# Tag Info

1

Somewhere between a joke and an essay, is Impure Math. http://www.snowman-jim.org/science/humor/impure-math.html

4

Here's a beautiful relation between $\pi$ and $e$, $$\sqrt{\frac{\pi\,e}{2}}=1+\frac{1}{1\cdot3}+\frac{1}{1\cdot3\cdot5}+\frac{1}{1\cdot3\cdot5\cdot7}+\dots+\cfrac1{1+\cfrac{1}{1+\cfrac{2}{1+\cfrac{3}{1+\ddots}}}}$$ It shouldn't be hard to guess who found this. As Kevin Brown of Mathpages remarked, "Is there any other mathematician whose work is instantly ...

4

I honestly just like the fact that $e + \pi$ might be rational. This is the most embarrassing unsolved problem in mathematics in my opinion. It's clearly transcendental and we have no idea how to prove that it's even irrational. They're so unrelated additively that we can't prove anything about how unrelated they are. Uh-huh. $e\pi$ might also be rational. ...

7

$$e^{\pi\sqrt{163}} =262537412640768743.99999999999925...$$

1

You may consider expressing Euler's identity as $e^{i\pi}+1 = 0$ instead of the way you have it, because 0 shows up.

14

$$\int_{-\infty}^\infty\frac{\cos x}{x^2+1}\operatorname d\!x=\frac\pi e$$ EDIT: Also: $$\int_{-\infty}^\infty e^{-x^2}\operatorname d\!x=\sqrt\pi$$

6

Some additional possibilities $$n!\sim\sqrt{2\pi n}\left(\frac {n}{e}\right)^n$$ The normal distribution is given by $$\phi(x) = \frac{1}{2 \pi}e^{(-1/2)x^2}$$ $$\int_{-\infty}^\infty\phi(x)dx=1$$ A personal favorite involving the Euler–Mascheroni constant. $$\int_0^\infty e^{-x}\ln^2x \,dx=\gamma^2+\frac{\pi^2}{6}$$ Also ...

13

I'm a fan of $$e^\pi-\pi=20{}$$(Well... almost...)

1

Cox concentrates on positive binary forms. I have given many, many answers on this site dealing with those or indefinite binary forms. Along with the first chapter of Conway, I like Duncan A. Buell, Binary Quadratic Forms. For me, the use has been that i have been able to write C++ programs to implement most of Buell. Oh, Stillwell's number theory book adds ...

1

Knuth's book: Surreal numbers builds Conways surreal number system up from the very beginning and steps through the very interesting proofs of it's basic properties. It's a bit more elementary and has a story element to it but there is certainly good mathematics in it too. Highly recommended.

1

John Conway's book: The sensual (quadratic) form is excellent and roughly fits the pattern you describe. Each chapter is about solving a new problem about quadratic forms, the machinery gets more and more sophisticated. Review: https://cms.math.ca/crux/v26/n3/page147-150.pdf

0

Relevant mathoverflow discussions: http://mathoverflow.net/questions/12709/are-there-any-books-that-take-a-theorems-as-problems-approach http://mathoverflow.net/questions/119621/learning-through-guided-discovery

0

Some obfuscation using linear algebra: Write $x + \frac{1}{x} = a$ and let $$p(\lambda) = \left( \lambda - x \right) \left( \lambda - \frac{1}{x} \right) = \lambda^2 - a\lambda + 1$$ be a polynomial whose roots are $x$ and $\frac{1}{x}$ and consider the companion matrix $$A = \left( \begin{matrix} 0 & -1 \\ 1 & a \end{matrix} \right).$$ The ...

