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Of course, I am generalising here. It may have been a teacher, a theorem, self pursuit, discussions with family / friends / colleagues, etc. that opened your mind to the beauty of mathematics. But this question is specifically about which books inspired you.
For me, Euler, master of us all is right up there. I am interested in which books have inspired other people.
Gödel, Escher, Bach by Douglas R. Hofstadter
Rudin, Rudin, Rudin. Baby analysis, Real and Complex Analysis, and Functional Analysis. His terse style is like no other as he makes you work for your understanding. Moreover, the exercises are fun, hard, and instructional.
I read as a child The Number Devil, by Hans Magnus Enzensberger. It covers the basics, but makes them fun.
While I concur that Gödel, Escher, Bach is a book of great beauty about great beauty, I actually read Hofstadter's "Metamagical Themas" first and it still holds special significance for me.
As for theorems/results, I'm not really a mathematician (more statistician), so I am incapable of appreciating the beauty in Galois theory or Lie groups. On my level, Cantor's diagonalization theorem was the first result which I felt was beautiful as opposed to something I had to memorize.
As an aside, I felt I crossed a line and could consider myself a student of mathematics, instead of one who studies mathematics, when I finally "groked" the concept of proof by induction.
The books that inspired me to actually find my own mathematical voice, start exploring my own derivations, and start coming up with my own results were
Ramanujan's Notebooks by B. C. Berndt (parts 1-5)
These are amazing in that they show results starting from early high-school level explorations down to some of the most amazing modular equations one might imagine. Nearly every result has some useful insight or proof to help reveal what Ramanujan was thinking, and seeing how natural and confident the progression in results is will surely inspire.
Long before that, though, I found the cute book
Calculus the Easy Way by Douglas Downing
in the summer between 6th and 7th grade. I had never been exposed to calculus and had no idea people thought about things like that. The fact that it was an actual story was great to me, and I spent a lot of time thinking about limits and the very small region behavior of functions. Later, I found Trigonometry the Easy Way which was also instrumental in my exploring functions and their behavior and these two certainly set me on the path in special functions that made Ramanujan's work so appealing to me later on.
I think most people who get into math have that type of experience, where it isn't really one book that exposes beauty, but a series of books that steers them to understand that the beauty is greater than their naïve self would ever have imagined. It's hard for me to even put these as my answer to your question, because those books seem so far removed from Michelson and Lawson's "Spin Geometry", Cassel's "Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2", or even Yoshida's "Hypergeometric Functions, My Love", all of which have deeply influenced my view of the rich interconnections in higher mathematics and from which I regular refer and seek inspiration from.
Aigner & Zieglers Proofs from the book contains a great number of examples of the beauty of mathematics.
Du Sautoys Finding moonshine tells how beautiful doing mathematical research can be.
Fermat's Last Theorem, by Simon Singh.
Please forgive the long answer, a bit of a story here.
When I was 15 years old I ran into some problems at home, and ended up living in a very small, very remote boarding school, filled mostly with young men having problems with the law (although this was not my reason for attending). As such, many people there were particularly lacking in mathematical ability, and I was perhaps the only person there with strong ability, as well as desire to pursue mathematics, although I was only beginning my study of trigonometry. As such, there were no courses really aimed at my level. I was lucky enough that one of the mathematics instructors was willing to teach me one on one during his lunch hour. Upon completion of the state standardized test that year, my mathematics instructor informed me that there was nothing else he could teach me. I was the first, and in his opinion, likely to be the last person to ever come through that school and make it to that course, let alone through it. But the boarding school required me to take summer courses, as during no time of the year was it acceptable for students to not be enrolled in some kind of academic program.
I reached out again to my algebra teacher and he told me that he would search his home for a book that I could use to further my studies. Although there was no course at that school by the name of "calculus," he would see to it that a special case would be made so that when I was ready to leave and return home, my transcript would indicate that I began taking calculus that summer.
It took him two weeks to present to me the book that changed my life (he was a hoarder, but when he said he would search his home, he did). I was given a very old, very battered copy of George Simmons, Calculus with Analytic Geometry. Within it's pages I found everything I needed to begin to understand what separated mathematics one does at a higher level from the kind of stuff you do in most schools - the use of proofs, the abstract thinking that goes into problems, the generalized approaches that lead to solutions to many problems. This book was well beyond anything I had ever found before, and it gave me inspiration to study mathematics beyond high school, and to stop doing the stupid things that I was doing to end up in that place so that I could focus my energy on the pursuit of knowledge. In that summer I probably completed about half of the book, and had a cursory knowledge of multivariate calculus within a year and a few months. Mr. (Dr?) Simmons allowed me to fall in love again with learning, and my algebra teacher's letter of recommendation pretty much secured my place at Stony Brook, where I currently study chaotic and dynamical systems, which is very closely related to calculus.
