The Man Who Loved Only Numbers
I really loved this one. It captures perfectly the quirkiness of Erdos, and his mathematician circles. Much more than a biography, this book popularizes a lot of incredibly interesting mathematics. It took me back 10 years to my math undergrad days.
The author does an incredible job of capturing the essence of many complex mathematical concepts in appealing and entertaining ways. It's instructive to compare the way this book tackles certain topics to the corresponding Wikipedia page, which is completely undecipherable by a non-mathematician, and often even by a mathematician that doesn't work in that domain. That said, I'm not sure that this book would be as appreciated by someone without a background in math. But it may serve as inspiration for a motivated, mathematically-inclined high school student. I'll find out!
Despite having spent 4 years taking a bunch of math courses, I learned a lot of new math from this book, and was reminded by some favorites.
Ramsey Theory: the inevitability of order in large quantities. We barely touched on Ramsey theory in 4th year Graph Theory, and it seemed really obscure. Not true! Draw any 5 points on a plane, and as long as 3 of them don't form a line, you are guaranteed to be able to form a convex quad using them as vertices. Take any sequence of 101 numbers, and you're guaranteed to find a subsequence of at least 10 increasing numbers.
Infinite series: a technical subject, but some really incredible results, like Taylor expansions of e (sum of 1/k!), pi (6 times sum of 1/k^2).
Cantor's analysis of various infinities (alpha numbers): firstly the beautiful argument about the countability of rational numbers, and then the famous diagonalization argument which proves that the set of real numbers is a bigger infinity.