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Most rigorous pure math textbooks

  1. May 8, 2009 #1
    I'm highly interested in pure mathematics, so I wanted to know some of the best textbooks that would prepare me for advanced pure math classes in college. I already understand most math that would be taught in high school, however I would like to re-learn it in a more rigorous way. (Our school's textbooks are pretty bad.) So far, my list looks something like this: (I chose many books from Howers's great thread on pure math textbooks, but changed some of the higher level math textbooks)

    Algebra 1:
    Elementary Algebra, Jacobs
    Algebra, Gelfand

    Geometry, Jacobs (I'm using the earlier editions of Jacobs's books by the way.)

    Algebra 2:
    Precalculus, Sullivan
    I've heard Precalculus with unit circle trigonometry is a good text, does anyone else know about it?
    Functions and Graphs, Gelfand

    Trigonometry, Gelfand

    Calculus (From here on out I can't remember the names of the textbooks):

    Rudin, Pugh

    Linear Algebra:
    Hoffman and Kunze

    Abstract Algebra:
    Dummit and Foote


    Please tell me, is this a good list? If you have any better textbooks or recommendations please tell me. Thank you for your help!
    Last edited: May 8, 2009
  2. jcsd
  3. May 8, 2009 #2


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    Apostol and Courant...
  4. May 8, 2009 #3
    Hoffman and Kunze is a nice book but I'm not sure if its the best choice for starting off without prior knowledge in LA. Maybe "Linear Algebra" by Serge Lang might be more suitable for you right now. Both are rigorous books, but I believe Serge Lang makes the book very readable.
  5. May 8, 2009 #4
    The name is Calculus, Spivak; you may also want to add Courant and Apostol(Volumes 1 & 2) as suggested above, those are very good books; Coxeter’s Geometry Revisited is also good along with Jacobs geometry (Hard to find anyway). You may want to note that most mathematics are NOT taught in high school... For calculus and beyond; taking some titles off my bookshelves:

    • Introduction to Calculus and Analysis (Volume I), Courant, Richard; Fritz John
    • Differential and Integral Calculus Volume 2, R. Courant
    • Calculus, Spivak
    • Calculus on Manifolds a Modern Approach to Theorems of Advanced Calculus, Spivak, Michael
    • Calculus. Volume I. One-Variable Calculus, with an Introduction to Linear Algebra. Second Edition, Apostol
    • Calculus. Volume II. Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probabi , Apostol
    • Linear Algebra, Hoffman & Kunze (Lang is often recommended as it is slightly easier)

    Beyond calculus:
    • Lang, Algebra
    • Rudin, Real and complex analysis
    • Functional Analysis, Riesz - Nagy
    • Hille, Complex Analysis
    Ahlfors, Complex analyhsis
    • Spanier, algebraic topology
    • Vick, Homology theory.

    Here is a pretty complete list of books for almost every branch of maths: http://www.cargalmathbooks.com/
  6. May 8, 2009 #5
    Your list is good, but you might find Jacobs algebra and sullivan's precalc boring - given your background. They are not pure math, more applied.

    Regarding pure math, at a highschool level there isnt much of it save for geometry. I mean what more can you say about algebra aside from its axioms? This is why the Gelfand books are really good, they give you hard questions that bring you to a high level. But the material is more or less the same. Use them if you feel a need to, because school books give people a false sense of security as the material is watered down and not at all rigorous. You can judge for yourself though: did you prove trig equations, do you know why you can factor and expand? I don't want to slow you down, so if you feel your fundamentals are strong you can begin with Spivak. For calculus and beyond, what you list are among the best books on the market.
    Last edited: May 8, 2009
  7. May 8, 2009 #6
    Thank you for your input. However, the textbooks my school offers are pretty terrible. They "explain" a concept by working out a problem, maybe two if you are lucky, and expect you to understand all the subtleties and irregularities of the concept. I'm afraid I may have missed something. I would like a more rigorous textbook that goes into greater detail, and I saw you had these recommended. What are some great high school algebra through pre-calculus textbooks then?
    Last edited: May 8, 2009
  8. May 8, 2009 #7
    I've read through the Gelfand series (Excluding his books on Calculus and beyond) and I found that they were terrific as far as depth, difficulty, and rigor. If you go through the series and mix a bit of practice (Thus why I put Sullivan in there, lots of exercise for Algebra and Trig) then your High School education could be considered complete. Although you really should get a book on proofs (Sorry I have none to suggest). You may want to skip Gelfand's Method of Coordinates and Functions and Graphs, leaving only Trigonometry and Algebra; the former two were VERY basic.

    I assure you that if you can work out through atleast 80% of the problems in those two books, then Calculus (Spivak's) should be relatively easy to grasp. Note that I also included Sullivan's book because it covers all, and a little more of what one would ever need to know hitherto calculus, as for Gelfand does not include quite as much.
  9. May 8, 2009 #8
    Pretty much all of Gelfand's are as rigorous as one can get at that kind of level. Normally I would recommend those easier Jacobs/Sullivan books to go along with Gelfand, because he can leave out things in the reading. Given your level though, you should just stick to Gelfand and supplement it with the 2 books by Santos D. on the web (link is the last one in that thread, under stage 1, elementary stuff).

    Edit: to clarify: Gelfand and Santos are much more in depth than what you'd see in highschool cookbooks, from algebra to precalc. They have the rigor and depth you seek. Use that logarithms book in the list as its the only one that actually explains what they are. You can also try finding some 60s books on algebra which are also very deep - but difficult to find (check used book stores). After either of these, you can say with confidence you have mastered highschool and can begin Spivak. Also, look into Velleman if you have trouble with proofs.
    Last edited: May 8, 2009
  10. May 9, 2009 #9
    I firmly agree with Howers here; Gelfand produces very rigorous and challenging books, but he leaves out a few things to make the book shorter, not many students are motivated to plow through bricks. I would also recommend pairing up Spivak with a book that has a few more exercises and supplied with answers. Several of the problems in Spivak's book will stop you from progressing for a good while; they are generally very challenging.
  11. May 9, 2009 #10


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    The AoPS books are also good since they incorporate a good deal of advanced problem solving which makes you think outside the box.
  12. May 9, 2009 #11
    "AoPS"? Can you elaborate?:confused:
  13. May 9, 2009 #12


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    art of problem solving

    http://www.artofproblemsolving.com/Books/AoPS_B_Texts_FAQ.php [Broken]
    Last edited by a moderator: May 4, 2017
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