What books should a high school calculus student read to learn more about truly beautiful mathematics?

Question

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?

Answer 1

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.

Answer 2

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

For realness: Real Mathematical Analysis by Charles C. Pugh

Answer 3

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!

Answer 4

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).