Book Reviews
Reading has always been one of my passions, and it has carried over nicely into the math world. There are so many wonderful math books available and I love to collect them. I like to spend hours scouting the library, checking out books sales, and trawling online recommendations. In my free time, I occasionally like to pick an interesting sounding book and just get lost in it for awhile to absorb some of its ideas. All of this searching has made me more mathematically mature, but it has also made me aware of just how difficult it can be to navigate the mathematically literature and find the right book. This problem is especially difficult for self learners, since they do not have the benefit of a professor or institution to guide them. Furthermore, the factors that make a book good for self study are different from those that make a book good to use with a course. With this page I want to make the process of finding and choosing a math book easier by providing reviews and recommendations. I would like to place extra focus on books that may not be widely known or that are especially suited to self study.
Of course, there are so many books out there and so much to be said about each one of them. I have selected some of my favorites to start off with and will be adding more over time. I will also be adding links to more in-depth reviews of some of the books over time. If you have feedback or recommendations, please contact me on twitter @MathOverload. Thank you and enjoy.
Use the links in the table of contents below to more easily navigate the page.
Pop Math and Math History
These books are light and fun reads about math. They focus on big ideas, history, and how math is used in society. Popular math books are generally written for a non-technical audience, and require little to no prior knowledge.
How Not to Be Wrong: The Power of Mathematical Thinking by Jordan Ellenberg: I enjoyed reading this book. The writing style is playful and there are many stories/real world connections weaved into the content. The book focuses on thinking mathematically in order to avoid some of the mental pitfalls that it is all too easy to fall into.
The Joy of X: A Guided Tour of Math, from One to Infinity by Steven Strogatz: This book starts from the notion of counting and builds up from there. Strogatz has a way of writing that is both entertaining and informative. Despite being a purposefully accessible book, The Joy of X gets into some wonderfully interesting and applicable mathematics such as Markov Chains (specifically how the PageRank algorithm works). Honestly, my only complaint about this book was that it build up my excitement so high and then had the audacity to end.
Unknown Quantity: A Real and Imaginary History of Algebra by John Derbyshire: I love this book. It is a lively history of specifically algebra (including linear and abstract algebra). The author alternates between chapters which intuitively explain the topics under discussion and chapters which focus on the historical events and figures involved.
Prime Obsession: Bernhard Riemann and the greatest Unsolved Problem in Mathematics by John Derbyshire: Another book by Derbyshire, this one reads like a mystery novel. Each chapter reveals a new piece of the puzzle, building the suspense until the full impact of the Riemann Hypothesis is revealed towards the end of the book.
Calculus
A Stroll Through Calculus: A Guide for the Merely Curious by Anthony Barcellos: Calculus textbooks are notorious for being large, intimidating, and unwieldy tomes. This neat little book is not one of those tomes. At about 200 pages, this book is designed to give students (especially those who may not be particularly comfortable with or enthused about math) an intuitive understanding of the main ideas of calculus. Unlike most calculus textbooks, this book begins with integration before presenting derivatives later on. Topics are built up using concrete, often visual examples. Most of the topics/ideas which would appear in a first or second calculus course are presented here. I recommend this book to people who wish to dip their toe into calculus, or those who are looking for a gentle primer before a more intense calculus course/book. People who are already strong in mathematics and are looking for a more rigorous introduction may want to just move on. The major weakness of this book (as a standalone textbook) is that it lacks exercises. The prerequisite for this book is high school algebra and trigonometry.
Calculus: Early Transcendentals by Jon Rogawski: Like most introductory calculus textbooks, this one is a beast of a book. Coming in at a little over a thousand pages, this book has all of the normal material covered in a standard single-variable, multi-variable, vector calculus sequence. This book has good explanations and uses mature mathematical language which will help prepare the reader for other courses. The book also has a good focus on applications. The formatting it excellent and helps make the book more readable. There is a huge number of exercises and solutions to about half of them. I used the exercises in this book to help study for the Math GRE. There is a large and comprehensive solution manual available to go along with this book, which is essential for those self learning math.
