# Mathematical Career Advice to a young 16 year wannabe mathematician

I am a '94 born aspiring mathematician, programmer and a little bit of drummer from Nepal. I have been an avid follower and lover of mathematics since my seventh grade. Although I do not even consider my self anywhere near the "genius" or "prodigal" zone, I really love and appreciate the beauty of mathematics. I am particularly interested in the unsolved problems of mathematics and would love to live to see someone (most preferably me :)) prove or disprove (the latter of which is very unlikely) the Rienmann's hypothesis.

I am currently doing my A-Levels (from Budhanilkantha School in Kathmandu, Nepal) and have taken an "Advanced Mathematics" course.

Even though i am very much interested in mathematics, I do not have much idea about it. 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 readong some books on these ( Ian Stewart's Cabinet has helped me greatly)

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

.....sorry for the long pointless backgroud, i really thought that it was necessary for me to mention these points

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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

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...

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Awesome answer. Echoes my opinion :) –  Nikhil Bellarykar Jan 8 '12 at 19:08
@paul_garrett I find this answer and the responses you given in a different questions fascinating. It would be interesting to have a fuller sense of what you are articulating here. In particular, you seem to have a vision of mathematics that cuts against the grain. –  Erik G. Jan 19 '13 at 0:24
not do the exercises? -1 –  Peter Sheldrick Jan 25 at 15:53
@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 at 16:23
But do read the exercises and try the ones that fancy you! –  kjetil b halvorsen Jan 25 at 17:42

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

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

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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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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I am also from Nepal. You seem to be doing quite well for your age. Now, do yourself a HUGE favor by getting these books:

1. Elementary Number Theory - D. Burton
2. Contemporary Abstract Algebra - J. Gallian
3. Topology Without Tears - Morris

I would insist that you start with either (1) or (3), preferably (3). You can find (3) online. That should keep you busy for a while.

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I recommend Dudley's number theory book of the same title, in place of Burton's. In my opinion it is better written. Moreover, the selling price of Burton (new, on Amazon, in the US) is roughly 28% of the average annual GDP of Nepal -- whereas Dudley's book is available for a tiny fraction of what Burton's costs. –  aglearner May 11 at 13:59