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24

The smooth structure on $\mathbb{R}^n$ is unique up to diffeomorphism, except if $n = 4$.


22

Why study set theory?      We like to think that mathematics developed from the need of our ancients to count things. I have four sheep, you have sixteen camels, my tribe has ten dozens of men, you have six hundred wives... etc. etc. But if you look closely, counting how many things you have of a certain type, first required you have be ...


20

If the $n$th Fibonacci number is prime then $n$ is prime, except that $F_n=3$ when $n=4$.


11

A generalization of the Four Color Theorem says that the chromatic number of a closed surface with Euler characteristic $\chi$ (the number of colors needed to color any map on the surface) is bounded above (sharply) by $$\left\lfloor \frac{7 + \sqrt{49 - 24 \chi}}{2} \right\rfloor,$$ except for the Klein bottle, which has Euler Characteristic zero, so that ...


11

Naive set theory: Set theory is the common language to speak about mathematics, so learning set theory means learning the common language. Another aspect is that of counting. Cardinality of sets is a very fundamental notion which be treated naively quite efficiently. Cardinality means counting, so learning set theory means learning to count (beyond the ...


8

The group of units of $\mathbb Z/p^n \mathbb Z$ is cyclic for every prime power $p^n$, except when $p=2$; then it's cyclic only for $n=1,2$.


7

$x^a-y^b=1$ has no solution in prime numbers, except $3^2-2^3$.


6

The duals of $L^p$ spaces: $$1\le p<\infty,\ \frac1p+\frac1q = 1\implies (L^p)^*=L^q,$$ but $$(L^\infty)^*\ne L^1.$$


6

$$\forall x,y \in \mathbb{Z_+}$$ $$|y^3-x^2| \ne 2 $$ except for $x = 5$ and $y=3$ Ie 26 is the only number between a square and a cube.


6

I talked about this literally today with my students, since we finally arrived to the discussion where the axiom of choice is openly assumed. First of all, in courses which are not set theory, the axiom of choice is usually chucked aside. Who didn't construct a sequence by recursion in calculus? Or proved that $f$ is continuous at $x$ if and only every ...


6

I will try to give you a VERY quick overview on the strategy of the proof of FLT. Of course I cannot avoid to use very technical tools, such Galois representation. If you don't know about these stuff, I hope you can at least follow the "shape" of the argument. The starting observation is the following: for $n\in\mathbb N$, let FLT($n$) be the statement ...


6

Project Euler: is a series of challenging mathematical/computer programming problems that will require more than just mathematical insights to solve. Although mathematics will help you arrive at elegant and efficient methods, the use of a computer and programming skills will be required to solve most problems. Solving Project Euler problems is an ...


6

Reading math books to learn math does, at least at times, feel a bit like reading a dictionary to learn a language. Linguists and polyglots will affirm that this is a bad strategy. Instead, they will recommend that you go find some people who speak the language you want, and just start talking to them! If you run into something you don't know how to say, ...


5

All spheres $S^n$ are simply-connected except $S^1$.


5

One source of free interactive demos is the Wolfram Demonstrations Project. You need to download the freely available Wolfram CDF Player if you don't have Mathematica. For very nice graphs, try the Desmos Graphing Calculator. The output looks really nice. See for instance an interactive example of drawing lines. Try also What's Happening in the Mathematical ...


5

For a list of great books, see The Mathematics Autodidact’s Aid.


4

There is no one answer to your question. It depends where you are in your mathematical journey. Take linear algebra for example. Perhaps one should begin with: (this may be too silly for some of us, not me) The Manga Guide to Linear Algebra Easy reading, fun, maybe good for highschool. A bit later, say after you've had some calculus at the university, you ...


4

Russell and Whitehead's Principia Mathematica has no formulas, but it's written in morse code. On a serious note, it's almost impossible to talk rigorously about anything but very simple mathematics without using mathematical notation at some point. Writing a math book without using mathematical notation is like writing a novel without using the letter ...


4

There is another good source for mathematics outside mathematics world: Feature Column from the AMS. One may try the following categories: History of Mathematics Math and Nature Math and Technology Math and the Arts Math and the Sciences Miscellaneous


4

The word I've most commonly seen used for a point at which a function is continuous but not differentiable is "kink", as in: "The function $f(x) = |x|$ has a kink at the origin."


4

For $\;n\in\Bbb N\;$ : $$\begin{align}&n^2=n\cdot n=\overbrace{n+n+\ldots+n}^{n\;\text{times}}\implies\\{}\\ &2n=\left(n^2\right)'=\left(\overbrace{n+n+\ldots+n}^{n\;\text{times}}\right)'=\overbrace{1+1+\ldots+1}^{n\;\text{times}}=n\implies\\{}\\ &2n=n\implies \color{red}{2=1}\end{align}$$


3

One very important subject of mathematics you must learn is Linear Algebra. Calculus is a MUST because it is the foundation for computation. I recommend this book because a provides you a structured foundation for proof strategy. It is one that all higher mathematicians (even me) have read. So, the path that I recommend to you is finish trigonometry, then ...


3

I think that number theory and group theory are two subjects that are easy enough to understand with no prior knowledge and most importantly are very good for learning styles in proof writing. Two good books. Friendly Intro To Number Theory Silverman ISBN 9780321816191 Edition 4 Book Of Abstract Algebra Pinter ISBN 9780486474175 Edition 2 I hope this ...


3

You do not need to know any calculus to do geometry. I think you are all set to read it. By the way, I am also reading Hartshorne!, but his other geometry book, also no calculus required there either.


3

Here is an extremely good source I have recently found: Panorama And in general, all the page Mathigon is outstanding well done.


3

You may be able to see some definitions of fractional derivatives and integrals with just an introductory real analysis background. But the general theory of fractional derivatives is a functional analysis topic. To get a brief glimpse into fractional integration/differentiation you can just take the various definitions provided in the Wikipedia article ...


3

For any prime $n$ except $2$, $x^n+y^n=z^n$ has no solution in nonzero integers.


3

A lot of answers to this question can be given by taking unique answers and negating them. The one that came to mind just now was "the number of critical points of a Morse function on an $n$-dimensional manifold is not equal to 2 unless the manifold is an $n$-sphere." We also have things like "a division algebra over the reals of any finite dimension except ...


3

Harmonic analysis means different things to different people, but to me it's describing the decomposition into irreducibles of the regular representation of a group $G$ on the space $L^2(G)$ of square-integrable $f:G\rightarrow\Bbb{C}$ via some appropriate "Plancherel theorem''. This may be something you're familiar with, but I'll say it anyway: the idea is ...


3

To expand on Ittay Weiss's post, and maybe this is not exactly what you are looking for, but set theory deals with the mathematical universe. More concrete, set theory has the ability to describe independence results. This, in my opinion, is the real importance in set theory. As Ittay alluded to above, consider $\mathbb{R}$. Does there exist a set $A$ such ...



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