I found abstract algebra to be really interesting. It kind of removes the concept of "numbers" and transitions it to operations on objects like sets or functions. You redefine operations like "addition" and "multiplication" to apply to these new objects, and see what happens. A whole bunch of interesting things fall out, many of them important to information encoding and error correction in practice.

For an example, see the concept of a Ring:

http://en.wikipedia.org/wiki/Ring_%28mathematics%29

Less directly related to math, I spent a joyous couple of hours back in college working through the Lambda Calculus wiki page, trying to every piece it. Once it "clicked" it became really fun to work through the examples.

http://en.wikipedia.org/wiki/Lambda_calculus

> Now I get scared whenever I see the "summation" symbol.

I'm not sure why, but I guess for me, removing the idea of "numbers" makes math suddenly fascinating. It's like dropping down a level to redefine what you thought was a well-defined concept (addition of numbers) as a generic operation for which numbers are simply one special case. Weird. :)

> book on mathematics that demonstrates the "beauty" of it

I'd say that you're asking wrong question. Yes, there're quite a few books intended to demonstrate the very thing you're asking for: beauty of mathematics. You are even likely to find some showcases interesting, might be somewhat beautiful. But essentially it won't be the same "beauty" that mathematicians see. It will be more like a thousand more-or-less scientific examples of how number φ appears in different places in our life, which isn't the essence of it obviously.

So if you're scared by "summation" symbol you're better to reconsider or forget about understanding "beauty of mathematics" forever. Because "beauty of math" is math itself.

About FFT: are you sure that you mean FFT? Because it's merely an algorithm to compute Discrete Fourier Transform. If you want to understand the Fourier Transformation itself, here's basic explanation: http://betterexplained.com/articles/an-interactive-guide-to-...

Anyway, here're some easy-to-read books about math considered good by various people:

  Hardy, "Mathematician's Apology"
  Hofstadter, "Gödel, Escher, Bach"
  Klein, "Mathematics and the Search for Knowledge" (he wrote many sci-pop books which I expect to be good, but I read only few of them, so I'll recommend only this one)
  Сонин, "Постижение совершенства" — a good one, but I don't know if it was ever translated to english, sorry