Between this teacher who had enough faith in me to teach me trigonometry one on one, and this excellent book which guided me through the rest of high school, I completed high school a year early, got into the university of my dreams, and have found a sort of calling in life - not only to do research and continue learning for its own sake, but to spread this passion in others, so that just maybe I can help someone else lost or stuck in a rut to find something to be passionate about. He let me keep that book. As a TA this semester, it's my number one reference, and I'd recommend it to any new student.
Well, I've been reading Pinter's A Book of Abstract Algebra and it's quite good. It really tries to bring out the intuitive meaning of the concepts covered, and really it's the best mathematical reading I've happened to stumble across. It's also the first mathematical book I've read for my own self-study, after the recommendation of both a teacher and a fellow student. When he spends time to reflect on the elegance and profundity of what is implied, it really shines.
Spivak's Calculus. It didn't really initiate my interest in math per se, but it was the first proof-based math book I read, and it arguably helped me get interested in that stuff without being too over my head. Rudin's Principles of Mathematical Analysis was also great, though I didn't read much of it.
The books of Martin Gardner. These "recreational math" books are not just great collections of puzzles and activities, they also include very accessible introductions to ideas from the whole spectrum of modern mathematics. Gardner makes more advanced topics hands-on and hints at tantalizing higher level connections in his puzzles. And they're very entertaining to boot.
I had several as a kid - I think they were The Unexpected Hanging and Other Mathematical Diversions, Mathematical Circus, and The Scientific American Book of Mathematical Puzzles and Games.
W. W. Sawyer's 'Prelude to Mathematics' is a great book that really opened my eyes. It can be read almost without any knowledge of mathematics.
I also believe that any mathematician ought to read 'Flatland'. It is a beautiful story it gave me my fist real intuition about higher dimensions.
From my recent experiences, Milnor's Topology from differential viewpoint; short, understandable, easy to read and containing surprisingly many ideas on 60 pages.
The two pages in the 1960 Compton's Encyclopedia article on "calculus": the left page was essentially on derivatives, the right page on integrals. A few excellent pictures. It showed a few of the many amazing things one can do with calculus, and made it appear obvious, simple, unburdensome. Two pages, simple ideas with huge potential. Wonderful. Yes, mathematics is powerful and magical.
(In contrast, any book that makes calculus difficult is inimical. Long books on calculus? Terrible. Too bad calculus has become a filter/weeder subject. Perverse.)
When I was about 12 years old, I read Euclid. The way the theorems were arranged, the way Euclid had proved the theorems were very beautiful to me. I didn't know that something so beautiful could exist in the world.
Burton's book Elementary Number Theory was the next book that I have regarded as a beautiful one. The books that I have read about number theory before that were very neat and rigorous in their treatment of topic but Burton's book not only treated the subject thoroughly but also gave historical introduction before each chapter. That was what made it a artist's work! I thoroughly enjoyed reading the book.
Landau's Foundations of Analysis is also a very inspiring book.
For me, the book was one I bet nobody else has heard of, the Engineering Mathematics Handbook of Jan J. Tuma. It is a reference work, with page after page of identities and formulas, organized by subject, and accompanied with hundreds of handsome two-color diagrams. It is aimed at the practicing engineer, who doesn't want to have to remember or derive the Taylor expansion of $\sinh x$; he wants to look it up. But that was not how I used it.
The first page starts with algebraic identities, such as the commutativity of addition, and then it works its way up through geometry, trigonometry, calculus, and differential equations, with excursions along the way into probability theory, statistics, and engineering subjects such as mechanics.
I spent hundreds of hours with this book, tinkering with the identities and seeing how they related. For example, there are two pages that tabulate the derivatives of common functions. One of these is $$\left(u^v\right)' = vu^{v-1}u' + u^vv'\ln v$$ and by substituting various choices of $u$ and $v$ you can see how the more familiar examples $$\begin{align}\left(a^x\right)' & = (\ln a)\cdot a^x\\ \left(x^n\right)' & = nx^{n-1}\end{align}$$ (which are also tabulated) are special cases of the general one.
Every entry was a puzzle to be solved. Page 85 contains a table of the derivatives of the elementary trigonometric functions, including a bald assertion that $(\sec u)' = -\frac{u' \sin u}{\cos^2 u}$. Well, I knew the definition of $\sec x = \frac{1}{\cos x}$; could I work out the correct derivative myself, using the chain rule, and would it be what the book said? Yes, I could! It was! But wait, someone else told me the derivative was $-\sec x\cdot \tan x$. Oh, but that is when $u=x$, and yes, it checks out.