0

Not sure how to show what the continued fraction equals but here's my attempt. Again similar in spirit to the algebraic method. Solving for $x$ in the first equation we have: $x = 4 - \frac{1}{x}$. Substituting this into the second equation: $(4 - \frac{1}{x})^{2} + \frac{1}{(4 - \frac{1}{x})^{2}} = 16 - \frac{8}{4-\frac{1}{x}} + \frac{1}{ 16 - ... 0 Here is a mildly disguised version of the same idea. Let$x=e^u$Then we have $$\cosh u=2$$ And we want $$x^2+\frac {1}{x^2}=2\cosh 2u=2(2\cosh^2 u-1)=2(8-1)=14$$ 1 Like you said, it is difficult to define what is a gueninely limitating result and what is just the discovery of. Both the irrationality of$\sqrt{2}$and the Halting problem can be stated in the form "there is no [...] such that [...]". I think any interpreted formal language is limited, and one can see maths as a formal interpreted formal language. One ... 0 Spectral theory gives a nice proof of the Plancherel Theorem. The operator studied is$L=\frac{1}{i}\frac{d}{dx}$on$L^{2}(\mathbb{R})$. The resolvent of$L$is an integral operator$$R(\lambda)f=(L-\lambda I)^{-1}f= \left\{\begin{array}{cc} i\int_{-\infty}^{x}e^{i\lambda(x-u)}f(u)du, & \Im\lambda > 0 \\ ... 0 The proof I came up with for the$L^2$isometry of the Fourier transform on$L^1(\Bbb R)\cap L^2(\Bbb R)$hinges on the$L^2$completeness of the Hermite-Gauss functions. It is well-known that these are eigenfunctions of the Fourier transform however the proofs using the Hermite-Gauss functions immediately jump to an$L^2$theory and neglect the integral ... 0 GeoGebra now includes a full Computer Algebra System (CAS) so covers (1), (2) and probably (3) of the original question :) 1 Consider${\Bbb R}^2$with a$\sigma$-algebra$\Sigma$generated by smooth curves of finite length. Construct a measure$\mu$so that$\mu(\gamma) = $length of$\gamma$if$\gamma$is a curve. This is a measure which is not$\sigma$-finite. The importance of this example is that the map$\mu:\gamma\mapsto\int_{\gamma}d\ell$is a natural consideration as a ... 1 Connect Hodge theory to Algebraic geometry, I think Lefschetz theorem on (1,1)-classes would be interesting, and a generalized one Hodge conjecture. Kodaira imbedding theorem, Kodaira Vanishing theory may also be relevant. 3 Here's one of my favorites. First, remember (part of) the more general version of the Hodge theorem: if$E_i$are a sequence of vector bundles on a compact smooth manifold, and$0 \to \Gamma(E_0) \xrightarrow{d_0} \Gamma(E_1) \cdots \xrightarrow{d_n} \Gamma(E_n) \to 0$is an elliptic complex ($d^2 = 0$and taking symbols gives an exact sequence), then the ... 2 (1) A Mathematician's Miscellany, by J.E Littlewood.A classic. Very interesting, very entertaining. (2) A Budget Of Trisectors, by Underwood Dudley. An exposition of a modern mathematician's interactions with assorted trisectors, circle-squarers,and assorted pseudo-mathematical oddballs. 1 It seems that Hartshorne gets the definition of immersion "wrong." On page 120, section II.5, he defines an immersion to be an open immersion followed by a closed immersion. One might argue that a better definition is that an immersion is a closed immersion followed by an open immersion. Hartshorne's definition creates problems, because according to his ... 4 When the fact is on unconventional mathematics, the first book that appears in my mind is Magical Mathematics of Persi Diaconis and Ron Graham. It is a book on card magics as well as rigorous mathematics, it is not simply basic combinatorics, but even the application of famous Fermat's Last Theorem in seemingly simple magic tricks! Gilbreath's Principle to ... 4 One of my favourite books is The Fascination of Groups, by F.$~$J.$~$Budden. This is certainly no ordinary introduction to group theory. Here's a sample from the preface, to give you a taste of the writing style: It takes 545 pages to cover what would be completed in most text-books in one to two hundred pages. But that is precisely its raison ... 1 Here are two answers that reached me trough other channels, one to question 1 (more examples of this phenomenon) and one to question 3 (is there a preferred choice as to which implication to prove directly and which using the other implication?). As for question 1, my friend dr. Q sent me another beautiful result about triangles called Ceva's theorem. In a ... 6 In the vein of books that meet at the intersection of art and mathematics a wonderful book that I've just discovered is: Anatolii T. Fomenko, Mathematical Impressions. The author uses detailed drawings to illustrate abstract mathematical concepts. And obviously anything by M. C. Escher. 4 Here's my list. William Dunham’s Journey through Genius: The Great Theorems of Mathematics. This is a book of gems. It mixes classical mathematical results with their background stories in a balanced manner. If I remember correctly, it contains an interesting geometric proof of Heron's formula, Cardano's solution to the cubic equation, and several ... 6 Visual thinking make me think of: Claudi Alsina and Roger Nelsen, When Less is More: Visualizing Basic Inequalities (2009) Roger B. Nelsen, Proofs without words 2. More exercises in visual thinking (2000) Roger B. Nelsen, Proofs without words. Exercises in visual thinking (1993) I am a great fan of the short-story master, Jorge-Luis Borges, who played a ... 4 Comment Gödel's Incompleteness Theorems applies under specific conditions : First incompleteness theorem. Any consistent formal system$F\$ within which a "certain amount" of elementary arithmetic can be carried out is incomplete. Thus, in order to conclude that "it show that mathematics is not unlimited, but has clear limitations", we have to make ...

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