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Looking for series of books(maths) arithmetic to calculus(or higher)

  1. Aug 19, 2011 #1


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    I am trying to learn maths at home and was wondering if anyone knew of any series(or group of books that you would recommend) of books with the structure:
    learn arithmetic
    practice arithmetic
    learn algebra
    practice algebra
    and so on up until calculus or higher

    I would prefer if the books didn't just tell me how to do something but rather made me try to work it out for myself, not just learn formula, apply formula and move on to next subject.

    If there isn't a series as such would anyone have a suggestion as to a group of books that could fulfill the same purpose?

    Also if anyone has any ideas as to how I could improve my plan, they would be very much appreciated.

  2. jcsd
  3. Aug 20, 2011 #2


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    you don't mention my favorite beginning topic, geometry but i will suggest that anyway.

    how about euclid's elements and euler's elements of algebra? after that you could try spivak's calculus.

    but since the books should be ones that you yourself enjoy and can read, it is better if you choose them yourself. for this try browsing a local college library in the math section.
  4. Aug 22, 2011 #3
    I think that you should stop worrying about figuring things out for yourself. It took the greatest minds in the world over 2000 years to get where we are today in math. Presumably, you don't have time to repeat that process. It will be enough of a challenge to learn math on your own, without forcing yourself to come up with the proofs for everything.

    I'm not saying to skip proofs; I'm saying that over the years, teachers have learned how much proof is appropriate at early stages, so don't feel that you have to prove everything, and don't feel that you should be able to come up with proofs by yourself. By all means, see if you can, but if you can't, don't feel you have to stop until you figure it out. It is enough that you can understand the proofs given. And some things that seem very simple and obvious turn out to be very subtle, so you are not expected to be able to prove them until you have taken some advanced analysis classes in college.

    You will hear a lot of sneers and condescension at "plug and chug" books, but almost every textbook in wide use is written for a spectrum of students. They have basic drill problems, but they also have more advanced problems that ask you to think, to understand the processes, and to prove results yourself. At the early stages you are talking about, that is more than enough --- some proofs are appropriate for that level, and some are not. You have to walk before you can run. You will eventually get to subjects like linear algebra, and analysis, where proving things will be the main part of the course, and those are typically the first courses (other than basic geometry) where that is true, even for gifted students. Too much proof, too early, can bog you down, or even discourage you to the point that you give up.

    So for arithmetic, like mathwonk said, just go to the library and find a book you like. There are probably more than enough free ones on the internet, too. Google Books has dozens of old textbooks from the public domain, and arithmetic has not changed in a long, long time. And there are all kinds of free websites that teach math --- Kahn Academy is one that comes to mind. The same goes for other subjects, up through calculus at least.

    When you are ready for it, i.e. after you have learned arithmetic, basic algebra, and basic geometry, I recommend that you find a cheap used version of Stewart's Precalculus -- he proves almost all his results, so if you work through the proofs carefully, you will satisfy your interest in understanding the principles, and you will have a very sold base for learning calculus and higher math.
    Last edited: Aug 22, 2011
  5. Aug 25, 2011 #4


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    So would arithmetic, geometry, algebra, pre calculus, calculus be the best way to go about it? And don't try to prove everything.

    I am starting at another school soon(My current school is rural, local and not very good(I live in Australia)) so I will be able to ask the maths teacher these questions. But in the mean time
    (this is not my internet, we can't afford it) would you be able to give me list of suggestions for books? I need to buy them for now as the closest library is tiny and the city is hours away, I have about $300(AUS).

    On arithmetic I found these which look interesting:
    Do they look any good, or has anyone used them/heard of them?

    On geometry I found these:




    On algebra I found these:
    https://www.amazon.com/Algebra-Dumm...9640/ref=sr_1_9?ie=UTF8&qid=1314251682&sr=8-9 What is algebra I?




    Thanks for the suggestions so far.
  6. Aug 25, 2011 #5
    The usual sequence in most schools I know of is arithmetic, algebra 1, geometry, algebra2, trig, pre calculus, calculus. Normally, that would take six years or more, so don't get discouraged if you can't do it in a year.

    But you sound intelligent and motivated enough to compress it. Probably you could start geometry as soon as you have learned enough algebra to be able to solve basic equations, and you could skip trig, since most precalculus texts start it from scratch. They also contain a pretty extensive review of algebra, so you may be able to start precalculus earlier than you think.

    Do spend as much time as it takes to thoroughly understand the subjects in your precalc book. Most calculus students who struggle, do so because their precalc foundation is shaky.

    As for your books, I'm pretty sure there are lots of websites where you could learn basic math for free, as well as lots of free books you can download from Google Books or the like. Most of them are quite old (which is why the copyright has lapsed), but subjects like arithmetic, basic algebra, and geometry have not changed much for centuries. Old trigonometry books will have chapters on using tables of functions, which calculators and computers have made obsolete, but otherwise they are fine, too.

    Your posts are so well written that it's hard to believe you don't know basic arithmetic. You might look for a book on Business Arithmetic or Business Math if you feel like you know your times tables, but want to learn more about applying arithmetic.