There is a section with diagrams of geometric figures, and the positions of their centroids, their moments of inertia, and so on. This was preceded by a page that gave the general formula for the centroid. If I applied the definition and carried out the calculation of the centroid of a parabolic segment, would I get the answer that was written next to the picture of the parabola? Yes! Could I also calculate the moment of inertia? No! (I never did figure it out, but I tried hard.) Another page gave the classification of quadratic curves by shape. If I put the coefficients of $xy-1=0$ or $x^2+y^2=25$ into the formulas, do they correctly identify the curves as a hyperbola and a circle? (They did!) Was the classification invariant under translations and rotations? (It was!) How did it work? (Parts were clear, other parts less so.) How does it treat degenerate cases, such as $x^2+y^2 = -25$? This is how I learned that quadratic curves have a hidden complex part.
Some pages were just mysteries. The definition of a hermitian matrix was presented without comment. The meaning was clear, but the purpose was not. I didn't find out why Hermitian matrices were interesting until thirty years later. There were tables of Laplace transforms. What were they for? But they were mysteries that exercised my mind, and when I encountered the answers, I smiled and said “Oh, so that's what that is about!”
The book would not have inspired many people, perhaps, but when I met it, it was the right thing at the right time, and I loved it then and still do.
The Music of the Primes by Marcus du Sautoy is an amazing book on the history of number theory and more particularly on the Riemann's hypothesis. Number theory being an abstract field of mathematics, du Sautoy describes its beauty in an artistic and almost poetic way. It is truly fantastic and surprisingly well-written.
For me, it's got to be Richard Courant's What is Mathematics. The way it showed why exactly trisection of an angle or doubling a cube is impossible using number fields. That proof just blew my mind. I actually bought it on an impulse, but that's got to be the best impulse purchase I've ever made.
"The World of Mathematics 1 to infinity" James R Newman. Read it like mad forgetting every thing else. Tried to byheart several paragraphs.
I can't exactly name one particular book that did it, but one that might be a contender is The M$\alpha$th Book By Clifford A. Pickover. Several pages were filled with beautiful renderings of fractals, and page $166$, for example, contains a description of the mathematical beauty of Euler's number, $e$. I could go on for hours about this book.
Edit: Italics not working, now removed.
'Science and Sanity: An Introduction to Non-Aristotelian Systems and General Semantics' by Alfred Korzybski; it not only opened my mind to the true value and beauty of mathematics, but also connected it to many other areas of science/philosophy.
While searching for a project for a CS discrete math class, the book Continued Fractions by C.D. Olds literally fell from a used bookstore shelf in front of me. This little monograph works through the basics of Pell's equation and infinite repeating continued fractions. The main result is rather neat and thus probably misled me regarding the nature of doing mathematical research.
Mathematical Constants by Steven Finch. This book evolved from a website that the author maintained in the 90's. I spent many hours looking around there, and learned of things like Khinchin's constant. The book is even better.
Finally, as an undergrad I used to poke around The Mathematical Atlas, maintained by Dave Rusin in the 90's. It included posts from the sci.math newsgroup like this one on cross products. An interesting glimpse into interplay between mathematicians, students, amateurs, and cranks. Edit: I should clarify that sci.math had its cranks - Rusin's site was strictly educational.
In my case it was Chebyshev and Fourier Spectral Methods by John Boyd. A great book! You feel like reading a novel while learning about approximation by polynomials. Reading this book gives you a feeling like you are having a conversation with the author.
Got me interested in Spectral methods and helped me learn to think spectrally.
From the Preface:
"The style is not lemma-theorem-Sobolev space, but algorithm-guidelines-rules-of-thumb.";
"The writing style is an uneasy mixture of two influences. In private life, the author has written fourteen published science fiction and mystery short stories. When one has described zeppelins jousting in the heavy atmosphere of another world or a stranded ex- plorer alone on an artificial toroidal planet, it is difficult to write with the expected scientific dullness."
In my case, it would be Figuring: The Joy of Numbers by Shakuntala Devi. I read this when I was in elementary school, and I regarded it as something mysterious and wonderful, with the seeming magic of the tricks contained within.
I remember answering a similar question. I wrote Mathematics and Imagination which I bought from an used book store. For the apt and only reason being, one, especially, a high school graduate does not necessarily associate "Creativity" and "Imagination" with mathematics. It forced me to rethink the subject.
Although I don't rate highly of that book, but I have been influenced by Metamath a lot from Gregory Chaitin.
Also I remember reading a Bengali book called "Manojder Odhbhut Bari" by Calcutta author Shirshendu Mukhopadhya. There one math dude solves complex mathematical multiplication in his head while traversing underwater. Neat!
Excursions in Number Theory was a beautiful book. Down to earth enough for the layman, but substantial enough for someone of intermediate mathematical skill. I was reading it while working my way through an open course on Quantum Mechanics. I'll never forget the moment when I realized that every multiplication I had ever done was a special case of matrix multiplication.
Oliver Byrne's rendition of the Elements of Euclid has to be on this list.
Ok, technically it is geometry rather than mathematics... but it is one of the most beautiful books I own for sure!
