# What books should I get to self study beyond Calculus for someone about to start undergrad mathematics?

I am struggling to pick out books when it comes to self studying math beyond Calculus.

My situation is as follows. I have taken all math courses at my school (up to Calc BC and AP Stats) and I have scored 5's on all of the exams. I am going to major in math at college next year but I really do enjoy learning in my free time and I am out of material. I have looked at some of the stuff on KhanAcademy, but the videos go really slowly and lack depth, so I would prefer something to read.

The only semi math book I have read for fun was Gödel, Escher, Bach which I greatly enjoyed. I am looking for anything, even if it reads similar to a textbook, that could further advance my mathematical knowledge in any way. Thank you.

EDIT 1: All answers do answer my question to some extent, so I will not be accepting an answer but rather using them all. Please continue to answer this question as I enjoy having more material to read.

• Linear algebra is the way to go. – Henricus V. Mar 26 '16 at 23:48
• Do you have any specific books on Linear Algebra? – schmidt73 Mar 26 '16 at 23:49
• Some abstract algebra is also good. This introductory book (not the graduate text) by Hungerford is quite good: barnesandnoble.com/p/abstract-algebra-thomas-w-hungerford/… . – M47145 Mar 26 '16 at 23:59
• I was in a similar situation. I just read a bunch of pop math books, for the most part. Like Dunham's Euler, Master of Us All and Journey through Genius. Read fun things; you'll have less time, in school, to goof around. – pjs36 Mar 27 '16 at 0:19
• @MathematicsStudent1122: challenging oneself can be good, but not if it leads to early discouragement. Sometimes it is better to take things one step at a time. – J W Mar 27 '16 at 9:15

Don't read anything too advanced, i.e. you should be able to understand everything, so you don't waste your time.

How to Prove It: A Structured Approach - Velleman

Numbers and Geometry - Stillwell

Calculus - Spivak

Linear Algebra Done Right - Axler

• Correct me if I'm wrong, but isn't Linear Algebra Done Right not meant to be an introductory book on linear algeba? Isn't it designed for someone having a second go at linear algebra? – ASKASK Mar 27 '16 at 7:33
• @ASKASK: The author does indeed say as much in the preface. That said, he does start from the beginning, so it might be approachable after reading Velleman or the equivalent. Still, starting with Poole, Lay or Strang would be gentler and introduce the reader to more applications than Axler does. – J W Mar 27 '16 at 13:41
• @schmidt73 First of all you need to read slowly and then understand all the theorems. When it comes to exercises, you should give serious attempt for every exercise. Exercises are very important, but there is no need to spend days on one problem set. Lastly, you don't need any "proof book" since Spivak explains everything. Before you start learning topics like topology, real analysis, or (linear) algebra, it would be good to read Velleman's book. – Hulkster Mar 28 '16 at 0:22
• "have never had a problem in a math book take more than 5 minutes" TBH I think that means you haven't been challenged enough in your courses so far. you should be prepared to spend much longer than 5 mins on a problem. otherwise you are not really learning from it. – Sasho Nikolov Mar 28 '16 at 5:18
• To second Yakk's comment: an hour is not an unreasonable amount of time to spend on an "exercise". I've not done anything from Spivak in particular, but in my first proof-based math course (having the benefit of an instructor, as Yak suggested), 30 minutes was on the low end of how long it would take to do a problem. – Eric Stucky Mar 28 '16 at 23:57

I'd recommend (elementary) number theory - save linear algebra for college.

Dover offers many inexpensive titles; you could buy several and read about the same topics from different points of view.

I particularly like Friedberg's offbeat Adventurers Guide to Number Theory. If you visit that book's page http://store.doverpublications.com/0486281337.html then Dover will show you other elementary number theory titles.

Two books turned me on to mathematics when I was in high school, and have been with me as a part of my professional life for years. The first, which I think I saw as a freshman, is Hugo Steinhaus's Mathematical Snapshots, reissued by Dover (http://store.doverpublications.com/0486409147.html). The second was senior year reading: Polya's Induction and Analogy in Mathematics. Free ebook (https://archive.org/details/Induction_And_Analogy_In_Mathematics_1_), paperback Princeton University Press from the MAA (http://www.maa.org/press/maa-reviews/mathematics-and-plausible-reasoning-volume-1-induction-and-analogy), hardbound 1954 edition from alibris (http://www.alibris.com/Mathematics-and-Plausible-Reasoning-Volume-1-Induction-and-Analogy-in-Mathematics-George-Polya/book/28098720?matches=22).

All the books are also available on Amazon, where you will find reviews.

