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I'm looking for a book for someone who only knows high-school level math that will show him what math really is and how amazing it is.
Do you know a book that could do that?

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I really loved "Fermat's Last Theorem" by Simon Singh (It's called "Fermat's Enigma" in the US). – Arthur Mar 20 '14 at 18:16
@Arthur "Fermat's Last Theorem" sounds enigmatic enough as it is. :D – Sabyasachi Mar 20 '14 at 18:27
What about not presume the readers knowledge, and give them a book that teaches them maths and math history side by side. Personal favorites by stephen hawking: God created the Integers, and On the shoulders of giants. – Sabyasachi Mar 20 '14 at 18:28
It may not be exactly what you're looking for, but Douglas Hofstadter's Gödel, Escher, Bach: An Eternal Golden Braid is an awesome book that can be appreciated by high school students (also for the topic treated, ranging from formal systems and the foundations of maths to human intelligence) – Alessandro Mar 20 '14 at 19:10

I am adding two personal favorites by Stephen Hawking here.

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Most of those books are either too academic or just not catchy. Here are my suggestions that would go for both experienced mathematicians and newbies:

1) Cedric Villani: Theoreme vivant ( Also available in german. Not sure if already in english. Includes detailed parts of Villani's research (in LaTex...). But reading the story of how to approach and finally win the prestigious Fields Medal in Mathematics and looking into the head of such a brilliant mind is fascinating for non-mathematicians.

2) Logicomix: An epic search for truth ( A brilliant graphic novel about the life of Bertrand Russell, touching deep insights into real mathematics and being catchy at the same time.

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Evan a person with a short attention span ought to be able to make it through Mathematics: A Very Short Introduction by Timothy Gowers. The author is a Fields medalist.

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The books by William Dunham are very readable (but require a fair bit of work by the reader, and some background). "Euler: Master of us all", "The calculus gallery", and perhaps best for the current question "The mathematical universe" and "Journey through genius" are the ones I've read, and can personally recommend.

Very nice is Nahin's "An imaginary tale: The story of $\sqrt{-1}$".

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I don't know of a book, but I know of a simple problem in applied math that could interest your friend. He could study a very simple physical system in some configuration space, and use basic mathematics to compute all the possible directions that this system can move towards in the larger space (there will inevitably be constraints, which you can further study). This can be done with the computation of "Lie Brackets" but don't be afraid of this fancy term; it is mostly just calculus and basic trigonometry and geometry from high school.

The simplest, classic first example would be the random car-parking example. Feel free to look it up.

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