    Be sure that the algebra books you read are for high school students. I don't know why, but there are MANY algebra books with titles like "Basic Algebra" or "Elementary Algebra" that are actually intended for upper division college students. If the book has chapters on rings or fields or groups, it might be that kind of book. You want a book that teaches you to solve simple equations.

    Schaum's books are very popular, but they are more for people who are reviewing what they once learned, or who want more practice. IMO they don't explain things well enough for someone who is learning a subject for the first time, let alone someone like you, who has indicated that he wants to really understand where things come from.

    Geometry might be the hardest subject for you to learn by yourself, because it is more about proofs than formulas, and if you get stuck, it's tough when there's nobody to ask. I would download several old books from Google Books so you can see different approaches; check the reviews in Amazon for suggestions on the best books for self study; and be sure to use this forum to ask for help when you get stuck. There is also an excellent interactive site on Euclid's Elements (the basic geometry textbook for 2000 years) here:


    Note that most high school geometry classes would only get through the first 3 or 4 of the 13 "Books" of Euclid.

    The only precalculus books that I have personal knowledge of are those by Stewart and Swokowski. I think both are very good, but maybe Stewart's is a little easier to read. You don't need a brand new one; look for a cheap used older edition on Amazon or Ebay. You should be able to find them for no more than ten dollars or so. If you can find good deals, get both of them -- sometimes if you don't understand one author's explanation, the other one makes it clear.

    You might even try starting with a precalc book. As I said, your posts are so well-written that it's hard to believe you don't know basic arithmetic, and Stewart starts at a pretty basic level on all the other subjects, except geometry, which you should study anyway since it will satisfy your desire for learning about proofs.

    You should probably expect to take at least two years before you are ready for calculus, by which time any recommendations made here will be dated. There are already several sites like MIT that offer full video courses for free, and many full online textbooks as well. There should be more and better every year.

    Best of luck to you.
    Last edited: Aug 25, 2011
  7. Aug 26, 2011 #6
    I was just looking at Khan Academy
    for something else, and noticed that they have tons of free lectures on arithmetic and basic algebra. I recommend you look at those before you spend much money on books.
  8. Aug 27, 2011 #7


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    If you are interested in Euclid's Elements, I have a set of free notes on the essentials from the first 4 books on my website, from a summer course I taught in July for very gifted children. It is similar to a college course, but moved faster. There are few diagrams, but Euclid's book has those.

    Books I-IV cover most of basic plane geometry by the way except for similarity. Actually there is a nice result in book III that is equivalent to the basic similarity result (Prop. III.35) as I point out in my notes. Books V-VI are about ratios and similarity, books VII-IX are on number theory, book X is about commensurable and incommensurable numbers, while XI-XIII are about solids (polyhedra).
  9. Sep 2, 2011 #8


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    First of all thanks to everyone for the help.
    I am going to start with Euclids elements and Eulers elements of algebra, in conjunction with Khan academy at school as they have internet there.

    What is the difference between algebra I and II, I live in Australia and we have foundation maths, maths methods and then if you want specialist maths(as well as some others).

    Also should I worry about learning probability and statistics(I have done some of them at school), are they useful(I want to study maths and/or physics at university)?

  10. Sep 3, 2011 #9
    I'm not sure that Euclid and Euler are your best choices for *primary* texts. They are great, important, historical works, but I think they are better for someone who wants to increase his knowledge of a subject he's already somewhat familiar with, than for someone who is learning the subject for the first time.

    You have to remember that many of the people who very kindly take the time to respond to requests like yours are super gifted in math, and read books for pleasure that average people would find very heavy going.

    I always tell people to save money by getting used books, because nothing much in basic math has changed in the last 50 years, but Euler is over 200 years old, and Euclid is over 2000 years old. That doesn't mean they are wrong, but hopefully in that time people have found simpler ways to explain some things. You might see if you can borrow the texts your school uses, and maybe ask the teachers of those classes what problems they assign for homework. That way, if you get stuck, you'll have someone to ask who is familiar with the problems.

    In any case, both Euclid and Euler go far beyond what is usually covered in high school, so if you get bogged down, don't assume you're not cut out for math. Just try a book that was written with high school students in mind.
  11. Sep 7, 2011 #10


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    Hi again,
    I started with Euclid and I prefer the online tutorial,(although I can't stand sitting for extended periods) to the book I was able to find(It is in Greek, with the translation written next to it).

    I have a maths teacher now(A useful one) and was talking to him about where to start and he said I should be able to start with a pre-calculus book as you suggested so I thought I'd try Stewart's and just wanted to check that this is the book that you were referring to before I bought it:


    Also has anyone here heard of Inquiry based learning, and if so do you know of any books that try to teach in that way?
    I have just discovered it and it sounds exactly like what I have been looking for.

    Last edited by a moderator: May 5, 2017
  12. Sep 8, 2011 #11
    Yes, that is the Stewart book, although it's a newer edition than the one I have. Judging from the table of contents, there are no important differences between the last couple editions, so I suggest you get a cheap used edition.

    Sorry, don't know about Inquiry based learning.
  13. Sep 8, 2011 #12


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