• That does seem very interesting thank you! – schmidt73 Mar 27 '16 at 0:12
• +1 on Dover. They have many awesome titles at great prices. – sjl Mar 27 '16 at 7:59
• Elementary number theory by David M. Burton is a good choice. – Dark_Knight Mar 27 '16 at 8:54

I've been putting together a (large) list of math book recommendations on my blog. (You could start with the basics if you're interested). Two books I would really recommend requiring only calculus that haven't been mentioned yet are

• generatingfunctionology by Herbert Wilf, and

• Concrete Mathematics by Graham, Knuth and Patashnik.

I think the first three chapters (out of 5) of Wilf are very accessible. The fourth using some complex analysis, but I don't recall it being too difficult. Concrete Mathematics is more challenging, but completely awesome. Both books contains exercise solutions in the back.

If you wanted something that was just easy reading I thought

• Symmetry and the Monster by Ronan

about the classification theorem for finite simple groups was a nice "pop" math book and can be read in less than a week.

I recommend Gilbert Strang's recent book Differential Equations and Linear Algebra. It's a clear, intuitive introduction to two central subjects in undergraduate math and gives a big picture view of how calculus and linear algebra can work together.

(But I also don't necessarily recommend focusing on just one book. An intro to number theory book could be great. Another option is Spivak's Calculus. Or Hubbard and Hubbard's vector calculus book. Or The Feynman Lectures on Physics. Whatever you think is fun.)

Depends on your interests really a bit but usually after calculus, linear algebra is taught. But since you're in high school I doubt you have finished vector calculus, which is studying calculus with multivariables in higher euclidean dimensions and you get to see the concepts you've learned so far taken a step forward.

Also since you plan on majoring in math it would be a good idea to look into books that teach you how to read and write proofs since that is a necessary skill for any math major. Abstract Algebra and mathematical logic are subjects that you can start looking into. Also a different flavor of math called Discrete Mathematics is something you could look into.

As for books, depending on what you want to look at I'd suggest

1. For Vector Calculus: Vector Calculus by J. Marsden and Advanced Calculus by Callahan
2. For Linear Algebra: Poole's Linear Algebra to start off with, but once you know how to understand and write proofs well enough definitely look at Linear Algebra Done Right by Sheldon Axler
3. For learning how to do proofs: "Proofs and Fundamentals" by Ethan Bloch (really fun to work through)

4. For Discrete Math: Discrete Mathematics and Its Applications by Kenneth Rosen :This book will cover the basic logic, set theory, combinatorics, etc. so it's my best recommendation I'd say.

Linear Algebra Done Right and Advanced Calculus are the tougher ones out of the books I've mentioned, but you sound like you are definitely ready for the topics in the rest of them. Good Luck!

For an accessible introduction to abstract algebra, if that's the way you want to go, I would definitely recommend A Book of Abstract Algebra, by Charles Pinter. I picked it up after a semester of linear algebra and found it wonderfully intuitive and understandable, being written in a conversational style that engages the reader and guides him/her along to understanding the material. Additionally, it is as inexpensive as you could possibly bargain for. I also highly recommend the lecture videos on abstract algebra as taught by David Gross at Harvard College. They move at quite a rapid pace, so you will need a reference to accompany them, or you will more than likely get lost eventually, as I did. However, they are still quite accessible and benefit from a lucid and effective style of presentation that I have yet to see matched anywhere else.

I would also definitely recommend any number of books in a "pop math" style, in particular Fearless Symmetry, by Avner Ash and Robert Gross. It definitely leans toward the technical end of the pop math spectrum, so you should expect to put in a fair amount of effort to understand what's going on, but the reward is that it exposes an otherwise recondite subject of mathematics in a relatively accessible fashion, and should really open your eyes to the world of abstract mathematics.

As a math undergrad: learn elementary logic and do as many proofs as you can. If I were to recommend a single book that prepares you best for undergraduate math, I would without any doubt suggest

How to Prove It by Velleman.

Buy it now, read through all of it, do all the proofs. Logic and proofs is the crucial part of your curriculum, the entry point to all other topics and courses you will be taking.

In your situation, it sounds like there's no hurry, so consider not rushing to dive too deep into university mathematics, unless of course you particularly want to have a head start. After all, you will get to see standard topics such as linear algebra later anyway.

Try some more popular mathematics books to get some flavor of what's out there. One I personally like as an introduction to topological ideas is Euler's Gem: The Polyhedron Formula and the Birth of Topology by Richeson. Or to warm up to abstract mathematical concepts, there's the oldie but goodie Concepts of Modern Mathematics by Stewart. You might also want to read about the history of mathematics, for instance in Stillwell's Mathematics and Its History.

