.

4.0 out of 5 stars Great but needs solutions to exercises for self-learners

P.R.A. · 21 July 2021

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.

5.0 out of 5 stars Useful

N. · 30 July 2025

Very useful book

5.0 out of 5 stars Proofs, Jay Cummings

A.S. · 16 June 2025

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

5.0 out of 5 stars Brilliant Book

A.C. · 3 June 2021

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.

Simplesmente fantástico

G.G. · 3 June 2025

Ainda estou no começo, capítulo 2, mas já pude perceber a qualidade do produto. O autor conversa com você, comete erros junto com você, tem um tom mais informal e engraçado, começando pela dedicatória a sua esposa, que Léo I livro inteiro com exceção das partes de matemática, rsrsrs.Simplesmente sensacional, um curso de matemática dado por aquele tipo de professor que transforma sua visão de um determinado assunto e que cruzamos raramente na nossa carreira de estudos.

Tolles Buch über mathematische Beweise

D. · 17 May 2025

Das Buch liest sich großartig. Der Author hat sich sehr viel Mühe gegeben die Themen interessant und verständlich zu erklären und zu veranschaulichen. Dazwischen immer wieder humorvolle Spitzen, sehr erfrischend!

Finally, a book that does its job, and does it exceptionally at that.

p. · 16 November 2024

I find that this book is sooo good all around the board. It has exercises which is always very important, and also the quality is perfect. Its big, wonderful, and cheap. What else can one ask for?

Excellent Introduction to Wider Mathematics

L.D.B. · 7 April 2024

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.

Good and Funny Math book

D. · 26 March 2025

Very good Math book

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More from this brand

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