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: 
I've been successful in using Courant and Robbins' **[What Is Mathematics? An Elementary Approach to Ideas and Methods](http://rads.stackoverflow.com/amzn/click/0195105192)** for adults who have not had a math class for a few decades, but are open to the idea of learning more about mathematics. 

![](https://i.stack.imgur.com/slRn1.jpg)

Some sections are too advanced for someone with only high school mathematics, and many more will appear that way to the person at first, but do not actually rely on anything beyond high school mathematics.
I've read and enjoyed some of the titles mentioned by Dunham (who balances motivations and mathematics really well; Euler: The Master of Us All was particularly good) and Stewart. Maor and Nahin also have some decent accounts. The former's book Trigonometric Delights sparked an interest in me on the history of maths back when I read it during high school, but it doesn't shy from actual derivations and mathematical reasoning.

Possibly the best book I've come across of this type however is Julian Havil's [Gamma: Exploring Euler's Constant](http://rads.stackoverflow.com/amzn/click/B009P09JLM). It was one of the first such accounts I'd read, and revisiting it recently I found it just as informative as ever. In place of biography alone (though there are plenty of fascinating historical and anecdotal titbits), Havil investigates connections through mathematics via the more mysterious of 'Euler's constants' (the Euler-Mascheroni constant as it's called). It's somewhat like Nahin's book on $i$, but Havil's treatment is more cautious and farther-reaching, if slightly more demanding.