Undergraduate Mathematics Bibliography

questions are quite popular on this website, which is, at least for me, one of the best places to get useful and insightful suggestions because it gathers a great number of experts in different areas of mathematics.

In my opinion, it would be very beneficial for the organization of the site (and for the many undergraduate students searching for guidance when facing a new course or looking for a good learning roadmap -- like myself) to make one thread that collects a big "Mathematics Stack Exchange Undergraduate Mathematics Bibliography" divided by categories just like the nice one proposed here, but surely more comprehensive (given the much larger number of contributors).

This thread should collect sparse material already available on the website but hard to find among the numerous questions and also new inputs. Ideally each entry should be briefly commented with matter-of-fact remarks.

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A complete guide to the best math books

Last updated: April 26, 2016.

I’ve divided the list into three categories:

You can spend forever reading book recommendations, but won’t learn any math doing that. No amount of research can tell you which is the perfect book, since there is no “perfect” book. Any book on this list will cover the most important material in its subject (and usually a lot more besides), and will leave you prepared to move forward. Spend some time reading these or other recommendations, but after an hour of research make a decision and get to work!

Basics: Calculus and Linear Algebra

Intended as a starting point for aspiring math students who have seen at least the basics of Calculus already. This includes calculus books, linear algebra books, and a section on books writing proof writing. If you want to be a serious math student, or are a particularly motivated self-learner, this is the place to start. Basics.

Core Subjects

Intended to list books covering the general topics one sees as an undergraduate math major through the first two years of graduate coursework in a typical program in the US. (This not well-defined, but hopefully the meaning is clear.) Anyone who has finished a subset of the books in the Basics section is prepared to start reading books on this list. Core Subjects.

Advanced Topics

Specialized topics. Areas not usually covered in undergraduate courses, but often show up as “topics” classes in grad school. This list is unbalanced in that it reflects my interests, and neglects areas I’m not interested in. I’ll try to recruit some friends to contribute to specialized areas I don’t know enough about to give valuable advice. Advanced Topics.

Post Script.

Some warnings about these recommendations.

  • This list is intended for people interested in pure mathematics. On a few occasions I mention math books that might be appropriate for engineers, people in applied math, or physics students, but you should know I’m none of those things. I’m set to start as a first year graduate student in a pure math department, and this list reflects that.
  • The books on this list are uniformly challenging. It is not meant as some rite of passage or test of fortitude, but stems from a belief that each book you read should stretch your mathematical ability. The idea is to read books and work on math problems that are just a bit too hard for you right now. (Don’t overdo it! If a book is really killing you, read something else!) This list is intended for people who are motivated to learn math: either on their way toward grad school, or self-learners who simply enjoy a challenge. (In some sense I’m both of those things. As I said, I am soon to be a first-year math graduate student, but I also spent a decade working after getting my undergraduate degree in finance and decided to go back to school to learn math a few years ago.)
  • For most books, there has been a proliferation of editions available in recent years. In general I’ve tried to link to the version I own or to what seems like the canonical version. I’ll do my best to keep the links fresh and to the newest edition. I’ve linked to physical copies of the books (I am biased toward actual paper copies of books since I don’t take my computer with me to study), and I’ve also tried to link to free (legal) copies of the books when available. Keeping all of these links up-to-date will be a bit of a challenge so please email me if you find a broken link or if a newer edition has come out.

Anyone who has spent time trying to find a list of the best math books to supplement classes or for self-studying has certainly run across the excellent Chicago Undergraduate Mathematics Bibliography (now with a new Github repository here). I owe the creators of that list a debt in helping to guide my studies the past four years. The debt this list owes to that one will be obvious to anyone reading both. Since that math bibliography is almost 20 years old now, I thought enough it was time for something new. Of course, in many cases the best math books of 20 years ago are still the best math books today.

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