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What are the 'best of the best' textbooks to help me learn math from the ground up?

  1. Oct 16, 2014 #1
    After 20 years of being a locksmith, I have decided that I want to get a college degree and I'll be starting next year! As part of my degree, I will be doing two math courses - one in calculus and the other in linear algebra.

    However other than addition and subtraction, I don't know much else! I'll need to work my way through K - 12 math textbooks doing topics such as arithmetic, algebra, counting & probability, geometry, number theory, calculus, etc before even touching first year college calculus and linear algebra textbooks!

    Could I please get some math textbook recommendations that are clear, proof-based and to the point? I have heard that some Soviet textbooks do what I want but I don't know too much about Soviet textbooks but it does sound interesting!

    I do prefer textbooks as I am a bit old fashioned and aren't the best when it comes to using technology! Money also is not a problem so please recommend as many textbooks as needed! If it's better to have a textbook for each field in math then so be it!
    Last edited: Oct 16, 2014
  2. jcsd
  3. Oct 16, 2014 #2
    Do you have any experience of calculus or advanced algebra, stuff like integrals differential equations and differentiation of functions or limits...? Or basic stuff like inequalitys(ok, probably yes) and so on...?
  4. Oct 16, 2014 #3
    It has been over 20 years since I've touched a math textbook so I'll need to start from square one! In fact what you just said to me was completely foreign.
  5. Oct 16, 2014 #4
    I would recommend you read algebra 1 or algebra 2 for dummies (don't laugh), they keep things short but it is a good introduction but, I would not recommend it as a reference. If you have read both or just number 2 you will be (hopefuly) capable of starting with calculus. I would not recommend you read calculus for dummies which is very messie and the examples are very bad, so I would suggest bridging the gap to university mathematics. From then on it's mainly trigonometry and geometry you should cover, but I don't know any book on that...(mocking starts in 3,2,1)
  6. Oct 16, 2014 #5
    Does algebra 1 cover K - 8 arithmetic (decimals, fractions, etc)?
  7. Oct 16, 2014 #6
    Yes,here is the index:

    1.Deciphering signs in Numbers
    2.Incorparating Algebraic Properties
    3.Making fractions and decimals behave
    4.Exploring Exponents
    5.Taming rampaging Radicals
    6.Simplifying Algebraic Expressions
    7.Specializing in Multiplication matters
    8.Dividing the long way to simplify algebraic expressions
    9.Figuring out factoring
    10.Taking the bite out of binomial factoring
    11.Factoring trinomials and special polynomials
    12.Lining up linear euqations
    13.Muscling up to quadratic equations
    14.Yielding higher powers
    15.Reeling in Radical and Absolute value equations
    16.Getting even with inequalitys

    It goes on until 21 but I would leave them out, those are just some formulas from geometry to calculate areas and so on.
  8. Oct 16, 2014 #7


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    I remember fondly the 'Schaum Outline Series' books... I don't know if they are still available. In those books, the theory was clearly explained, in compact paragraphs, there were a few of problems already worked out, and a lot of them to be solved by the reader...
  9. Oct 16, 2014 #8
    The second book does matrices and arrays, systems of equations, cramers rule. And elaborates on functions, like radical functions and it covers polynomials.
  10. Oct 16, 2014 #9
    @NTW I do want the textbooks to be 'to the point' but I don't want a summary. Are those books a summary?

    @moriheru I'll have a look into those books. They keep things short, as you said, but is it a summary? I'm trying to find textbooks that are 'clear, proof-based and to the point', I'm not sure if I'm going to find anything!
    Last edited: Oct 16, 2014
  11. Oct 16, 2014 #10
    I don't quite get what you mean, but if you mean covering a large range of basic mathematical topics, yes it does cover most of the basic topics, as you can see by the index. But you shouldn't expect a fully fleshed and detailed book, it's just an introduction, but it introduces you to a large number of topics that you will all need and teaches you the basics, so you can read the details or learn them in university.
  12. Oct 16, 2014 #11
    Well yes then I would say suggest reading them and they do give you examples but if you want exercises you should by the companion work books(which are cheap). And it has the odd joak. Even thoe now I prefer books dry as dust(Quantum mechanics and kaluza klein theory), I did like the joakes.
  13. Oct 16, 2014 #12
    Thank you @moriheru. I welcome any other recommendations!
  14. Oct 16, 2014 #13
    A ditionary of all formulas is going to come in handy once youve got to calculus. Believe me you will need it! I don't think there are any better and worse dictionary, just more detailed, so I don't have any recommendations.
  15. Oct 16, 2014 #14
    I would strongly suggest I.M. Gelfand's Algebra and Trigonometry (two separate books). I think these would be exactly what you want: old-school, Soviet, clear rigorous texts. After you're done with those, you would do well to go through Lang's Basic Mathematics.

