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I love mathematics! Unfortunately, I don't know as much about it as I would like to. I honestly spend a large portion of my free time reading further in my Calculus textbook, and it's very interesting. The world of mathematics is truly beautiful - but unfortunately, for me, it's incomplete.

I'd like to learn more about real mathematics. When I say "real", I don't mean that parts of math aren't real; I just mean to say that mathematics in it of itself can present really puzzling challenges. Like, for example, the paradox of filling Gabriel's Horn with paint. Or perhaps, finding equations to analyze probability matrices for Markov Chains. Or even something as simple as using differential equations to describe pursuit curves.

Mathematics is all around us - but what books are there that can truly capture its meaning? What books should a high school calculus student read to learn about truly beautiful math?

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closed as primarily opinion-based by Daniel W. Farlow, Rebecca J. Stones, ASB, user147263, Claude Leibovici Mar 21 '15 at 5:27

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.

  • $\begingroup$ So you are seeking "applied mathematics" rather than "pure mathematics?" $\endgroup$ – Thomas Andrews Mar 21 '15 at 4:00
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    $\begingroup$ Might I recommend Problems for Mathematicians Young and Old by Halmos for enjoyable problemsets. Another popular book is Proofs from the book which showcases several classic proofs in many ways. If you are looking for a treatise on a new topic, then perhaps a book on number theory, real analysis, combinatorics, or topology might be enjoyable to you. $\endgroup$ – JMoravitz Mar 21 '15 at 4:02
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    $\begingroup$ $$\begin{array}{l}\text{When old age shall this generation waste,}\cr \text{Thou shalt remain, in midst of other woe}\cr \text{Than ours, a friend to man, to whom thou sayst,}\cr \text{"Beauty is truth, truth beauty," – that is all}\cr \text{Ye know on earth, and all ye need to know.}\end{array}$$ $\endgroup$ – Will Jagy Mar 21 '15 at 4:08
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    $\begingroup$ I had some friends who (as did I) read books by William Dunham (Euler: Master of us all, Mathmatical Universe) at about that time and found the style of writing very accessible and engaging. $\endgroup$ – asd Mar 21 '15 at 4:48
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    $\begingroup$ The book "$e$" The Story of a Number by Eli Maor is quite nice. $\endgroup$ – André Nicolas Mar 21 '15 at 4:52
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I happen to think Coxeter's Regular Polytopes is truly beautiful, and should be accessible enough for a well-prepared calculus student with the patience to work through the material.

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When I was in this situation, I chose Spivak's Calculus. It's a difficult book, but he motivates the material well, and the writing is very enjoyable to read. The exercises at the end of the chapters are where it really shines. They're the first time I've seen truly interesting end-of-chapter problems that I actually wanted to work in a math textbook.

Instead of doing things like calculating limits, you'll be doing things like proving the squeeze theorem. And there's a whole lot of interesting problems involving concepts I'd never heard of in my standard Calc I and II courses.

It's not for the faint of heart. But it's an excellent softer introduction to real analysis, and I learned a lot from it (and this is coming from an engineering major--the insight I got from the book, while not necessarily applicable to engineering, was extraordinarily enriching).

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For true beauty: An Introduction to the Theory of Numbers by G. H. Hardy

For realness: Real Mathematical Analysis by Charles C. Pugh

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Well I guess it all depends on what kind of math you are interested in. I would suggest, since you are a high school Calc student. Start looking at linear algebra, it is a solid base for mathematics since it is more of an introduction to proofs and vectors. After that its kind of up to you. There is so much you can venture into after that. Vector calculus, Differential Equations, or if you want to explore more theory, abstract algebra is a great starting place, you will begin to learn more about how numbers are generated. Another very interesting subject is discrete math and graph theory. Applied Combinatorics is a great area to look into and will introduce you to patterns and graphs. I hope this some what answers your question and gives you a great start!

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