Calculus: Early Transcendentals by Briggs, Cochran, and Gillette: This book is almost exactly like the book by Rogawski, but I do like its coverage of vector calculus a bit better. While the book has less of a focus on applications, the language is simpler in some areas and the exercises are pretty good. The solution manual is not as good as Rogawski’s.
Calculus by Micheal Spivak: This is the book that is often recommended to give people a rigorous introduction to calculus. The book is written in the style of a higher level math textbook, with an emphasis on proof. It is hard for me to give a proper review of how this book is as a first exposure, since by the time I first saw it I had already learned the material elsewhere. However, this book is so commonly recommended that I couldn’t not mention it.
Proofs
These books are intended to introduce the reader to the language of mathematics. They focus on the techniques of proving statements and are used as a form of transition from lower mathematics to higher mathematics. For recommendations for a self study curriculum, look here.
Book of Proof by Richard Hammack (available free here). This is a commonly recommended textbook and for good reason. The writing is clear and involves a lot of worked examples from various areas of mathematics. Every section of the book has a set of exercises with it and the solutions to the odd-numbered exercises are in the back of the book. What is excellent about these solutions is that they include full proofs, an invaluable resource for self-learners. The book has very little in the way of unnecessary fluff in it. The author notes that chapter 3 can be skipped, but if you are already reading the rest of the book you might as well read chapter 3, too. This book follows the typical approach to proofs where the student is asked to forget, temporarily, most of their previous knowledge in mathematics in order to build a new foundation from scratch. The upside of this approach is that the book has very few prerequisites. A student should have a solid foundation in basic algebra.
An Introduction to Proofs and the Mathematical Vernacular by Martin V. Day (available free here). This book is shorter than the Book of Proof and it takes a different stylistic approach. While many proofs courses ask the reader to set aside their previous knowledge, this book specifically does the opposite. The book assumes a solid knowledge of calculus and some linear algebra, even going so far as to introduce some less commonly taught concepts results in calculus as a training ground for working with proofs. Two downsides with this book are the pacing and the minimal amount of problems/solutions. The book moves quickly from one topic to the next, and while I like how the book presents things it might work better as a sort of second exposure to proofs. Although the book has a number of worked examples and some problems at the end of sections, it has no solutions. Solutions are absolutely essential when learning proofs since they may be the only form of feedback available to the self-learner.
Mathematical Reasoning, Writing, and Proof by Ted Sundstrom (available free here). This book is very long and detailed, holding the reader’s hand through every step of the process. It is a great reference textbook when reading another textbook, or a source for problems since it has solutions to the odd-numbered exercises at the end of the book. The writing is extremely clear and I feel the book has good pacing. It follows a similar progression of topics as the Book of Proof does, with only a few minor differences.
Introduction to Mathematical Thinking by Keith Devlin: This book is more of short booklet than a full fledged textbook. However, it has some great pearls of wisdom and provides a nice introduction to logic and proofs. I really enjoyed reading this book and recommend it as a supplement to any other proofs course or textbook.
Number Theory
An Introduction to Number Theory with Cryptography by James S Kraft and Lawrence Washington. This is a great introductory text for number theory, especially if you also have some interest in programming. Of the seventeen chapters there are two chapters devoted to cryptographic applications of the material being covered. Every chapter also ends with projects and ‘computer explorations’ which are specifically intended to be solved using programming. The writing is clear and easy to read. The reader should have some basic knowledge in proof techniques. There are no other major prerequisites for the bulk this book.
Algebra
A Book of Abstract Algebra by Charles Pinter (review): This was actually my first “real” math book, and what inspired me to continue learning math. The book is extremely accessible (I did not even have any formal exposure to proofs when I read this book) and strikes a nice balance between theory and applications. The body of each chapter presents the theory and then exercise sets following the chapter introduce applications such as coding theory, automata theory, and a little bit of game theory. The writing itself is clear and the proofs are easy to follow, making this a great book for hobbyists or self learners. The book has no major prerequisites beyond basic algebra, other than a basic familiarity with set notation. Highly recommended.
