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So i completed an year ago my schooling and i am pretty good at maths well at my level and i am very interested in maths and want to learn as much maths as possible and i like stuff like number theory. what got me really fueled up is the fact that i watched a video on numberphile (its a youtube channel on math stuff) about prime numbers and i got goosebumps when i heard that they have a formula that approximately calculates n'th prime number i thought prime numbers follow no pattern and bad thing i did not understand that formula .... well not to go with the story and all so the thing is i want to study more of mathematics by my own so i need you all to recommend me good books to get firm hold of every topic in math i not very hifi stuff that i want to start low and then become a jedi that does not mean i need books for noobs to understand what my math level is consider these things so that you may help me in a better way to recommend books

  • i do know $e^{iθ}= \cos θ + i\sin θ$ and its proof using expansions of $\sin x, \cos x$ and $e^x$
  • i do have basic knowledge about parabola, circle, straight lines(and pair of straight lines), ellipse and hyperbola(although i don't like them idk why)
  • basic trignometry

now my thinking levels

  • i once tried to calculate a algorithm or formula that would give equation of circle that would have maximum overlap with a parabola and passes through its vertex(failed miserably)

    • tried to calculate function that would give distance(not displacement you have to follow along curve) between any two points on curve of sine graph(not failed actually a computer would be able to calculate using that infinite sum to some accuracy but still failed)

well that should be enough i would ideally like books that would assume i know basics and then follow up but books that do start from very basic and get to pro are good to but not the ones that do not tell me what i already know but are just filled with much difficult problems based on what i do know i want although problem solving is must for math and i am not saying i don't want that in books you recommend i do want them and lots of problems but you get it what i am trying to say well go for it !!!!

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closed as too broad by user91500, SchrodingersCat, Daniel W. Farlow, heropup, user149792 Feb 6 '16 at 22:15

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ tl;dr: "Just finished highschool and have a good understanding of most highschool level topics, and would like you to recommend books on slightly more advanced topics." $\endgroup$ – fosho Feb 6 '16 at 15:10
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    $\begingroup$ @user306284 does that mean you recently graduated from college/university? It's important that we know how much math you know if we're going to recommend something at your level. $\endgroup$ – Omnomnomnom Feb 6 '16 at 15:14
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    $\begingroup$ @user306284 I did in fact read read your whole post, and I am not mocking but rather trying to help you get more answers. $\endgroup$ – fosho Feb 6 '16 at 15:14
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    $\begingroup$ @user306284 As Daniel said, what you've posted isn't enough information. For example, your first bullet could indicate that you had a particular interest in the one famous identity. It could also indicate that you have a basic understanding of the theory behind Taylor/MacLauren series. It could also indicate that you have a deep understanding of the theory behind analytic functions. $\endgroup$ – Omnomnomnom Feb 6 '16 at 15:21
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    $\begingroup$ @user306284 The best indication of what you know is which bodies of mathematical techniques you've learned ("basic rules of differentiation", "integration by parts", those sort of things). I'd say that your inherent ability doesn't tell us what you should learn next; rather, it tells us how quickly you learn. $\endgroup$ – Omnomnomnom Feb 6 '16 at 15:29
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A good starting point, covering lots of topics in first/second year college math, are the various lecture notes by William Chen. Well written, clearly explained.

Kenneth Bogart's "Combinatorics through Guided Discovery" is a nice introduction to it's topic, geared towards self-study.

If you want to be dazzled by a broad variety of ingenious reasoning, perhaps Aigner and Ziegler's "Proofs from THE BOOK" (Springer, 5th edition 2014) is what you are looking for. You need some higher math to fully understand it.

If you want a flavour of how math became what it is today, Dunham's "Journey through genius" (Penguin Books, reprint 1991) could be a good starting point.

In any case, "I want to learn more math" is awfully broad. Today's mathematics has splintered into dozens, if not hundreds, of subtopics, each of whose detailed exploration can (and does) take a lifetime. Do you want to get a leg up for college? Go for Chen's notes. Want to learn more in depth, explore some types of stuff that aren't in high-school main fare? Go for Bogart's. Both available for free. You might also check out open culture's textbooks.

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You say you like number theory so, try A Friendly Introduction to Number Theory as a start. If you like this, follow it up with Elementary Number Theory, David M Burton.

Also you may be interested in Linear Algebra, and this would be a good step into more advanced maths. For this you can try Linear Algebra and Its Applications, 4th Edition.

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