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!
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.
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.
Best Answer
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.