Elements of Modern Algebra by Linda Gilbert and Jimmie Gilbert: This is a bit more of a standard style textbook. It covers all the basic material on groups, rings, and fields and does so well but without any extra frills. This book does not have the same kind of motivating exposition that Pinter’s book does, but it does go into the material in a bit more depth and with a bit more rigor. The book has good proofs and straightforward writing. Something I really like about this book is that includes two introductory chapters, one covering material you might see in a proofs class and one covering basic number theory. These chapters do a lot to make the book feel more self contained. The book is easily accessible for undergraduates. To make the most out of it you will want to have some good exposure to basic proof techniques.
Algebra: Chapter 0 by Paulo Aluffi: This book is great for what it is, and that is a second introduction to abstract algebra and a first introduction to category theory. While you could, technically, learn abstract algebra from this book, it would be much more effective to get a first exposure to algebra from another book/course before moving on to this book. This book introduces categories early on and uses them as a foundation for the rest of the book. This makes clear the universal properties of the objects under consideration clear and really makes clear, on a deep level, why things are structured the way they are. My ability to see the connections between different areas of mathematics was certainly improved by this book. While I highly recommend this book (especially to students planning to go to graduate school for mathematics) be aware that it is challenging and cannot be breezed through.
Linear Algebra
Elementary Linear Algebra by Howard Anton: This book is a great book for a first course in linear algebra. One of the great tragedies of math education (in the US) is that linear algebra is not introduced earlier in the curriculum. On a practical level, so many applications in STEM are executed using linear algebra. On a theoretical level, proofs in many areas of math often involve reducing something to a problem in linear algebra. This book is accessible and does not require the reader to know any calculus for most of the book. The book begins with a focus on vectors, matrices, and how to compute with them. The emphasis is on finding solutions to linear systems. In the latter half, the book transitions to focusing on proving things about vector spaces and working with inner product spaces. There is some discussion of applications to topics like chemistry, economics, optimization, networks, Markov chains, and many others. Overall, I recommend this as a solid first linear algebra textbook.
Linear Algebra Done Right by Sheldon Axler: I will come right out and say that I love this book. I love the style, the formatting and the content. This book is meant as a second introduction to linear algebra, although it is technically self-contained on the topic. The book requires a good knowledge of proofs and some basic familiarity with abstract algebra (specifically fields). This book places the focus on linear maps between vector spaces, and especially on linear operators. When presented this way, with linear maps as the central concept, many of the familiar concepts such as invertability, eigenspaces, and subspaces seem like natural structures rather than special properties of matrices. The book doesn’t describe any direct applications, per se, but it was easy to see where the theory of linear operators applies to other areas of mathematics such as differential equations and Fourier analysis. The tone and pace of the book flows nicely. The first time I read it, I went through it almost like a novel before going back and reading it properly. The book has good exercises and is very suitable for self study.
Differential Equations
A First Course in Differential Equations with Modeling Applications by Dennis Zill: I used this book in my first course on differential equations. It is a good textbook, with nice coverage and interesting exercises. There is a student solution manual available which is a benefit for those who want to self study differential equations. Technically, only two semesters of calculus are required to be able to use this book, but I would recommend waiting until you have a solid grasp of linear algebra. Linear algebra makes the theory behind differential equations make sense. Topics like superposition of solutions, eigenvalues, linear operators, and annihilators will help to deepen your understanding. There is a short discussion of numerical methods i nthe final chapter which could provide a nice bridge into a numerical analysis course/book.
Differential Equations with Boundary Value Problems by Dennis Zill and Micheal Cullen: This book is identical to the book above (also by Zill) but has six additional chapters covering partial differential equations. If you know you are going to also study PDEs then it would be worth it to get this book, or if not you can save a bit of money by getting the slimmer book. Don’t buy both. All of the compliments about the other book carry over for this one as well. The same advice goes about the important of having a solid grasp of linear algebra. In my view, you should be very familiar with inner product spaces before talking about things like Fourier series.
