## 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.

Thanks a lot for books list and solutions of Apostle Calculus.

It will be great if you can also suggest some probability introductory book(s).

I am looking for text similar to similar to Apostle Calculus or Hoffman Linear Algebra, so that I can revisit it later time to time as and when I need to refresh my knowledge or clear some misconception without worrying about quality of the content.

Regarding Courant – Fritz Calculus I, you didn’t mention but there is a solution manual by Albert Banks. All the exercises plus more are included. It’s online here: https://archive.org/stream/ProblemsInCalculusAndAnalysisAlbertBlank/Blank–Problems-in-Calculus-and-Analysis#page/n133/mode/2up

The solutions are often brief but not as brief as Apostol. The problems are difficult though; the chapter on techniques of calculus has problems relating to differential equations. The brevity of the solutions makes me struggle, when the material requires some prerequisites. So the presentation isn’t as linear as Apostol’s (in the preface it is encouraged for the math student to skip around sections if so inclined), nor are the solutions as understandable as yours are. I will say though, the (Courant) problems seem well thought out- they have exposed weakness’s in my understanding (for example, lots of examples of applying Leibnitz’s Nth derivative formula, rather then just proving the theorem by induction, say).

Great find! Thanks. I wasn’t aware that the old solutions were still around and available online. I only knew about the volume 1 solutions getting dropped in the new Springer edition. I’ll update the post in just a minute to add this information in.