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I am going overseas for the summer, I need a book or two so I can learn about mathematics (overviews, engineering applications, history, connection with other branches of science) without actually working on much of the problem during when I am transiting between cities via air or waiting for my next train or killing time in a hotel room.

In the past this has worked out pretty well for me. I read about Riemann Zeta Function with a book by John Derbyshire, I read about knot theory using Farmer and Stanford, I read about connection between mathematics and physics with Roger Penrose's Road to Reality not to mention a dozen of books on interesting historical and contemporary characters such as Ramanujan, Perelman, and John Nash which gave me motivation to study nested series, differential geometry and game theory respectively.

Since I am an engineer, I am particularly interested on gaining a subjective understanding of concepts in pure math. I wonder if there are some books out there that provides an overview to advanced concept in pure math and their potential applications without having to dig too much into the notation and underlying mechanic, just so I know that it exists and will look back more deeply when I am interested later on. But for now I am open to suggestions!

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  • $\begingroup$ Not sure how grounded you want it to be, but I recommend Godel, Escher and Bach. $\endgroup$ – muaddib Jun 8 '15 at 14:47
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    $\begingroup$ amazon.com/Verbal-Fractions-Without-Pencil-Lesson/dp/0913063150 $\endgroup$ – zeraoulia rafik Jun 13 '15 at 2:32
  • $\begingroup$ Illegal Immigrant, It's always better to do the problems, whether your just want to read them. If you want to learn more math, especially more challenging math like you're suggesting, certainly you should do the problems, sometimes just to understand them. It doesn't matter whether most of the work is in your head, if you wrote something down, you will probably retain more information than if you just read. Even a small notepad would do. $\endgroup$ – Kbot Jun 18 '15 at 1:53
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The Princeton Companion to Mathematics has all that you asked for and more, although it might be a bit large to take on a trip. The book is divided into several sections, describing the main ideas of modern mathematics, the historical development of mathematics, the lives of prominent mathematicians, important theorems and problems, and the influence of mathematics in other fields.

The Mathematical Experience has less in the way of actual mathematics, but it will give you a phenomenal sense for how research mathematics works, and hence how the people who know the most about it tend to think of pure math. This could be particularly useful for someone like yourself who (from what I can gather) comes from a background focused more on applications of mathematics.

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