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Real Analysis: A Long-Form Mathematics Textbook
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. Examples often drive the narrative and challenge the intuition of the reader. The text also aims to make the ideas visible, and contains over 100 illustrations. The writing is relaxed and includes periodic historical notes, poor attempts at humor, and occasional diversions into other interesting areas of mathematics.The 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. The first appendix contains a construction the reals, and the second is a collection of additional peculiar and pathological examples from analysis.The author believes most textbooks are extremely overpriced and endeavors to help change this.
375 pages, Paperback
Published July 30, 2018
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554 people want to read
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Displaying 1 - 13 of 13 reviews
December 27, 2022
Ive been trying to self study real analysis. Here’s a list of some analysis books I've either read or skimmed through:
- Understanding Analysis by Abbott
- fundamentals of mathematical analysis Haggarty
- analysis with an introduction to proof by Lay
- introduction to real analysis by Silva
- introduction to analysis by Mattuck
- many, many more that i could get my hands on
This is the best one i could find. The author actually explains the motivation behind the proofs, and shows you how to *think* about them so you can derive the theorem and similar results on your own. This is in stark contrast to the presentation in many other books where a pretentious “polished” proof is shown with absolutely no hint as to *how* it was actually derived in the first place.
Not so with this book. The author shows how the derivation is actually done by a human and not a computer.
There are also plenty of illustrations in the book to provide an intuitive understanding of the proofs before giving the formal derivation.
Each chapter comes with an introductory section providing the motivation for the topics about to be studied, which gets you curious about what you’re about to learn. This is the first math book I’ve read that I actually want to *read* for its own sake, not just use as something to “study” from.
The only shortcoming is that there are no solutions to the exercises in the book, which often include crucial concepts (none that are needed for later chapters though; you can easily read the entire book and work on the examples without doing the exercises). The author does provide hints and partial solutions to some of the exercises on his website but i wish they were more comprehensive for those looking to self study.
I also wish there was a section on metric spaces but really, i cant complain. This is such a well written book; it deserves nothing less than 5 stars. I haven’t listed the many books i tried to work through to understand analysis, this one just clicked for me very quickly. I even prefer it over Abbott’s as a first book on mathematical analysis, and I’ll be buying his other book on proof writing as well :)
- Understanding Analysis by Abbott
- fundamentals of mathematical analysis Haggarty
- analysis with an introduction to proof by Lay
- introduction to real analysis by Silva
- introduction to analysis by Mattuck
- many, many more that i could get my hands on
This is the best one i could find. The author actually explains the motivation behind the proofs, and shows you how to *think* about them so you can derive the theorem and similar results on your own. This is in stark contrast to the presentation in many other books where a pretentious “polished” proof is shown with absolutely no hint as to *how* it was actually derived in the first place.
Not so with this book. The author shows how the derivation is actually done by a human and not a computer.
There are also plenty of illustrations in the book to provide an intuitive understanding of the proofs before giving the formal derivation.
Each chapter comes with an introductory section providing the motivation for the topics about to be studied, which gets you curious about what you’re about to learn. This is the first math book I’ve read that I actually want to *read* for its own sake, not just use as something to “study” from.
The only shortcoming is that there are no solutions to the exercises in the book, which often include crucial concepts (none that are needed for later chapters though; you can easily read the entire book and work on the examples without doing the exercises). The author does provide hints and partial solutions to some of the exercises on his website but i wish they were more comprehensive for those looking to self study.
I also wish there was a section on metric spaces but really, i cant complain. This is such a well written book; it deserves nothing less than 5 stars. I haven’t listed the many books i tried to work through to understand analysis, this one just clicked for me very quickly. I even prefer it over Abbott’s as a first book on mathematical analysis, and I’ll be buying his other book on proof writing as well :)
December 15, 2023
I love u Jay Cummings
August 17, 2020
i would have really appreciated this book back when i underwent "Trial by Fire" aka Analysis I. bartle's unassuming blue volume as used by GT was actually pretty solid prose, but the exercises were lacking and uneven, and it doesn't even cover Lebesgue, without which can you really claim to be integrating as opposed to kinda whipping up a hearty Riemann–Stieltjes soup of ℝᵖ. well, this doesn't push into measures either, but during those first weeks when you're beating your head against convergences and continuity, this would have been a wonderful guide (at significant cost savings, natch). cummings is rigorous, but expansive and didactic as well.
kids today have it too easy. we need a good war to thin out their numbers.
kids today have it too easy. we need a good war to thin out their numbers.
July 6, 2025
The GOAT textbook. Gotta rate this for the jokes but honestly the textbook reads insanely well and somehow lets me build a relationship with the author.
June 9, 2024
Real analysis has long served as a pons asinorum in students’ mathematical progression, dissuading would-be math majors and prospective graduate students from continuing their studies.
This is, in part, an unavoidable consequence of what professors call a “lack of mathematical maturity”: to truly grasp the nature of, say, a compact set, you must sit with subtle definitions and theorems for a long time. Not everyone possesses the requisite focus and discipline for such work.