If your budget stretches far enough, you might enjoy browsing through the giant tome: The Princeton Companion to Mathematics, edited by Gowers et al. I see that there is now a companion volume: The Princeton Companion to Applied Mathematics, edited by Higham et al. Don't worry about the parts you don't understand, just dip in here and there or skim through to get the lay of the land. As you progress through your studies, you can revisit whatever seems most interesting or relevant.

Go through the excellent self study video courses at MIT's open courseware; http://ocw.mit.edu

Take the ones labeled SC

• 18.01SC Single variable calculus with David Jerison
• 18.02SC multi variable calculus with Dennis Aroux
• 18.06SC linear algebra with Gil Strang
• 18.03SC differential equations with Arthur Mattuck

All great teachers. Besides the videos MIT provides course notes and other materials. Buy the books and do problems for the best experience, but you'll learn a lot just by watching these exceptional lectures.

There is also a video course 18.04 on Probability and Statistics by Jeremy Orloff and Jonathan Bloom. That's a hugely important subject needed in many disciplines. I just left it out of the list above since I haven't personally watched that lecture series.

OCW also hosts a series of lectures from Herb Gross where he makes a quick concise review of topics in Calculus, Linear Algebra and Differential Equations. Search OCW for "Calculus Revisited"

Coursera.org and edx.org also have great resources

• Woo, there is a video lecture series with Strang? Thanks! – Tobia Tesan Mar 28 '16 at 9:25
• Actually, Dr. Strang made quite a few video lecture series at MIT OCW. He was a pioneer in providing video content for MIT OCW that laid the groundwork for today's MOOCs. – Alex Blakemore Mar 28 '16 at 14:02

I would recommend "A Transition to Advanced Mathematics" by Douglas Smith. This book begins with covering the basics from logic that every mathematics student should know and goes over different proof strategies. It then covers more basic concepts essential to many undergraduate math classes such as elementary set theory, relations, functions, partitions, equivalence classes, and some basic stuff on cardinality. It then ends with two chapters which give a small taste of analysis and abstract algebra.

According to this site I found on the internet (http://mathtuition88.com/2014/10/19/undergraduate-level-math-book-recommendations/), some books you can read are:

• Calculus by Spivak

• Complex Variables by Brown and Churchill

• Abstract Algebra by Rotman

These 3 books should be enough as a starter to expose you to both Analysis and Algebra before you start university.

Firstly, well done on taking the initiative of seeking out extra material to read. All of the books below are elementary and have no prequisites beyond high school level mathematics.

Personally I cannot recommend R.P Burn's books highly enough. His books are different from most because they are comprised soley of hundreds of problems. The goal of this approach is to guide the reader in discovering mathematics for themselves and creating a sense of ownership of the mathematics - what's known as constructivism. Check some of these out:

1. Burn, R. P. (1992). Numbers and functions: Steps into analysis. Cambridge: Cambridge University Press.
2. Burn, R. P. (1985). Groups, a path to geometry. Cambridge: Cambridge University Press.
3. Burn, R. P. (1982). A pathway into number theory. Cambridge: Cambridge University Press

The last book that I am suggesting is one that I wish I had read when I started university three years a go. It deals with category theory which will provide a great framework to view mathematics in. The exposition of this book is delightful and engaging.

1. Lawvere, F. W., & Schanuel, S. H. (1997). Conceptual mathematics: A first introduction to categories. Cambridge: Cambridge University Press

Here is a review of book 1 from the Mathematical Gazette and this website has a collection of reviews of book 4.

You can try your hand at some combinatorics which is a very good aid to learning math. The second book is a wonderful introduction to group, rings and fields. The third reference is an excellent Calculus text which, I think is beautifully written.

All these three can be read by a high school student but are used in Universities. Someone with less proof-writing experience might start with Spivak and move on to (b).

I'd read up on some introductory historical/mathsy texts which introduce more advanced topics as an easier read, e.g. The Calculus Gallery and such like. Other "popular" maths books are an easy read and can give you food for thought before reading the rigorous texts.

Newman's The World of Mathematics is an amazing survey of the whole field.

It's only four volumes long $(\sim2500\,\, \text{pages})$, so it can only hit the high points.

In case you want to learn Differential equations, then Differential Equations, 3rd Ed. by Shepley L. Ross is a good start. You would probabily be learning Probability in your undergrad. course. For that An Introduction To Probability Theory And Its Applications by William Feller is a good start.

Beside for some fun applications of mathematics you can always see Invitation to discrete mathematics by Jiří Matoušek.

You can also try ‘A Synopsis of Elementary Results in Pure and Applied Mathematics’ by G.S. Carr.

The Nuts and Bolts of Proof by Antonio Cupillari is a really good book also to learn about proofs. The proofs are straightforward and unambiguous.

Mathematical Proofs (3rd edition) by Chartrand is also very complete, and easy to follow.