Ordinary Differential Equations: An Elementary Textbook for Students of Mathematics, Engineering, and the Sciences by Morris Tenenbaum and Harry Pollard: This is an older but classic textbook. It was written in 1963, but the treatment of the material still holds up. Dover publishes it now, which means that it is cheaply available. At roughly 800 pages, this book is long but that space is well used with coverage of nonlinear differential equations, perturbation techniques, and more applications than most ODE textbooks have. The exposition is fantastic, although the formatting is nothing to be impressed by. The exercises are comprehensive and many have good hints, however the book does not have solutions listed which could be an obstacle for the self-learner.
Real Analysis
Real Analysis: A Long-Form Mathematics Textbook by Jay Cummings: If you are looking for a gentle first introduction to real analysis, this would be the book I would recommend for you. The book is written in a clear, conversational tone with ample amounts of motivation and illustrations. There are even some decent jokes throughout the book. Almost all of the proofs are preceded by ‘scratch work’ where the informal justification for why something is true is laid out before being converted into a formal proof. The book cover the material you would expect in most undergraduate real analysis courses, going up to just touch on the idea of a measure. The book has numerous exercises, however it has no solutions. The lack of solutions is a bit of a bummer for people trying to self study real analysis, but the clarity of the writing and examples still makes this a great choice.
Analysis with an Introduction to Proof by Steven Lay: This is a bit of an older book, but well worth the hunt. This book was gifted to me by a friendly professor early on in my math journey and it helped me out immensely. The focus of the book is on real analysis, but with almost 90 pages dedicated to introductory proofs material this book could work as a first exposure to upper-level mathematics. All of the standard undergraduate topics are covered, although the book does not get into any measure theory. Every section has practice problems with answers at the end of the chapter. There are also many high quality exercises. The exercises do not have solutions present, but many of them do have hints in the back of the book. This is a great book for self-study. I would even recommend this to someone who enjoys math as a hobby, since some of the exercises are fun to think about. I won’t spoil it, but there was one exercise involving marbles which stumped me for a while but was so satisfying to finally solve. As an added bonus, the hardcover is quite aesthetically pleasing on my shelf.
Principles of Mathematical Analysis by Walter Rudin: This book is recommended all of the time by people online and offline. It is affectionately known as “Baby Rudin” as opposed to “Papa Rudin” (Real and Complex Analysis) and “Grandpa Rudin” (Functional Analysis). In terms of praise, I’m not sure I can put it any better than this review here. The book presents the material in a straightforward manner at a high level. If you take the time to go through this book, you will be prepared to move on to more advanced material. The first eight chapters cover the standard material. The last three chapters cover multi-variable analysis, differential forms, and measure theory. Although I have not personally used this book for self study, the number of people who do let’s me assume that it is a good (although perhaps challenging) book for self-study. Every chapter has exercises but no solutions. There are solutions available online, however.
Advanced Calculus of Several Variables by C. H. Edwards, Jr: This book could arguably be placed under the calculus category since it could theoretically be used as a first exposure to multi-variable and vector calculus, but given its reliance on linear algebra and proofs I feel it is better thought of as a second, in-depth exposure to the topic. The book proceeds at a nice pace and has a good amount of exercises. The book makes use of manifolds, differential forms, and provides some introductory material on the calculus of variations. This book is published by Dover so it is cheaply available.
Measure, Integration & Real Analysis by Sheldon Axler: (Available for free here) This is a relatively recent book by the same author of Linear Algebra Done Right. This book would be best used as a second exposure to real analysis and first exposure to measure theory. The writing is clear and the proofs are easily followed. The formatting is beautiful just like his other book. The last two chapters cover Fourier analysis and probability theory using measure theory. Best of all, the book is available for free. Plus there is a free supplement with review material from basic real analysis that could be an effective prerequisite for the book. This book is highly recommended if you are looking for a good introduction to measure theory that is suitable for self study.
