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

Penned By Jay Cummings
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Proofs: A Long-Form Mathematics Textbook (The Long-Form Math Textbook Series)

Penned By Jay Cummings
4.85 58,475 Ratings
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Editor's Note:
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. ~Jay Cummings

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Length 511 Pages
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Identifier (ISBN) b08t8jcvf1
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Jay Cummings
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Jay Cummings

Jay Cummings is the author of several novels. Originally from the East Coast, they now live in the Midwest with family. In addition to writing, Jay Cummings enjoys gardening, cooking, and reading way past bedtime. Their work often explores the complexities of everyday life and human relationships.

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Sarah Lopez
Sarah Lopez September 30, 2024
"Took a moment to build, but the payoff was spectacular. The middle had a deliberate pace, but the climax was phenomenal. I would highly recommend this to discerning readers."
Steven Wright
Steven Wright February 26, 2025
"Absolutely brilliant. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. I will undoubtedly revisit this masterpiece."
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Michelle Roberts
Michelle Roberts March 14, 2025
"Absolutely brilliant. I adored how the story unfolded. It kept me captivated without sacrificing elegance. I am now a lifelong fan of this author."
Melissa Harris
Melissa Harris April 14, 2024
"Such a rich narrative. The prose is exquisite. It’s accessible yet undeniably sophisticated. A true literary gem."
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Linda Hall January 28, 2025
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Nicholas Lopez February 01, 2024
"Exceeded all my expectations. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. A true literary gem."
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"Such a rich narrative. The author orchestrated a breathtaking finale. Truly a work of art. I will undoubtedly revisit this masterpiece."
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"Masterpiece from start to finish. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. I will undoubtedly revisit this masterpiece."
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"Such a rich narrative. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. I am now a lifelong fan of this author."
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"I was completely blown away. I became deeply invested in the protagonist's journey. A masterclass in storytelling. A true literary gem."
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"Exceeded all my expectations. The prose is exquisite. It’s accessible yet undeniably sophisticated. I am now a lifelong fan of this author."
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"I was completely blown away. The author orchestrated a breathtaking finale. Truly a work of art. Worth every moment spent reading."
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"Honestly one of the most premium reading experiences lately. The author orchestrated a breathtaking finale. Truly a work of art. An effortless 5 stars."
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"Honestly one of the most premium reading experiences lately. I became deeply invested in the protagonist's journey. A masterclass in storytelling. An absolute must-read for connoisseurs of the genre."
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"Such a rich narrative. The prose is exquisite. It’s accessible yet undeniably sophisticated. I am now a lifelong fan of this author."
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Steven Roberts
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"A highly sophisticated read. The middle had a deliberate pace, but the climax was phenomenal. Quite impressive overall."
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"A profoundly satisfying read. The prose is exquisite. It’s accessible yet undeniably sophisticated. An absolute must-read for connoisseurs of the genre."
Steven Roberts
Steven Roberts April 29, 2025
"Masterpiece from start to finish. I became deeply invested in the protagonist's journey. A masterclass in storytelling. An absolute must-read for connoisseurs of the genre."
Linda Brown
Linda Brown April 14, 2025
"Took a moment to build, but the payoff was spectacular. The protagonist was compelling, though some subplots felt slightly overshadowed. A wonderful companion for a quiet weekend."
Anthony Harris
Anthony Harris July 19, 2025
"Took a moment to build, but the payoff was spectacular. The protagonist was compelling, though some subplots felt slightly overshadowed. A very distinguished novel."
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Carol Harris December 11, 2024
"I was completely blown away. The prose is exquisite. It’s accessible yet undeniably sophisticated. I am now a lifelong fan of this author."
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"A very strong entry. The middle had a deliberate pace, but the climax was phenomenal. I would highly recommend this to discerning readers."
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"Exceeded all my expectations. I became deeply invested in the protagonist's journey. A masterclass in storytelling. A true literary gem."
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