So you want to read about mathematics?

There are three kinds of math books:  Problem Books, Essays about Math, and Subject-specific Texts.

In most cases, links are to the Seminary Cooperative Bookstore in Hyde Park; you can also buy these books from amazon.com, but why not support an independent store with a terrific math section?  Math books tend to be expensive, so check with your library (or Mr. K, if you're at Payton) to see if you can borrow these.

Problem Books

There are lots of collections of problems from specific contests: the Contest Problem Book series contains AHMSE/AMC and AIME problems; the ARML Contest books, etc.  Besides these...

The Art of Problem Solving volume I (easy) and volume II (more challenging) are terrific collections of problems with some text to help you learn new ideas and techniques.

Paul Zeitz's The Art and Craft of Problem Solving is similar in spirit, but a little more challenging.

Arthur Engel's Problem Solving Strategies has much less text and many more problems; it's a great introduction to Olympiad-level mathematics (and is based on the training regimen used by the German IMO team).

Mathematical Olympiad Challenges is written by Titu Andreescu, who used to coach the US IMO team, and is probably the world expert on high-level problem solving.  Like the other books, each section describes a particular idea or strategy, followed by a bunch of problems; these problems are substantially harder.  There's a new second edition that's supposed to be more accessible, but Mr. K hasn't worked on it.

Essays about Math

Martin Gardner's "Mathematical Games" column in Scientific American magazine is the gold standard for writing about mathematics.  Some of the best of these are collected in The Colossal Book of Mathematics, but really, you can't go wrong with any of his books.

Ian Stewart's books are all excellent, but two notables are What is Mathematics?, a systematic exposition of essential mathematical ideas (the first edition, by Courant & Robbins, is a classic; Stewart's update does justice to the original), and The Mathematical Maze, which is full of curiosities.  Again, however, it's hard to go wrong.

KAM-Geometry students read Dunham's Journey Through Genius, which is outstanding and quite challenging; also well worth reading are The Mathematical Universe and Euler: The Master of Us All.  KAM-Analysis students read Gamow's One, Two, Three,...Infinity, which is one of the books that fundamentally shaped the way I look at the world (and math).

I've been reading Brian Hayes's Group Theory in the Bedroom and really enjoying it; the essays connect math and science in beautiful and unexpected ways.

Finally, Douglas Hofstadter's seminal Gödel, Escher, Bach and Metamagical Themas are philosophical and mathematical both; to get a sense for Hofstadter's take on things, the second title is an anagram of Mathematical Games, which he took over from Gardner.

Subject-Specific Texts

Really, it depends on what you want to learn and how you want to learn it.  Just a couple of ideas: