# Tag Info

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Let there be $5$ different characters $\{\_\ , a,b,c,d\}$ to form a string of length $n$. There are $5^n$ of those strings. Apart from the string of all $\_$'s $"\_\ \_\ \_\ldots \_"$, all other strings can be grouped into exactly one of four groups in this way: Take the first character along each string that is not an $\_$, which can be one of $a,b,c$ or ...

11

Using base-$5$, this is pretty obvious: $$10^n_5 - 1 = \underbrace{444\ldots4_5}_n = 4\times \underbrace{111\ldots 1_5}_n$$

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Algebraic proof: Consider the field $K:=\mathbb{F}_{5^n}$. We shall show that the group of units $K^\times =K\setminus\left\{0\right\}$ has a subgroup $H$ of order $4$. By Lagrange's Theorem, $4$ must then divide the order of $K^\times$, which is $5^n-1$. Now, this group $H$ is given by the subset $\{1,2,3,4\}=\mathbb{F}_5\setminus\{0\}$. The result ...

9

Some of my favorite results in topology are: The Borsuk-Ulam Theorem: Given a continuous function $f:S^n\to\mathbb R^n$, there exists a point $x$ such that $f(x)=f(-x)$. My favorite result is that somewhere on the planet, there are antipodal points which have the same temperature and pressure. For this reason, this is sometimes called the Meteorologist's ...

6

There are a couple of applications in PDEs that I am quite fond of. As well as verifying that the Laplace operator $-\Delta$ is positive on $L^2$, I like the application of integration by parts in the energy method to prove uniqueness. Suppose $U$ is an open, bounded and connected subset of $\mathbb{R}^n$. Introduce the BVP \begin{equation*} -\Delta ...

4

Look at the following for $n = 2$: $$\bullet \bullet \bullet \bullet \diamond \\ \bullet \bullet \bullet \bullet \bullet \\ \bullet \bullet \bullet \bullet \bullet \\ \bullet \bullet \bullet \bullet \bullet \\ \bullet \bullet \bullet \bullet \bullet$$ The black dots visualize $5^2 - 1$. Now one easily sees a division of this in sets of $4$ and therefore ...

4

I recently came across Prof. Tokieda's series of lectures on Topology at the African Institute for Mathematical Sciences and found the lectures to be very good as well as very entertaining. The first lecture covers a number of nice examples using Möbius strips.

3

The question: Prove that there are infinitely many solutions in positive integers to $x^2+y^2=z^2$ is not Pythagoras' theorem, because Pythagoras' theorem is a theorem about side lengths of right-angled triangles. Consequently, it doesn't make sense to talk about a non-geometrical proof: it's a statement about geometry. If you want to prove your ...

3

Fourteen Proofs of a Result About Tiling a Rectangle collects $14$ proofs of the fact that a rectangle tiled by rectangles each of which has at least an integer side has an integer side.

2

Robin Chapman gives many proofs of $\sum n^{-2}=\pi^2/6$.

2

Elisha S. Loomis, The Pythagorean Proposition, contains $370$ proofs of the Pythagorean theorem. ERIC has a PDF of NCTM reissue of the $1940$ second edition.

2

The book "The Fundamental Theorem of Algebra" by Fine and Rosenberger (link) contains detailed discussions of at least six proofs of this theorem, all rooted in different areas of mathematics. Links to other papers (not all in English) compiling various proofs of the theorem can be found at this MathOverflow question. H. W. Kuhn gave a combinatorial proof ...

2

Consider $$(1+x+x^2+x^3+\cdots)(1+x+^2+x^3+\cdots)$$ Coefficient of $x^k$ in this expansion is just $k+1.$ Because $$1.x^k+x.x^{k-1}+x^2.x^{k-2}+\cdots+x^k.1=(k+1)x^k.$$ Therefore $$(1+x+x^2+x^3+\cdots)^2=1+2x+3x^2+4x^3+\cdots.$$ Now evaluate (you can use the same procedure) $$(1-2x+x^2)(1+2x+3x^2+4x^3+\cdots)=?$$

