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Let me introduce myself. My name is Filip and I am going to 10th grade now. My school deals with electrotechnics and computers (programming, hardware etc.) I was always good at math but not quite interested. Lately math became my obsession! I don't know why, but I am crazy about it.

Now I have a little problem which you could help me with. In school we are learning some basic stuff. Things like geometry, systems of equations, quadratic equations/functions, complex numbers, logarithms and logarithmic (in)equalities and trigonometry. This is all great but I want to go further, I want to learn more and more. I am very interested in number theory, linear algebra, limits, integrals, probability and everything else. A friend of mine recommended this page and I decided to ask this question here.

Can you help me with literature and order in which I should learn this "advanced" math? Also before I start, I would like to do a review of stuff that I know at this point (I listed it above) so I could also use some book that covers those things.

I hope that you won't find this question silly and that you will be able to help in my further education!

Thank you, Filip

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It is great to hear that you are so interested in mathematics! I would suggest you to first learn calculus (if you have not already). There are many excellent texts on calculus and since I am not very familiar with them all, I will let other users recommend such texts. After calculus, the next step could be to learn linear algebra and real analysis. I would recommend Principles of Mathematical Analysis by Walter Rudin and Linear Algebra Done Right by Sheldon Axler. In theory, you could read the last two books before learning calculus but in practice this may not be appropriate. – Amitesh Datta Jul 17 '11 at 13:19
While @Amitesh's recommendations are great if you're really serious about getting into math, I'd recommend to look into books that go at a more leisurely pace. Two great examples that come to mind are Courant-Robbins and Aigner-Ziegler. – t.b. Jul 17 '11 at 13:30
Hello Filip, your name (as well as the school's curriculum and teachers' attitude) remind me of my country. Would you happen to be from Croatia? As for suggestions, terrific text for calculus is Spivak's Calculus, which is rigorous enough that some people call it an analysis text, while maintaining high readability. – user5501 Jul 17 '11 at 13:35
(meta: "We could speak on our native language here but I believe stackexchange rules forbid it." - Not really "forbid", but it's common courtesy here to deal in English since it's what most people speak here.) – J. M. Jul 17 '11 at 13:43
@Bhargav: In general, posts that do not answer the question should be posted as comments, not answers. However, because you do not have 50 reputation points yet, you can only comment on your own questions and answers. So you didn't do anything wrong; the "add comment" button will only appear for you once you gain 50 points. Asking and answering questions are the main (but not only) ways of getting points. – Zev Chonoles Jul 17 '11 at 14:43
up vote 8 down vote accepted

You should pick up a first-year university book in an elementary topic. "Elementary" here means that it doesn't assume any particular knowledge from you beforehand, not necessarily that it's easy. Some such topics are linear algebra, calculus, abstract algebra, discrete mathematics, elementary number theory and graph theory. Most other topics will assume you already know linear algebra and calculus.

I would suggest the book Discrete Mathematics by Biggs. It has a very clear style, and it goes through a lot of interesting topics in set theory, elementary number theory, abstract algebra, algorithms, graph theory, combinatorics, and so on. I throroughly enjoyed this book when I read it. My one complaint is it doesn't go very deeply into any topic, but it will give you a taste of a lot of things.

Some further advice: Pick only one topic and one book at a time, and focus on that. Make sure you learn from a book and not from some mix of sources online: a book will be much better organized to learn from than clicking links on Wikipedia. Read slowly and read every page from the start. Don't be discouraged if progress is slow: taking one hour to read one page is not uncommon. Do as many exercises as you can. Once you have a solution or a proof of an exercise, write it out, even if you think it seems too easy or silly. It will make you think through your arguments better.

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Thank you a lot! Now I see direction which I have to follow and everything is easier now. I hope I will make a good progress. This place is great, I mean I got right answers so quickly, you people are great! – user13420 Jul 17 '11 at 17:34

I like the Art of Problem Solving books that do a nice job of teaching problem-solving technique. Here is a link to them.

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There are decent online video lectures these days that should let you learn basic but essential mathematics at your own pace.

The Khan Academy may be useful and is very accessible. If you would like to sample some basic university level lectures, you may find the MIT Open courseware useful as well, although it may be a bit over your head (it depends on how strong you are; I've known plenty of people your age that could handle it, but they are quite strong). Neither gives anything too fancy, but it should be a good way to get the sort of math that is used in most technical fields and both certainly cover material above where you are now.

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