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So over the past couple of years, I've been studying what I'd like to call "olympiad math"(so not subjects that require a lot of theory such as linear algebra, calculus, real analysis, etc but instead, those which require little theory but use a lot of "tricks" and "manipulation" such as number theory, elementary geometry, combinatorics, etc. Basically, the sort of problems that come up in math olympiads) to prepare for the IMO board exam which, as you might know, is in a few months.

But after that, I don't think that's something I'd like to continue and even up to this point, this was more of a hobby for me than anything. Instead, I'd like to start studying what I call school math; basically things such as calculus, linear algebra, etc but I've no idea where to even begin so I was wondering what books would you recommend?

A couple things to note:

$1.$ I'm a CS student so I'm mostly interested in calculus and linear algebra but please note that although I already know a little bit of calculus and linear algebra, I'm by no means good at them.

$2.$ I'm currently in grade 9 but I do grade 12 CS and grade 10 math so unfortunately, going up another grade for me is not an option.

$3.$ Other than "olympiad math", I know up to(and including) grade 10 math.

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  • $\begingroup$ @mathworker21 Not IMO iteself(that I think is in 10-11 months) but the board exam is. $\endgroup$ – Borna Ahmadzade Aug 23 '19 at 2:49
  • $\begingroup$ @mathworker21 Yeah sorry I meant the board exam:) $\endgroup$ – Borna Ahmadzade Aug 23 '19 at 2:51
  • $\begingroup$ I think the fact that you believe that number theory “requires little theory”, means that you have a limited perspective on number theory. $\endgroup$ – Joe Aug 23 '19 at 2:54
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    $\begingroup$ @Joe I meant the number theory required for IMO and by "little", I meant relative to something like calculus or real analysis. If I were to get into things like cryptography, then you'd be absolutely right but although IMO does require some theory, it's by no means as much as the amount of theory needed for let's say calculus or analysis $\endgroup$ – Borna Ahmadzade Aug 23 '19 at 2:57
  • $\begingroup$ Have you tried Khan Academy? It’s a great, free resource for math K-12 and has AP Calc, and a little on college math such as Diff Eq and Linear Algebra. $\endgroup$ – Joe Aug 23 '19 at 3:07
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I don't think there's a lot of sense in learning things that will be in your high school classes. You might want to take in some elementary set theory. Once you understand that, you can start plunging into any of the modern formal math subjects -- axiomatic linear algebra, group theory, graph theory, and so on. You don't have to dive too deeply into any of them.

One thing that really excited me when I was your age was reading through and making sense of the works of Raymond Smullyan. He writes logic books with a very approachable framing story that eventually leads to very deep and influential topics in logic and computability. There tends to be a lot of overlap in his books, particularly the first third, but I think any of his logic books are worth reading. My personal favorites are To Mock a Mockingbird and Satan, Cantor, and Infinity. I also like Forever Undecided, although that is particularly challenging.

Also, if you like coding, there are avenues for developing your catalog of languages and skills with algorithms and dynamic programming. Things like the UVa Online Judge can be fantastic things to dive into.

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  • $\begingroup$ Yeah I really like logic actually. There’s this book about logic(which is unfortunately only in my language, Persian) about logic in which the author discusses different things(everyday arguments, politics, etc) and arrives at contradictions and how you could win an argument. I’ll definitely check out To Mock a Mockingbird! $\endgroup$ – Borna Ahmadzade Aug 23 '19 at 13:07
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The typical school math up to college is Algebra I → Geometry → Algebra II → Precalculus → AP Calculus AB/BC, or Calc I/II in college → Multivariable calculus → Linear Algebra. I do not know which of these you have learned already, or not, but these are sort of the "general" math, sort of like general chemistry. Considering you are in ninth grade, I do not suggest you learn past calculus, as if you take a STEM major in college, you will be required to take classes anyway. I highly suggest you take AP calculus BC as a class, or self study for the exam if you feel up to it, and instead learn some math in a non traditional field, like graph theory, game theory, or topology. These topics are very, very interesting and will not be a required class for you later on.

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  • $\begingroup$ I definitely agree with doing what you’re interested in, but I wouldn’t recommend topology in 9th grade, especially when you say that you’re not good at linear algebra or calculus. $\endgroup$ – Joe Aug 23 '19 at 3:03
  • $\begingroup$ Eh, basic topology is quite fascinating. I totally agree with you when it comes to actual analysis especially in the form of equations, but considering they said it would be as a hobby, its not necessary to go very complicated with things. $\endgroup$ – Gabe Aug 23 '19 at 3:05
  • $\begingroup$ I agree that point set topology could be fun for an advanced student, but it would have to be from a book/website catered to an audience that didn’t have the mathematical maturity necessary for a normal topology class. $\endgroup$ – Joe Aug 23 '19 at 3:09
  • $\begingroup$ Yes, totally. It would be hard to find a book or website like that, now that I think about it. $\endgroup$ – Gabe Aug 23 '19 at 3:14
  • $\begingroup$ It shouldn't have to be. Intro point-set topology is just glorified set theory. The hitch is that you probably need at least precalc to have the intrinsic motivation to learn about closeness and appreciate the examples. $\endgroup$ – Matthew Daly Aug 23 '19 at 3:41

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