2

Consider the integral: $$I = \int_0^\infty \frac{\arctan(x)}{1+x^2} dx\,.$$ By evaluating this integral through a change of variables: $u=\arctan x ,$ and $du= {1\over 1+x^2}dx,$ $$I=\int_0^ {\pi\over 2} u du= {\pi^2\over 8} \$$ Now,we notice that: $$\arctan(x)=\int_0^1 \frac{x}{1+x^2y^2} dy\,$$ Utilizing this fact, we can see: $$I = ... 2 In the n-dimensional vector space F_5^n, every non-zero vector generates a one-dimensional vector space with 5 elements. Each pair of such vector spaces intersect only at the zero vector. Therefore, F_5^n can be divided into one-dimensional sub-spaces (all containing the zero vector). If the zero vector is removed, we have partitioned remaining ... 2 "Drawing a figure" is a bit tricky for higher dimensions, but one could consider expanding an "n- cube" with "edge length" of 4 . To make it a bit easier to work with, we would extend each "edge" by  \ \frac {1}{2} \  in both "directions", rather than simply adding 1 . For a line segment , we have  \ 4 \ = \ 5 \ - \ 2 \ \cdot \ \frac{1}{2} \  , ... 2 It is more likely to find subtle, nonobvious applications of algebraic graph theory to graph theory. A celebrated early example is the Friendship Theorem https://en.wikipedia.org/wiki/Friendship_graph#Friendship_theorem which has a nice proof using the spectrum of the adjacency matrix. More recently the critical groups of sandpiles on graphs are a major ... 1 Perhaps not the most elementary, but in Hatcher's Algebraic Topology text (freely available) in the first chapter on the Fundamental Group there is an explanation of two linked circles which lets you intuitively and visually explain a fundamental group which is isomorphic to the additive group of integers (infinite cyclic). Though you might not even get ... 1 I remember watching The fault in our stars which is a great movie, but the wrong math at a very emotional scene, ruined it for me. Shailene Woodley (who has cancer) says this to his boyfriend who also has cancer- “I am not gonna talk about our love story because I can't. So, Instead I am gonna talk about maths. I am not a mathematician but I do know ... 1 Here is an opinionated and detailed 100 page Study Guide to logic textbooks, updated fairly regularly. 1 The problem is that your criteria are contradictory (and to a lesser extent, subjective): you want "the very basics," and you say you don't want a treatise on elementary number theory or (abstract) algebra, yet your very first criterion is "Show why things are the way they are." These are not mutually compatible requirements. The reason why they are not ... 1 Here is a question on this site with a whole bunch of proofs of \sum(n+1)x^n=(1-x)^{-2}. 1 Mathematics: The Loss of Certainty by Morris Kline fits your requirements. You can view extracts from the book on Amazon.com. 1$$x \to 0 , \cos x \to 1 \\\cos^m(ax) \sim 1-\frac{m*(ax)^2}{2} $$My proof for this: Based on combining Taylor expansion of \cos x and this x \to 0, so, (1+x)^m \sim 1+mx. Example for that formula:$$\lim_{x \to 0}\frac{\cos^3(5x)-\sqrt[3]{\cos(3x)}}{1-\sqrt{\cos x}}=\\\lim_{x \to ...

1

Here's a way of thinking about it. To be clear however, this does not prove the Pythagorean theorem. Given two 2-dimensional vectors $\mathbb{x},\mathbb{y} \in \Bbb R^2,$ with entries $\mathbb{x} = (x_1,x_2)$ and $\mathbb{y} = (y_1,y_2),$ we define a norm as, $$|\mathbb{x}| = \sqrt{x_1^2 + x_2^2}$$ We say two vectors are orthogonal if,  x_1y_1 + x_2y_2 ...

1

For your new question: Take triplets $(t^2-1)^2+(2t)^2=(t^2+1)^2$, in which $t\ge 2$ in order that $x,y,z\in \mathbb Z^+$

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You could try to use something like this, but as you'll see, it would be a circular argument, because in the derivation of these results the Pythagorean theorem is already used. You could just Define arc-length to be the linked expression, but then I think you'll loose your high school students.

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