    After that, start calculus. I'd suggest Spivak's Calculus, but you can look around and decide then.
    Last edited: Oct 16, 2014
  16. Oct 16, 2014 #15
    I just looked up Gelfand's Algebra and Trigonometry - it seems perfect! Would I be able to use them as my main textbooks or are they meant to be supplementary textbooks? Do these textbooks cover all the algebra and trigonometry that is covered in high school?

    Could you recommend any books similar to that for arithmetic (K - 8) and the other topics I mentioned? I'll also have a look into the other texts you mentioned.
  17. Oct 16, 2014 #16
    You can use them as main texts. Serge Lang's Basic Mathematics also covers algebra and trigonometry in somewhat briefer detail. When you're done working with these three, you'll understand these topics better, and in far more detail than they are taught in a regular high school.

    These should be sufficient to get you the prerequisites to begin calculus. However, these are more challenging than most books, but I'd advise you to persist and get through- you'll have a richer and deeper grasp of the subject.

    EDIT: As for arithmetic, you don't need to get a separate book.
  18. Oct 16, 2014 #17
    Thank you @Feryn. I welcome any other suggestions! Like I said, the more, the better!
  19. Oct 16, 2014 #18
    In addition to the Gelfand books, Kodaira wrote some high school texts too. I haven't seen more than whatever previews are on Amazon or Google though. Judging by their contents it looks like precalculus & more basic stuff, which seems to be what you're looking for:

    I would also recommend something like Schaum's for mass quantities of problems to solve. There's probably a book with a title like "(so many thousands) of Solved Problems in Precalculus".
  20. Oct 17, 2014 #19
    @fourier jr Do you mind telling me what topics are done from K - 12, in order? For example, arithmetic, prealgebra, Algebra I, Geometry I, Algebra II, Geometry II, Proofs & Logic, Discrete Math, Calculus and Geometry III.

    Are those all of the topics studied and is that the best order to do it in?
  21. Oct 17, 2014 #20

    Simon Bridge

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    The trouble you will run into is that different people have different learning styles - the best text books will be those which best match your learning style. So it is impossible to pick the best of the best for you. What people are doing is trying to suggest text books that are the least likely to be rubbish.

    ... similar to the above: you should start with something closest to what you already know.
    Not all the curriculum subjects get studied by every student - even if they are majoring in mathematics.
    Most subjects are taught simultaneously at the same level, and lower levels are taught before higher ones.
    The exact details depends on the school (and the country you are in).

    However; it is a good idea to find an online course or course summary to guide you.
    There are plenty to choose from. You should have a look at several to see what sort of thing fires you up the best.
  22. Oct 17, 2014 #21
    @Simon Bridge I haven't found any online course or course summary to guide me. Also, maybe I shouldn't have said 'best of the best' but how about just any recommendations to help me learn math from the ground up using textbooks that are clear, proof-based and to the point?
    Last edited: Oct 17, 2014
  23. Oct 17, 2014 #22
    People have suggested you very good books. Just take whichever you like most, and get started with it. If none takes your fancy, take a look at the textbook listings. Don't be too concerned about the order you're going in, as long as you can follow it.
  24. Oct 17, 2014 #23
    @Feryn I already ordered my first book! I cannot wait to start learning again!
  25. Oct 17, 2014 #24
    Ah, good! Which one?
  26. Oct 17, 2014 #25
    @Feryn I decided to go with "Basic Mathematics" by Serge Lang. It looks good! I plan to move on to more complex books afterwards. I just have a quick question though, I am preparing for an engineering degree. When I have the choice between two good textbooks - one skill based, the other theory based, which one should I go for?
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