Complex Analysis
Complex Variables and Applications by James Brown and Ruel Churchill: This book is commonly used as a textbook in many undergraduate complex analysis. I used it in my course, and while the material is presented in a straightforward and complete manner, I turned to another book (below) to help me learn. However, going through this book carefully I really feel like I gained a rigorous understanding of how things work in complex analysis. The standard topics are covered very well, and I especially enjoyed the sections on the Cauchy-Riemann equations, the Cauchy integral formula, and the applications of residues. Overall, if you get this book you will be well served by it but it requires work to get the most out of it. To make the most of this book, you should have completed multi-variable calculus and some basic familiarity with proofs.
Fundamentals of Complex Analysis for Mathematics, Science, and Engineering by Saff and Snider: Single variable complex analysis takes place in 4 dimensions so it is difficult to visualize and build an intuition around. This book proved to be a lot better at providing intuition than Churchill’s book was so I used this text as a supplement during my complex analysis course. I really, really liked how the book presents the Cauchy integral formula in two different ways, once using ‘deformation of contours’ approach (which will look familiar to anyone who has studied some algebraic topology) and once with a vector analysis approach. There are a number of helpful illustrations. There are not a lot of exercises in the book but there are lists of suggested further reading in every chapter which makes this book a good starting point for self study.
A First Course in Complex Analysis with Applications by Dennis Zill: If your motivation for studying complex analysis is less about the mathematics and more about the applications (to things like physics and electrical engineering) then this may be the book for you. The book is intentionally written specifically for people without a ‘strong’ mathematics background. To use this book you only need to have completed the calculus sequence (and maybe seen a little bit of differential equations and linear algebra). The book still goes through proofs but the emphasis is on the intuition, applications, and computations. Every chapter has exercises and an applications section. The exercises also have some dedicated computer problems, which will help you learn how to use the material for actual applications involving computers. I highly recommend this book, especially for self-study.
Numerical Analysis
Numerical Analysis by Timothy Sauer: I used this book in my numerical analysis class and felt it was a solid book on the subject. The formatting is nice and modern. There are ample exercises. The text was clear and well motivated. The proofs and examples were generally easy to follow, with concentration. We covered the first five chapters out of twelve in the course and then I continued to study on my own after the course ended. The book outlines three potential concentrations that the book can be used for by selecting different combinations of chapter. The first three chapters are considered core material, and then the read can choose between “traditional differential equations”, “discrete mathematics, orthogonality, and compression”, and “financial engineering concentration”. Or the reader could choose my favorite option: all of the above. Overall, learning this subject felt empowering since I was learning how to apply math to efficiently solve real, tangible, complicated problems. I would recommend this book.
Topology
Topology by James Munkres: This book is considered by many to be a sort of standard introduction to the topic of topology, and I see no reason to disagree with that assessment. The writing is clear and generally accessible to anyone with some good experience with proofs. Concepts are explained using a variety of examples and illustrations. There are a large number of exercises which range in difficulty from routine to challenging. The book starts out with basic point-set topology but later gets into algebraic topology with concepts like the fundamental group of a space and covering spaces. Some exposure to group theory is recommended for those later sections. I used this book to help self study while I was learning about algebraic topology and it helped me tremendously.
Topology: A Categorical Approach by Bradley, Bryson, and Terilla: (Available for free here) The main idea of this book is to provide a categorical approach to topology, perhaps as a second exposure to topology. It reminds me, in spirit, of Algebra: Chapter 0 by Aluffi (See the algebra section of this page), so I imagine that many of the things which can be said about that book can also be said about this book. Since this book just came out literally the same day as I am writing this, I have not read the book yet. However, it is on the top of my list to read when my copy arrives this weekend. I will update this review as I read it. Based on the preface and a quick perusal, the book should be accessible to some one with a firm grasp of linear algebra, abstract algebra, and some strong mathematical maturity. It does not seem to require any previous knowledge of category theory or topology, although I’m sure having at least some exposure to both would be useful.