Mathematics Books

  • #1

I am new to the forums but I will be specific. I am deeply in love with mathematics, even after having survived through 5 years of HS(implying the quality of mathematics education in HS). I have thus decided to rework all my current knowledge, and definitely surpass it. For background info, I am very skilled at maths, I self-study as a hobby in my free-time (when not at school). I know a good amount of calculus, trigonometry, algebra, geometry, proofs and problem solving although I assume that I know none. Basically I am looking for a list/series of books that have changed the way you think about maths, starting from ground-up(Covering, hopefully, every aspect of math; Algebra-->Geometry-->Trig-->Precal-->Cal/linear algebra and so on, up to, and not exceeding, graduate level). Thank you very much.


P.S. I have heard about Calculus by Michael Spivak and was looking into it. Difficulty and price are not variables to be considered, I can easily cover for both.
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Answers and Replies

  • #2
Gelfand: Algebra, Method of Co-ordinates, Trigonometry. His series were the math books used in the advanced high schools of Russia, and they stand as the rigorous books on the subject. They will probably be review for someone at your level, but there are enough challenging problems and topics not usually covered to make them a learning experience. Unfortunately I don't know any good precalc books, I'm pretty sure they don't exist. Besides that, you can check out other high school books like Niven's books in New Math Library (google it), Courant's What is Mathematics, Coexters Geometry Revisited, etc.

Once you have a firm knowledge of basic math, Spivak will teach you calculus the right way. The rest you can easily find on these forums to prepare for grad school.
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  • #3
Thank you very much for your reply. As I mentioned, I'm willing to dedicate any amount of time on a book. I would much rather understand every aspect of math than to plug and chug using memorization. As they say, "Give a man a fish feed him for a day. Teach a man to fish, feed him for a lifetime." Haha. I am looking forward to more suggestions.
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  • #4
The art of problem solving Volume 2 gives a nice overview of high school level math with a lot of challenging problems.
  • #5
The math book I used (and still using) that introduced me to proofs is "Introduction to Calculus and Analysis" by Richard Courant and John Fritz. The book gave me a better idea of what math really is. I highly recommend it.
  • #6
I have heard of that one as well, along with Spivak's Calculus and Apostol's two volumes. I've read many posts, and after reflection I have decided to purchase all four books:biggrin:. Can't wait to get those books, along with all the others I just ordered!
  • #7
Not a very smart move. It starts well.
  • #8
What do you mean? I don't think I'm going very wrong...I did mention money was of no issue.
  • #9
What do you mean? I don't think I'm going very wrong...I did mention money was of no issue.

Buying all 4 was a bad move because it might be difficult to read them all at the same time and/or bicker over which one is the best.

I suggest you stick with Spivak or Apostol. Courant is good, but Spivak made it obsolete. Courant has this magic quality to his books, but its not something I would recommend for learning the subject. Its pretty outdated and what not.

Its stupid making ambitious plans and buying all these books. They will only end up on the shelf. Its better to buy 1-2 books at a time and work through them. I hope this will not happen with you.
  • #10
Well, I'm getting Courant because I love the way he writes. Spivak will be the one I learn from, and I will go back through Apostol's two volumes for review. Of course I'm not going to be reading all of them at the same time, but Spivak-->Apostol seems good to me.

  • #11
Any other suggestions for must-read mathematics books? For one, I propose "What is Mathematics", Courant. Mathwonk's thread is also good, but the books presented there are mostly textbook/rigorous learning, as I am looking for the more "novel-side of mathematics", so to speak, you know, for bed:tongue2:
  • #12
When I see people buy two somewhat similar books I always see them read one, get tired, want to learn something else, and move on.

Courant and John is an excellent book to learn from. But someone said it right: Spivak made it obsolete. Although I do prefer that over Spivak or Apostol for reference.

I find that mature mathematicians usually prefer Apostol or Courant over Spivak, and I personally find Spivak a very irritating book to keep. His "Calculus" is too long winded. Brevity of "Calculus on Manifolds" ... well, just legendary isn't it.

What I usually recommend to undergrads who wants challenge, is to get first volume of Apostol and Spivak's "Calculus on Manifolds." Those two books are enough. But in the end, I prefer Courant and John.

You probably want a linear algebra book to go with that. Judging from your book selections so far you cannot go wrong with Hoffman and Kunze. I REFUSE to teach linear algebra from any other book because that's the only way it should be learned (Halmos and Friedberg, Insel and Spencer are OK too.)

If you want a bedtime reading you probably want "The Princeton Companion to Mathematics" edited by Gowers. I think it is one book that everyone should have.

