Fun, interesting, slightly advanced books

I came across a really interesting thread in the Internet where the author was asking for fun, but serious Maths book recommendations. I saw plenty of excellent books being recommended there and thought I'd start a similar thread over here :)

Some characteristics the books should have

1. They should be readable and friendly.

2. They should be dealing with topics that are not commonly taught at the school or undergraduate level.

I don't want books dealing with very commonly taught topics like differential equations, or Calculus by Spivak, which though elegant is not what I'm looking for.

Here are some examples of books I liked to make myself clearer:

1. Visual Complex Analysis by Tristan Needham - I like this book for the simplicity and the sheer beauty of it all.

2. Cauchy-Scwhartz Inequalities by Micheal Steele - Mathematical inequalities are not a very advanced topic, yes. But, it's also not taught extensively.the author describes this book as a master class where mathematicians come to fine tune their skills, and that's exactly what it is. I loved it.

3. Concrete Mathematics by Graham Knuth and Patashnik - Again a very readable book. It doesn't deal with mathematics that is very advanced, but then again it gives a very beautiful and new treatment to it.

   4. Flatterland _ Ian Stewart A wonderfully readable book.

I hope this gives you a clear idea. If you have any questions regarding what I'm looking for, you could ask in addition to your recommendations. I don't mind books on problem solving and the history of mathematics too.

Solution 1:

School level:

1. Intuitive Topology, V. V. Prosolov.
2. Stories about Maxima and Minima, V. M. Tikhomirov.
3. Mathematics can be Fun, Y. Perelman.
4. What is Mathematics?, Courant and Robbins.

College:

1. Proofs from the Book, Aigner and Ziegler.
2. On Numbers and Games, Conway.
3. The sensual (Quadratic) Form, Conway.

General:

1. A Mathematician's Apology, G. Hardy.
2. Logicomix: An Epic Search for Truth, A. Doxiadis.

Solution 2:

Cited several times on Math.SX, Street-Fighting Mathematics, by Sanjoy Mahajan, can be an example of fun but serious Maths book.

Solution 3:

I recommend Quantum Computing Since Democritus by Scott Aaronson.

It's an informal, fun, very enlightening overview of many topics related to computational complexity and quantum computing.

Some samples that give a bit of the flavor of the book:

From p. 110:

Quantum mechanics is what you would inevitably come up with if you started from probability theory, and then said, let's try to generalize it so that the numbers we used to call "probabilities" can be negative numbers. As such, the theory could have been invented by mathematicians in the nineteenth century without any input from experiment. It wasn't, but it could have been.

From p. 19:

First, though, let's see how the incompleteness theorem is proved. People always say "the proof of the Incompleteness Theorem was a technical tour de force, it took 30 pages, it requires an elaborate construction involving prime numbers," etc. Unbelievably, 80 years after Godel, that's still how the proof is presented in math classes!

Alright, should I let you in on a secret? The proof of the Incompleteness Theorem is about two lines. It's almost a triviality. The caveat is that, to give the two-line proof, you first need the concept of a computer.

From p. 140:

From my perspective, Richard Feynman won the Nobel Prize in Physics essentially for showing BQP is contained in PP.