Books of interesting mathematics aimed at mathematicians

Question

I'm currently in my third year of an undergraduate degree, mainly doing courses in abstract algebra. I'm enjoying my degree, but every so often when I'm swamped with difficult proofs, I feel like I need some extra inspiration to give myself more mathematical energy and remind me why I got interested in maths in the first place - otherwise I wear myself out. I do things like watching videos on numberphile, but I feel like this is aimed at a much lower level than I'm at.

My question: What is a good resource that features advanced mathematics and isn't shy about going quite deep, but is still fairly recreational in its nature? I don't really care what form this resource takes, it could be blogs, books, videos etc.

Any help would be greatly appreciated.

From one udergrad student to another - i thoroughly enjoyed Godel Escher Bach (Douglas Hofstadter). I can not give this book enough praises. — Oria Gruber, Feb 24 '15 at 22:57
Have you seen Basic Notions of Algebra by I. Shafarevich? Here's an MAA review. "...It's all too easy, when teaching or learning algebra, to lose sight of the forest. I can't imagine a better way to reorient oneself than reading through this book. Shafarevich provides reliable and interesting guidance, and he often has something new or unexpected to show us." (Fernando Q. Gouvêa) — Bill Dubuque, Feb 24 '15 at 23:09
I recommend the book Quantum Computing Since Democritus by Scott Aaronson. — littleO, Feb 26 '15 at 21:52
Thanks for all the answers! I have a lot of reading to do now :) — James, Mar 01 '15 at 15:28

score 5 · Answer 1 · answered Feb 25 '15 at 16:58

The Princeton Companion to Mathematics by Tim Gowers et al. is an astounding work. It has an entry for almost any mathematical topic you can think of, and you can get an feel for the flavor of all the areas of mathematics via the typically lucid, not-extremely-technical (for an undergraduate with some mathematical background, anyway) expository articles on the major topics. It's also full of biographies of mathematicians, historical exposition, and essays by leading mathematicians on doing and loving mathematics. Highly recommended.

score 4 · Answer 2 · answered Feb 24 '15 at 23:24

I really like John Conway's Symmetries of Things, which you'll be in a great position to appreciate, taking abstract algebra. It's divided into three sections, the first of which is nice and easy, and things escalate quickly. It's an odd, but good, mix between pop math and a classification of symmetry groups in small dimensions (cylinders, the plane, $3$- and a bit of $4$-space).

Journey through Genius by William Dunham is a fun book that's much more laid back, but still full of very good mathematics, and the mathematicians known for each topic.

Finally, for a very laid back book, I've come across two biographies of Paul Erdös. I know one is called The Man Who Loved Only Numbers by Paul Hoffman. I can't remember the other, but they were both good, even after I'd read one, so either should be a fun read. I don't think you'll learn much math here, although you may learn of some theorems/topics.

@RobertCardona Yes, that's the one! Now I can't recall which was better, they were both quite entertaining, thank you! — pjs36, Feb 25 '15 at 23:12

score 4 · Accepted Answer · 2015-02-26T22:25:55.343

My personal favorite actual math book in the context of your question is Silverman and Tate "Rational Points on Elliptic Curves."

Another excellent math-type book is Havil "Gamma Exploring Euler's Constant."

On a less technical level, here is a list from the Cambridge Math Dept. with an excellent selection:

http://www.maths.cam.ac.uk/undergrad/admissions/readinglist.pdf

score 1 · Answer 4 · answered Feb 25 '15 at 16:44

Following books may help you: 1.One,Two,Three....Infinity- by George Gamow

2.Fun and fundamentals of Mathematics- by Jayant V Narlikar and Mangala Narlikar.

These books take you through basic concepts of mathematics to the complex ones unveiling the beauty of mathematics.

Books of interesting mathematics aimed at mathematicians

4 Answers4