Other than that something like perhaps Krantz's "Mathematical Apocrypha" interest you. Probably there are more books than this, ask if you want more advice.
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  • #13
Thank you very Much Unknot. As far as books, right now I just ordered most of Gelfand's books, I really like his way of teaching. I also ordered "A Real and Imaginary History of Algebra", "What is Mathematics", as well as a few others. I would really like a good book about Pi, as well as a dictionary to mathematics. What I'm looking for isa book that I can look up a symbol and know what it means, for example: A|B; if I wanted to know what '|' meant, then I would look it up...

  • #14
If you are really willing to learn higher-level mathematics I say you probably won't enjoy lot of books directed at general audience. If you learn properly those will feel too watered-down and not interesting to you. But I suppose at your level you can still enjoy some of those.

Book about Pi? What do you want to know about it?

Dictionary to mathematics - well, I think there are some but to be frank these are never too useful. There's a reason why people who study mathematics use textbooks as references. I have never seen a book with just mathematical notations in them - it is something that you have to learn one by one, not just look up.
  • #15
So I assume getting one of those "Dictionary to Mathematics" is probably a waste (they're mostly definitions anyway). Not sure about Pi, I think I may have enough books for the moment. For the higher level math, of course I don't want a book directed at the general audience, as long as the book is good(good being clear, rigorousness is encouraged) and does not go astray. What do you recommend past linear algebra? (Assuming I went through Gelfand-->Spivak's Calculus-->Hoffman Linear Algebra)

Many thanks,

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  • #16
Well, I do have an e-mail I sent to a student who wanted to self-study - so I just suggested a sequence. Let me rewrite that here. If I had to start over, I would:

0) Learn whatever high school math
1) Learn calculus
2) Learn linear algebra (with Hoffman you will get some taste of abstract algebra too)
3) Learn multivariable calculus
4) Learn classical analysis (Rudin PMA is a good choice, but if you want to self study you probably want to look at other options. What really matters is how strong you are mathematically)
5) Learn abstract algebra (lot of choices again. Dummit and Hungerford come to mind.)
6) Learn complex analysis (Ahlfors is the classic - now I don't know many other books, but I hear Stein's new book is good)
7) Learn real analysis (people would normally mention Royden, Folland, etc...)

These are the basics. I think everyone should know it and that will take you to the graduate level. A question I got was "where do I pick up number theory? where do I do topology?" Well, I really think to get good introductions for other areas you need to know those seven things very, very well. Sure you can learn differential geometry after doing calculus. But I don't believe in that.

You should always keep in mind that blindly following someone's program might really frustrate you, especially when you don't have lectures to go with your readings. So I would wait and see what other posters say.

[EDIT: I must mention that even getting to multivariable calculus is a challenge for a lot of people. If you are not THAT interested in math, just going over a classical analysis book is a real achievement. It rarely happens that the same person comes to you after doing those suggested to see what he/she needs to after that.]
  • #17
I'm not very worried about that. I had started a calculus book recently, and I love mathematics with a burning passion:shy:. That's why I decided to refine my skills by going a little back, and then taking a more rigorous approach, thus making me a much better mathematician, so-to-speak.
  • #18
Although I agree that the aforementioned books are decent books (Spivak is enjoyable) to read and to ultimately possess, it might be best to not have a naive approach of getting 'totally' armed with books. Yes, money might not be a factor in your case, but just stick with a couple of texts for now and when you're finished with them, you will automatically realize what you actually want to progress in.
  • #19
Oh, I definitely agree. I am only getting a few, under 10 for this year. But my ultimate goal is to have a list of books on hand, so that I don't have to get in a wild scurry for books when I get an urge to get more books...Reading is my passion, why not. :wink:

  • #20
To make it clear, I only buy what I'm going to read, cover to back. I don't buy all the books named here either, I consider them suggestions, and when I see a trend of a specific book, I usually buy it. I'm not blind, nor dimwitted :wink:


  • #21
One has to wonder why do you want to learn all that math when you have all that money. :rofl:

You're going to have to make a choice I believe. It's either Spivak or Apostol/Courant. Both Apostol and Courant start with the theory of integration, with Apostol's approach being better because it is axiomatic. Courant gives good insight and is quite verbose, Apostol is straightforward. Apostol's explanation of the techniques of integration are also better than Courant's, IMO. Courant's treatment of continuity is superior to any that I've seen in an elementary calculus textbooks. His introductory chapter is also very nice, demonstrating very useful results and inequalities, along giving explanations of very important notions. So I would definitely recommend reading Courant's first chapter, followed by Apostol's treatment of integration. But Courant uses a characterization of the integral through Riemann sums, which is definitely more popular than Apostol's, so you might want to read a little into the 2nd chapter once done.

As for other good books, there are

Mathematics: Its Content, Methods and Meaning

Aleksandrov, Kolmogorov and Lavrent'ev

It gives an overview of many important branches of mathematics.

