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20

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.


10

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, ...


10

I read as a child The Number Devil, by Hans Magnus Enzensberger. It covers the basics, but makes them fun.


7

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.


7

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 ...


6

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, ...


6

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 ...


6

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 ...


6

One list of families of prime numbers is https://oeis.org/wiki/User:Charles_R_Greathouse_IV/Tables_of_special_primes which I have collected over several years. Most of these families are infinite though knowledge of their size is varied. Perhaps worth noting is that, for any $n$, there are infinitely many primes with more than $n$ 1s in binary, since ...


5

You really should be looking for definitions of the exponential function $e^x$, not definitions of $e$. Here are the most important ones that come to mind: $e^x$ is the unique function $f(x)$ satisfying $f'(x) = f(x)$ and $f(0) = 1$. $e^x$ is the inverse of the function $\displaystyle \ln x = \int_1^x \frac{dt}{t}$. $e^x$ is the power series ...


5

You didn't give concrete examples of the sort of arithmetic you stumble over, besides long division, so I don't know how to give you specific advice. So instead, I'll offer a few very general slogans for becoming more fluent in mental arithmetic. (I call them "slogans" because they are just that: not rules, or axioms, or anything formal. I suppose I could ...


5

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 ...


5

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.


5

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.


5

"The World of Mathematics 1 to infinity" James R Newman. Read it like mad forgetting every thing else. Tried to byheart several paragraphs.


5

Daoud is not richer than Fatma.


4

A relation over geographic regions: New York State is within the bounds of the United States. New York City is within the bounds of New York State. Manhattan is within the bounds of New York City. Manhattan is within the bounds of Manhattan; where else would it be? Because it is within New York City, it must be within the bounds of New York State, and ...


4

'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.


4

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 ...


4

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.


4

Generally what I think makes such series so good is that the format forces the authors to explain non-trivial and often non-elementary mathematics in accessible and inspiring way. The concentration of mathematically "clever" and "cool" both fascinates and challenges. They also often expose parts and perspectives of mathematics that are largely missing in ...


4

$$e=\lim_{n\to\infty}\sqrt[\large^n]{\text{LCM}[1,2,3,\ldots,n]},$$ where LCM stands for least common multiple.


4

My "theorem": The statement Everybody loves my baby, but my baby loves nobody but me is about a pair of lovers It is so simple and so obvious, even my grandma will understand it. And no matter how much you explain the simple logic calculation which shows that we are talking about a single narcissist here, half the class of first-semester logic students ...


4

There have been many category theoretic advancements, e.g., in the work on Deligne's "Weil II version" of the Riemann Hypothesis over finite fields. A reference is the book "Convolution and Equidistribution Sato-Tate Theorems for Finite-field Mellin Transforms" by Nicholas M. Katz. In general, category theory has advanced many fields, like noncommutative ...


4

I wouldn't say that either of these results is unexpected, but I agree that inverse $\tan$ wouldn't spring to mind to somebody who only knew the power rule for integration. Let's say we've got $\int \frac{1}{\color{red}{1+t^2}}\color{green}{dt}$. Let $t=\tan(\theta) \iff \color{purple}{\theta=\arctan(t)}$. Now, ...


4

There is a great deal of similarity between the two results: $\begin{align} &\displaystyle \int \dfrac{t}{t^2 + 1} dt = \dfrac{\ln (t^2 + 1)}{2} + C = \dfrac{\ln(t-i)(t + i)}{2} + C &= \boxed{\dfrac{\ln(t - i) + \ln(t + i)}{2} + C}\\ &\displaystyle \int \dfrac{1}{t^2 + 1} dt = \arctan t + C = \dfrac{1}{2i}\ln\dfrac{t-i}{t+i} + C ...


3

Let me illustrate the case you give at the beginning of the post. Each function will be developed as an infinite series built at the origin. So, $$\dfrac{1}{t^2 + 1}=1-t^2+t^4-t^6+t^8-t^{10}+t^{12}+O\left(t^{13}\right)$$ from which we can derive (just multiply each term by $t$) $$\dfrac{t}{t^2 + 1}=t-t^3+t^5-t^7+t^9-t^{11}+t^{13}+O\left(t^{14}\right)$$ and ...


3

Any real number can be computed somehow. More formally: For every real number, there exists a finite-length program that computes that number. Since real numbers are uncountable while computable numbers are countable, that just can't be the case. This limitation comes from the fact that we're stuck using finite-length programs. Infinite-length ...


3

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 ...



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