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I am high school math enthusiast I am very much interested in mathematics. I try to learn stuffs from the internet but I can't seem to find the resources and am also confused about where to start.

I am 16 and have recently finished my eleventh grade. I know some basic single variable calculus, complex numbers...(Euler's formula and stuffs) and also have some idea on partial derivative, double and line integral and differential equations. Besides these, I do have some faint notion as to what the Riemann's hypothesis is, Cantor's work on his continuum hypothesis, Godel's incompleteness theorem, Poincare's conjecture, Golbachh's conjecture and stuffs though I lack the ability to explain them mathematically. As you can see my mathematical knowledge has some severely big blank gaps. This is because I have had no one tutor me and have learn't most mathematics by myself, on the internet, watching documentaries and reading some books on these

In Mathematics; Number Theory, Topology and Abstract mathematics are few of my topics of interest and I would like to go further into these subjects and possibly pursue these subject as my career ; Number theory being my first priority

So could you guys please advise me on what should I do now, where do I resume my mathematical journey, what next??? I would really appreciate you time and help on this regard.

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closed as primarily opinion-based by LeGrandDODOM, Sujaan Kunalan, Krish, Jonas Meyer, Mark Fantini Apr 8 at 2:51

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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Cross-posted from MO -- with an accepted answer. –  t.b. May 29 '11 at 16:18
    
+1 because it's my problem too –  Bruno Alano Aug 7 '13 at 3:11

7 Answers 7

up vote 13 down vote accepted

I would add a dissonant note: many of the "popular" sources are invidious, making mountains of molehills. One should not wait 200 pages to learn what a group homomorphism is.

My advice would be to not feel obliged to comply/obey the implied commands in textbooks and other school-math. Instead, try at every moment to move toward a thing that interests you, that captures your fancy. Make yourself the ultimate judge of what is good or bad/boring.

Do not accept the school-math notion that one will look at only one or two books each year. Obviously no real mathematician got anywhere doing any such thing. The key point is to not do the exercises, most of which (all the more an undergrad or beginning grad level) were contrived to satisfy publishers and/or traditional expectations.

One more time: trust your own judgement. Demand that sources convince you.

... but realize that you may have failings/weaknesses...

A charmingly complicated business. :)

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Awesome answer. Echoes my opinion :) –  NikBels Jan 8 '12 at 19:08
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not do the exercises? -1 –  Peter Sheldrick Jan 25 '14 at 15:53
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@PeterSheldrick, indeed, I disrecommend "doing all the exercises in whatever text one is using", especially in most standard texts, because the significance of the details being addressed, if_any, can only be clear later, much of the time... and very often "the exercise" is obvious at the moment its significance arises (if ever). And then there are make-work exercises... Anyway, my answer was also intended to provide some non-unanimity about the mythology of "do all the exercises". –  paul garrett Jan 25 '14 at 16:23
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But do read the exercises and try the ones that fancy you! –  kjetil b halvorsen Jan 25 '14 at 17:42
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@kjetilbhalvorsen, Indeed! Reading the exercises to see what's there, to see if anything strikes one's fancy, is a great thing! Thanks for making the point! –  paul garrett Jan 25 '14 at 19:33

Just realized that this question is old; oops, I've already written an answer. Guess I might as well post it. I completely agree with Paul Garret when he says:

My advice would be to not feel obliged to comply/obey the implied commands in textbooks and other school-math. Instead, try at every moment to move toward a thing that interests you, that captures your fancy. Make yourself the ultimate judge of what is good or bad/boring.

Yep, exactly. This is so, so true. I highly doubt that any great mathematician became that way by mindlessly memorizing stuff that others claimed to be important. You have to hack your own way through the wilderness to some extent. Yes, you should allow yourself to be guided by the paths that others have built, but don't just mindlessly follow them.

This takes self-knowledge. You need to work out:

Am I a theory-builder, or a problem solver?

Everyone is a little bit of both, of course, but its good to know which way you lean. So ask yourself: which do you prefer?

  1. Frameworks/logic/conceptual clarity/beautiful proofs?
  2. Solving challenging problems and developing new techniques for doing so?

The way to get good at maths depends on which way you lean. If you have a preference for (1), then I would suggest that you grab a hold of the stuff that isn't 100% well-conceptualized in the stuff you're learning at school, and start trying to "work it out" or "fix it." For example, you may say: okay, the definition of asymptote given is pretty lame. Okay, so what are the correct framework(s) for understanding asymptotes, and what are the correct definition(s)? You should spend some of your time pondering this yourself, and some of your time trying to research what others have said about the issue, both on the Internet and also in books.

On the other hand, perhaps you're a problem solver. If so, you have to find problems that really grab you. I don't necessarily mean problems like the Riemann hypothesis which are exciting because they're famous. Rather, yous should problems that are genuinely exciting to you, because you find them intellectually stimulating. That way, even if you don't solve the problem (and you often won't), you'll still find that you've learned important things.

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An excellent way of learning is just hanging out here... here (and elsewhere on SE) you'll see a wide variety of problems, solution attempts, and a selection of things on which to hone your estetics (yes, math is much about "beauty"). If you can, check out Proofs from The BOOK, it is a collection of outstanding proofs. The books by William Dunham on the history of mathematics are extremely nice. The Dolciani Mathematical Expositions series by the Mathematical Association of America is also worth checking out. Reading through this should give you a sense of what interests you most.

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I agree with utdiscant, there's so much you can find on the internet. I would DEFINITELY recommend you looking at some courses online. You don't even have to take them, most colleges nowadays put their coursework and lessons up online. Good luck in your journey of mathematics :)

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It's good that Mathematics has attracted you. As for advice please read terry tao's blog

Where to start? This really depends on your interests. In high school if you like calculus, then please start learning some Real analysis on your own, by reading books such as:

  • Elementary Analysis by Ken Ross

  • Calculus by R. Courant.

If you like Algebra then please start reading basic text's such as:

  • Contemporary Abstract Algebra by Gallian

  • First Course in Abstract Algebra by J.B.Fraleigh.

Note: Mathematics is a very hard subject and you will have up's and downs. Persevere and you shall have success.

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+1 for mentioning Terry Tao's Blog. –  Eric Naslund May 29 '11 at 16:30
    
@Eric: Well, his name appear's almost everywhere :) True Genius –  user9413 May 29 '11 at 16:32

In my opinion, a good way to learn something about mathematics is to buy yourself a good textbook and read it from the beginning to the end. If there is something you don't understand, you can still go to math.SE and ask for help.

I read Concrete Mathematics from Don Knuth, Oren Patashnik and Ronald Graham. I find it quite enlighting, but it may be more suitable for people who are interested in computer science.

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imho the best txtbook ever, apart from some Russian probability stuff... –  sigma.z.1980 Jul 7 '11 at 0:43
    
@sigma.z.1980 It's good to know, that other people have the same opinion as me. –  FUZxxl Jul 7 '11 at 16:11

There is no perfect answer for that question. My advice, would be to look into a linear algebra course on the internet, or maybe just a long basic math 101 course at university level. In that way, you will get a lot of basics, before having to choose your path yourself.

http://ocw.mit.edu/courses/mathematics/

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