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Dear MathOverflow community,

In about a year, I think I will be starting my undergraduate studies at a Dutch university. I have decided to study mathematics. I'm not really sure why, but I'm fascinated with this subject. I think William Dunham's book 'Journey through Genius ' has launched this endless fascination.

I can't wait another whole year, however, following the regular school-curriculum and not learning anything like the things Dunham describes in his book. Our mathematics-book at school is a very 'calculus-orientated' one, I think. I don't think it's 'boring', but it's not a lot of fun either, compared to the evalutation of $\zeta(2)$, for example. Which is why I took up a 'job' as as a tutor for younger children to help them pass examinations. I wanted to make money (I've gathered about 300 euros so far) to buy some new math-books. I have already decided to buy the book ' Introductory Mathematics: Algebra and Analysis' which should provide me with some knowledge on the basics of Linear Algebra, Algebra, Set Theory and Sequences and Series. But what should I read next? What books should I buy with this amount of money in order to acquire a firm mathematical basis? And in what order? (The money isn't that much of a problem, though, I think my father will provide me with some extra money if I can convince him it's a really good book). Should I buy separate books on Linear Algebra, Algebra and a calculus book, like most university web-pages suggest their future students to buy?

Notice that it's important for me that the books are self-contained, i.e. they should be good self-study books. I don't mind problems in the books, either, as long as the books contain (at least a reasonable portion) of the answers (or a website where I can look some answers up).

I'm not asking for the quickest way to be able to acquire mathematical knowledge at (graduate)-university level, but the best way, as Terence Tao once commented (on his blog): "Mathematics is not a sprint, but a marathon".

Last but not least I'd like to add that I'm especially interested in infinite series. A lot of people have recommended me Hardy's book 'Divergent Series' (because of the questions I ask) but I don't think I posess the necessary prerequisite knowledge to be able to understand its content. I'd like to understand it, however!

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    $\begingroup$ Please take what Harry says with a grain of salt,Max.He thinks anything with motivation is not mathematics and that's not good for beginners. $\endgroup$
    – The Mathemagician
    Commented Jun 14, 2010 at 21:10
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    $\begingroup$ Max, I think that the sooner you stop worrying about questions like "But isn't Topology more of a graduate subject?", the better. Most Bourbaki's books do not make good first reading for the subject, that's true, but there are topology books that can, and should be, read while still undergraduate. There is no such thing as undergraduate/graduate subject, there is mathematics and something else. $\endgroup$
    – Vladimir Dotsenko
    Commented Jun 14, 2010 at 21:34
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    $\begingroup$ 1. Some of the answers you have received are a little odd, IMO. I see some fairly difficult graduate level books being recommended for a high school student who knows some calculus. (Homological algebra? Really?) I honestly don't know what to make of this. 2. Generic advice that may or may not be something you need to hear: Don't get discouraged. Math is hard for everybody. Be persistent, but if you have a book you can't make progress on, don't feel the least bit of shame in turning to a more elementary treatment or going back to learn prerequisite topics or whatever it takes. $\endgroup$
    – Michael Benfield
    Commented Jun 14, 2010 at 22:10
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    $\begingroup$ Aack! If you want italics, on MO put underscores _ or asterisks * on either side, not dollar signs. In TeX, use {\em text }. Dollar signs make the computer process whatever's inside as math, as if you had all those variables to multiply together. The classic example is $difference$ versus difference — notice the spacing around the f s. (In the default TeX font, the correct look is $\textit{difference}$.) The spacing is even weirder for words with ffi: $spiffier$, $\textit{spiffier}$, spiffier. $\endgroup$
    – Theo Johnson-Freyd
    Commented Jun 14, 2010 at 22:10
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    $\begingroup$ I looked at the table of contents for the book the OP linked to ("Introductory Mathematics:...") and they define things like sets, functions, injective, bijective, complex numbers, vector spaces, etc. I think it would be considerate if people kept this in mind when suggesting books. Some people have not done this, and I would agree with Mike Benfield that you shouldn't be discouraged if a randomly chosen book from the answers is too difficult to understand right now. Many of them will still be difficult after several years of studying math. $\endgroup$
    – Peter Samuelson
    Commented Jun 15, 2010 at 0:33

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I found "On Knots" by Louis Kauffman to be very inspiring when I was in high school. It's not a book to be read linearly, but rather you should hop around from section to section. As your mathematical career progresses you'll be able to understand more and more of it.

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