Linear Algebra, A modern introduction (3rd edition), by David Poole is perhaps the best introductory book on linear algebra I have yet to encounter. It is almost impossible to not be able to follow the author's line of reasoning. Also, if your college/university happens to be using Beauregard's book for linear algebra, find another (better) book right away!!!. Mainly because that book leaves out a ton of essential details that would make your understanding of the subject much better.

"How to solve it" by George Polya is also a very good book, from what I heard.

Also, may I ask what program you are going into? That would make it much easier to suggest mathematical books that are relevant to your program of study. Because, needless to say, Mathematics is vast.

As someone who is at the other end of his studies, that is I have virtually finished them, I would suggest as a great entry level subject, number theory.

For ages, it has been perhaps the sole area of mathematics, where "amateurs" can produce elegant results, while many of it's most dificult-and open-problems can be understood by someone who has a basic knowledge of mathematics from school.

To that end, I would suggest "104 Number Theory Problems" which gradually, via various problems of increasing difficulty, introduces the subject in a beatiful manner, particularly adapted at self-study.

Courant and Robbins - What is Mathematics? is an excellent read at about the same level as Stillwell's Numbers and Geometry which I also recommend.

Nobody mentioned complex analysis (functions of one complex variable). It is a beautiful subject, and perhaps there are books out there accessible to you. A quick search gives some online notes like http://www.unc.edu/math/Faculty/met/complex.pdf but I don't know if they are good or appropriate for you.

Another piece of advice is to try to contact some mathematics professors at the university that you will be attending to ask for their advice on the matter. This way you will have some idea of what some faculty in your university are interested in, and will get a sense of how open they are to talking to undergraduates.

As a math major in my senior year, I would say that you'll have plenty of time to study subjects such as Abstract Algebra and Analysis.

If I could go back to my last year in high school, I would invest more time into gaining and shaping mathematical reasoning than dutifully study certain specific subjects.

I would go through these books:

How to Prove it by Velleman: Imagine yourself as a soccer player. This book serves like those drills on short passes and ball control. They are the basics, and when you become good you'll do them automatically, but you will only do them automatically if you had first spent several hours practicing the basics.

The art and craft of problem solving by Paul Zeitz. Challenging. The former book focused on the basic logical steps that constitute a mathematical proof. This one focuses on the ideas, and on being creative. It covers the basics of Number Theory, Geometry and Combinatorics and suggests wonderful references that I suggest you also look up.

Discrete Mathematics by Laszlo Lovász. This one is fun and easy to read. You'll get introduced to a myriad of topics in discrete mathematics which are fun and you won't usually study them at your typical courses in college.

Finally, I would like to recommend Topics in the Theory of Numbers by Paul Erdös. This book is rather challenging, but in my opinion gives you a feeling and intuition for Number Theory unlike any other, with beautiful and ingenious proofs and challenging exercises.

Focuse on cramming the first, and enjoy the others. If you spend a considerable time on these books before going to college you'll be ready to start studying the traditional subjects from more advanced textbooks. You will have the required maturity for Baby Rudin and you'll sail through Axler's, then I hope you'll be fine to tackle Spivak's Calculus on Manifolds and enjoy one of the best experiences as a math major.

In addition to all the books mentioned I would like to add two other

1. Street Fighting Mathematics, by Sanjoy Mahajan. This book presents a rank of approximation techniques to attack real life problems in a effective way.
2. Algebra, by Michael Artin. Presents linear, bilinear and abstract algebra in a great, self contained and well presented exposition

I would like to recommend the following books in the order given here for self study. Definitely, they will prepare you with good foudation: 1. Naive set theory, Halmos 2. Lectures in abstract algebra, vol I,II, Jacobson 3. Mathematical Analysis, Apostol 4. General topology, Kelley 5. Principles of Mathematical Analysis, Rudin 6. Real and complex Analysis, Rudin 7. Functional Analysis, Rudin 8. Smooth manifolds, Sinha

I highly recommend "Transition to Higher Mathematics" by Dumas and McCarthy. http://www.amazon.com/Transition-Higher-Mathematics-Structure-Advanced/dp/007353353X Here is a link to a pdf of it as well http://www.math.wustl.edu/~mccarthy/SandP2.pdf

This is the textbook for a math course at Columbia University called Introduction to Higher Mathematics. The course description is "Introduction to understanding and writing mathematical proofs. Emphasis on precise thinking and the presentation of mathematical results." The book will teach you to write proofs while giving you a survey of subjects in higher math, such as abstract algebra and analysis.

The goal of the course is to prepare Calculus students for higher level, proof-oriented classes such as Modern Algebra and Modern Analysis

I recommend a survey of math and its language, which is offered by the The Princeton Companion to Mathematics.

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