Question: I'm not a real Mathematician, just an enthusiast. I'm often in the situation where I want to learn some interesting Maths through a good book, but not through an actual Maths textbook. I'm also often trying to give people good Maths books to get them "hooked".

So the question: What are good books, for laymen, which teach interesting Mathematics, but actually does it in a "real" way. For example, "Fermat's Last Enigma" doesn't count, since it doesn't actually feature any Maths, just a story, and most textbook don't count, since they don't feature a story.

My favorite example of this is "[Journey Through Genius](http://rads.stackoverflow.com/amzn/click/014014739X)", which is a brilliant combination of interesting storytelling and large amounts of actual Mathematics. It took my love of Maths to a whole other level.

**Edit:** 

A few more details on what I'm looking for.

The audience of "laymen" should be anyone who has the ability (and desire) to understand actual mathematics, but does not want to learn from a textbook. Obviously I'm thinking about myself here, as a programmer who loves mathematics, I love being exposed to real maths, but I'm not going to get into it seriously. That's why books that show actual maths, but give a lot more exposition (and much clearer explanations, especially of what the intuition should be) are great.

When I say "real maths", I'm talking about actual proofs, formulas, or other mathematical theories. Specifically, I'm not talking about philosophy, nor am I talking about books which **only** talk about the history of maths (Simon Singh style), since they only talk *about* maths, they don't actually show anything. William Dunham's books and Paul J. Nahin's books are good examples.

Answer: 
One's that were suggested to me by my Calculus teacher in High School. Even my wife liked them and she hates math now:

* [The Education of T.C. Mits: What
modern mathematics means to you](http://rads.stackoverflow.com/amzn/click/1589880331)
* [Infinity: Beyond the Beyond the
Beyond](http://rads.stackoverflow.com/amzn/click/1589880366)

Written and illustrated(Pictures are great ;p) by a couple: Lillian R. Lieber, and Hugh Gray Lieber. These books were hard to find before because they went out of print but I have this new version and like it a lot. The books explains profound topics in a way that is graspable by anyone without being dumbed down. 

[Godel's proof](http://rads.stackoverflow.com/amzn/click/1589880331) is one I enjoyed. It's was a little hard to understand but there is nothing in this book that makes it inaccessible to someone without a strong math background.

Keeping with Godel in the title, [Godel, Escher, Bach: An Eternal Golden Braid](http://rads.stackoverflow.com/amzn/click/0465026567) while not just about math was a good read (a bit long ;p).

[The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics](http://rads.stackoverflow.com/amzn/click/0060935588) It describes the Riemann Hypothesis and people who were involved with it somehow. My favorite part was learning about the people who attempted to solve it. Many I never heard off before this book. (Side not: I'll have to read pguertin suggestion, sounds in like a similar but more profound book).
Definitely, "[Elementary Differential Equations](http://rads.stackoverflow.com/amzn/click/0471319988)" by W.Boyce and R. Di Prima, as it covers a mathematical subject which is used in many applied science disciplines in a way that makes it understandable to non-mathematicians. 

**Edit** (1/2/2015): I recently got involved with mathematics of social networks. A nice introductory book for anyone interested in this fascinating topic is "[Networks, Crowds and Markets: Reasoning about a highly connected world](http://rads.stackoverflow.com/amzn/click/0521195330)" by D. Easley and J. Kleinberg. A preliminary draft version of this book is available [here](http://www.cs.cornell.edu/home/kleinber/networks-book/networks-book.pdf).