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Mathematics Books

  1. Feb 7, 2009 #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.
  2. jcsd
  3. Feb 7, 2009 #2
    Gelfand: Algebra, Method of Co-ordinates, Trigonometry. His series were the math books used in the advanced highschools 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 highschool 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.
  4. Feb 7, 2009 #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.
    Last edited: Feb 8, 2009
  5. Feb 13, 2009 #4
    The art of problem solving Volume 2 gives a nice overview of high school level math with a lot of challenging problems.
  6. Feb 14, 2009 #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.
  7. Feb 14, 2009 #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!
  8. Feb 14, 2009 #7
    Not a very smart move. It starts well.
  9. Feb 14, 2009 #8
    What do you mean? I don't think I'm going very wrong...I did mention money was of no issue.
  10. Feb 15, 2009 #9
    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 reccommend 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.
  11. Feb 15, 2009 #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.

  12. Feb 17, 2009 #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:
  13. Feb 19, 2009 #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.
  14. Feb 19, 2009 #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...

  15. Feb 19, 2009 #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.
  16. Feb 19, 2009 #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,

    Last edited: Feb 19, 2009
  17. Feb 19, 2009 #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.]
  18. Feb 19, 2009 #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.
  19. Feb 19, 2009 #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.
  20. Feb 19, 2009 #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:

  21. Feb 19, 2009 #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:


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