Foundations and Fundamental Concepts of Mathematics

Howard Eves

Definitely a book anyone interested in understanding the real foundations of mathematics should read.

The Mathematical Experience
Reuben Hersh and Phillip Davis

Very good book that kept me reading from cover to cover.
  • #22
Just curious - how has Spivak made Courant-John obsolete?
  • #23
I was wondering the same. Would you say Spivak is the more rigorous of the three? For example, if three identical (Yes, even math skills) twins were to start each book; twin 1 with Spivak, twin 2 with Apostol...Would any of them come out with a better understanding? That's what I'm trying to deduct here. I was planning to use Spivak and then Apostol as reference (Reference being when one book doesn't satisfy you, you look it up in the other book, which presents it in a different way...).


  • #24
Well, about Courant-John being obsolete, I mean that it is a really old book. It's 45 years old! Spivak provides more streamlined and modern approach. If I was to teach a calculus course I would be hesitant using anything other than Spivak. People simply have better experience with it.

But again, for single variable I prefer Apostol's approach (integration first). For multivariable I think the second volume of Apostol (which I read, a long, long time ago) is not as good. But we do have a brilliant book "Calculus on Manifolds". It's a tough book, and most students hate it, but they soon realize (with more mathematical maturity) that it is very well written.

Another thing is I have not yet heard of schools still using Courant-John as a textbook. Spivak and Apostol dominate introduction to mathematically rigorous calculus. There are many reasons I am sure.
  • #25
For one thing there isn't a lot of set-theoretic language in Courant/John, an unpopular approach these days. That Spivak has made Courant/John obsolete in my opinion is false, Courant/John cover a lot more ground than Spivak in the first volume alone.
  • #26
Courant/John cover a lot more ground than Spivak in the first volume alone.

This is very true, but I think Spivak covers everything necessary anyways.
  • #27
"For one thing there isn't a lot of set-theoretic language in Courant/John"

If you meant by this that for example Courant writes "if x is a real number" instead of "if [tex]x \epsilon R[/tex], then yes, this is true. To give the basics of set theory, one can always read the beginning of Apostol's book. I love Courant-John and I've also read Apostol, so it just seems weird to me that one would call it obsolete.

I guess it's silly to bicker over which one of the three is the best as they're all excellent books.
  • #28
Which one would you agree upon that gives the broader spectrum of knowledge? It's my main focus, I don't want to miss out on important ideas/topics by reading a particular book. I really had heard that Spivak's Calculus on Manifolds was amazing too.

  • #29
Here are the table of contents for Courant/John (first volume), Apostol, and Spivak, in that same order:

You will be able to navigate yourself to the table of contents after following the links.

For what it's worth, I found Courant/John more difficult to read than Apostol. I had to stop with Courant, read some Apostol, then after that go back to Courant. That's just me though.
  • #30
I've scanned through the tables of contents, and I've noticed that Apostol seems rather "dull", it's a long list and the way it's presented makes it look longer. I have also noticed that Courant has applications of calculus, and that Apostol has a bit more than just calculus itself. Spivak on the other hand has a VERY short table of contents. The contents of all three books were more or less the same, and Courant seemed to have more cover on power series and infinite sums. I am still supporting the idea of buying two of these three, right now Spivak and Courant seem like good choices.

Thank you very much JG.

  • #31
To reply to Unknot's post, my aim is really to get as much as I can. Obviously I won't be reading 5 different calculus books in hope to get all I can get from calculus, so I'm really trying to find one or two books that can cover the necessary, and more. :smile:


  • #32
Also Apostol is the only one out of the three to cover DE's (in the first volume at least). In the 2nd edition, he also covers some linear algebra, but I think it's lousy, that's why I have the 1st.
  • #33
Did you find Apostol to be dull or never-ending? The way something is presented usually greatly helps out, and Apostol's book did not seem very "inviting". :wink:
I was also wondering, what about multivariate calculus? All three books don't seem to cover it really. Is Spivak's "Calculus on Manifolds" as good as his "Calculus"? (No, I'm not obsessed with Spivak, I just hear a lot of good about him:biggrin:)

  • #34
I find Apostol's motivation of the integral the best from a pure mathematician point of view, and Courant's from an applied mathematician point of view. You shouldn't worry about multivariate calculus for now, and it's difficult to learn it properly without any linear algebra under your belt. I recommend either Hoffman or Friedberg, but maybe you should start with Lang (because the other two are more advanced and deal with abstract vector spaces and vector spaces in F^n where F is arbitrary, whereas Lang only does R^n). To learn multivariate calculus, there is no better learning source than Rudin's Principles of Mathematical Analysis.
  • #35
Courant also covers Fourier series, which I don't think the other two do.

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