But this is also —in the most charitable interpretation— a very much avoidable holdover from an earlier time, when typesetting challenges made figures cumbersome to produce and publishers set strict page limits. Leaf through Cauchy’s Cours d’analyse (1821) and you will not find a single graph, table, or illustration. The same holds for Rudin’s Principles of Mathematical Analysis —written 150 years later— and many other advanced calculus and real analysis textbooks used around the globe.
Although texts like Stephen Abbott's Understanding Analysis show that these conventions were in already beginning to change for the better, Cummings's Long-Form Mathematics Series marks a sea change.
Let’s take a moment to reason heuristically and do some scratch work before launching into the formal proof, you can almost hear Professor Cummings say.
Take a break and watch this video on infinite series—you'll thank me later.
Having trouble understanding pointwise convergence? Plot this function in Desmos and adjust the slider bar.
Here’s a clever pun and a historical aside to break up this particularly dry section.
These stylistic choices do pedagogical wonders for the text.
There are drawbacks to Cummings's approach. Piecing your way through Kant's Critique of Pure Reason builds intellectual stamina in ways a less opaque text never could; so too, future math PhDs are likely better served by a book written in the traditional sparse, laconic style. But for the unwashed masses trying to navigate a course that has dashed the hopes of prospective physicists, economists, and statisticians for generations, Real Analysis: A Long-Form Mathematics Textbook is a godsend.
This is, in part, an unavoidable consequence of what professors call a “lack of mathematical maturity”: to truly grasp the nature of, say, a compact set, you must sit with subtle definitions and theorems for a long time. Not everyone possesses the requisite focus and discipline for such work.
But this is also —in the most charitable interpretation— a very much avoidable holdover from an earlier time, when typesetting challenges made figures cumbersome to produce and publishers set strict page limits. Leaf through Cauchy’s Cours d’analyse (1821) and you will not find a single graph, table, or illustration. The same holds for Rudin’s Principles of Mathematical Analysis —written 150 years later— and many other advanced calculus and real analysis textbooks used around the globe.
Although texts like Stephen Abbott's Understanding Analysis show that these conventions were in already beginning to change for the better, Cummings's Long-Form Mathematics Series marks a sea change.
Let’s take a moment to reason heuristically and do some scratch work before launching into the formal proof, you can almost hear Professor Cummings say.
Take a break and watch this video on infinite series—you'll thank me later.
Having trouble understanding pointwise convergence? Plot this function in Desmos and adjust the slider bar.
Here’s a clever pun and a historical aside to break up this particularly dry section.
These stylistic choices do pedagogical wonders for the text.
There are drawbacks to Cummings's approach. Piecing your way through Kant's Critique of Pure Reason builds intellectual stamina in ways a less opaque text never could; so too, future math PhDs are likely better served by a book written in the traditional sparse, laconic style. But for the unwashed masses trying to navigate a course that has dashed the hopes of prospective physicists, economists, and statisticians for generations, Real Analysis: A Long-Form Mathematics Textbook is a godsend.
August 16, 2020
Very intuitive!
December 17, 2021
This and Stephen Abbott's Understanding Analysis, best books for a first course in Real Analysis EZ
July 24, 2024
This is such a great book. I know some people will think the humor is cheesy or dumb but it's not. It can be really depressing going through a hard proof but the humor and light heartedness helps tremendously. I read this along side Understanding Analysis by Stephen Abbot. My main book was Abbot's book and so whenever I get stuck, I go to back to this book and re-read the chapter to get help with what I missed. Very generous book! I love it.
March 18, 2022
Intuitive! Best!
March 31, 2023
best
This entire review has been hidden because of spoilers.
May 10, 2024
Readable introduction to real analysis with explanations of proofs and motivation. A good review of the topic.
July 29, 2024
I bought this book together with its companion, 'Proofs: A Long-Form Mathematics Textbook.' They make a great pair. However, this one felt weaker compared to the other.
Generally, the point of using a tricycle is so that, at some point, you can ride a bicycle. This book was good, but it felt like another tricycle — arguably a bicycle with training wheels. I understand the need for textbooks like this one, but having read one of them, I felt the second one was too lengthy.
If you are studying independently and trying to grasp epsilon-delta proofs, then by all means, this book is excellent. However, if you're trying to be time-efficient on top of that, there are probably better options available (although, they will inarguably be less entertaining).
Generally, the point of using a tricycle is so that, at some point, you can ride a bicycle. This book was good, but it felt like another tricycle — arguably a bicycle with training wheels. I understand the need for textbooks like this one, but having read one of them, I felt the second one was too lengthy.
If you are studying independently and trying to grasp epsilon-delta proofs, then by all means, this book is excellent. However, if you're trying to be time-efficient on top of that, there are probably better options available (although, they will inarguably be less entertaining).
March 16, 2025
Jay Cummings saved my life.
Displaying 1 - 13 of 13 reviews