Questions tagged [proof-without-words]
For questions concerning the creation and understanding of pictorial proofs.
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Is it possible to explain geometrically why $\arctan (1/2) +\arctan (1/3) = 45$ degrees?
$\arctan(1/2)$ seems to be some strange, irrational angle, and the same goes for $\arctan(1/3)$, but those two angles seem to sum up to $45$ degrees. This seems like a mystery to me even though I can ...
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Geometric proof that $1 + 2+ 3 + \dots + n=\frac{(n+1)n}{2}$
I understand that the sum $1 + 2+ 3 + \dots + n$ takes the form $\frac{(n+1)n}{2}$.
This can be shown symbolically: $1+2+3+\dots+n$, written backwards is $n+(n-1)+(n-2)+(n-3)+\dots+3+2+1$. If you ...
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Showing in general way that three square pyramids can combine to form a cuboid having same dimensions as square pyramid Visually
I know that volume of a square base pyramid of bottom square area $S$ and height $H$ is given by $\frac{1}{3}SH $ , and volume of a square base cuboid outside having dimensions exactly that to be $SH$ ...
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Can you calculate $1+2+3+4+\ldots+k$ geometrically?
I am learning Integrals from What is the sum $1+2+3+4+\ldots+k$? - Week $11$ - Lecture $2$ - Mooculus
and I still don't get how to calculate $1+2+3+4+\ldots+k$ geometrically:
According to the video, $...
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A Proof with no words that $\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}=2$
Question
What are the words to describe the method in the image below? (from Nelsen's Proofs without Words II)
Attempt
I was thinking and could define the sequence $u_1=2; u_{n+1}=f\circ g^{−1}(u_n)$ ...
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How to prove that $(\sum_{i=1}^n a_i)(\sum_{i=1}^n b_i)= \sum_{i,j} a_ib_j$? [closed]
How to prove that $(\sum_{i=1}^n a_i)(\sum_{i=1}^n b_i)= \sum_{i,j} a_ib_j$? Is there any way to visualize the sums on both sides.
user907912
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What is a cubic triangular number and what is a pictorial justification why a number is classified as such?
Can someone help me understand the question posed, what is a cubic triangular number, and what is the spatial reason why a number is classified as such?
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Can all mathematical proof be represented visually? [closed]
Can all mathematical proof and all mathematical be represented visually?
When defining visually I mean "could potentially be expressed visually", such as graphically or geometrically so the ...
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Intuition for reflections/symmetry, shifts, parity, and periodicity of sine and cosine functions at allied angles 0, $π/2$, $π$, $3π/2$, and $2π$
I have proved the table given above, for example, $\cos \left( \dfrac{\pi }{2}-\theta \right) =\sin \theta $ and $\cos \left( \dfrac{\pi }{2}+\theta \right) =-\sin \theta $ using sum/difference ...
user845875
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Geometric proof in Concrete Mathematics
On page 32 of Concrete Mathematics by Graham, Knuth and Patashnik, they demonstrate that the sum of a geometric progression is
$$
\sum_{k=0}^n a x^k = \frac{a-a x^{n+1}}{1-x}.
$$
In the margin next ...
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Solution of linear equation obtained by partially differentiating a second degree general equation in two variables
Let's consider a second degree equation in two variables say,
$$ S\equiv ax^2+by^2+2hxy+2gx+2fy+c=0 $$
where $a, b, c, f, g$ and $h$ are constants. To be clear, I'm dealing with conic sections here.
...
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Prove that $\forall n \in N $ , $ \sum_{i=1}^n i = \frac{n(n+1)}{2}$, counting in two ways the number of shaded squares in the diagram
Prove that $\forall n \in N $ $$ \sum_{i=1}^n i = \frac{n(n+1)}{2}$$ counting in two ways the number of shaded squares in the diagram.
I have been thinking about this, but I can't understand how ...
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How to prove a tiling of a hexagon must form a 3D cubic stack?
How to prove that a tiling of a big hexagon consisting of triangles, using only $2$-triangle tiles (three possible orientations), must resemble a continuous, convex (for each small cube), manifold, $3$...
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Can someone explain this “visual proof” of $(1 + 2 + 3 +…+ n)^2 = 1^3 + 2^3 + 3^3 +…+ n^3$?
I see a lot of people saying this visual proof is beautiful and I really want to be able to understand it. If anyone could help me out, I'd really appreciate it! Thanks in advance!
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Is there a graphical proof that $9n+1$ is triangular if $n$ is?
It is straightforward to prove that if $n$ is a triangular number then $9n+1$ must be.
Is there a systematic decomposition of a triangle made of $9n+1$ dots into nine triangles of $n$ dots plus one ...
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Geometric proof that the product of the $x$-intercepts equals the $y$-intercept for a monic quadratic
I know you can prove that the product of the roots of the monic quadratic $x^2+a_1x+a_0$ equals the $y$-intercept $a_0$ by comparing its coefficients to the coefficients of $(x-m)(x-c)$ where $m$ and $...
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Regular hexagon divided into triangles
Problem: Give an regular hexagon and an interior point of this, join this point with each vertex. The hexagon is divided in $6$ triangles, paint the triangles alternately. Show that the sum of the ...
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Prove $\sum_{n\geq1}\frac{2^n (1-\cos\frac{x}{2^n})^2}{\sin\frac{x}{2^{n-1}}}=\tan\frac{x}{2}-\frac{x}{2}$
How to prove for $|x|<\pi$:
$\sum_{n\geq1}\frac{2(1-\cos(\frac{x}{2^n}))}{\sin(\frac{x}{2^{n-1}})}=\tan(\frac{x}{2})$
$\sum_{n\geq1}\frac{2^n (1-\cos(\frac{x}{2^n}))^2}{\sin(\frac{x}{2^{n-1}})}=\...
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geometric series diagram query.
I was going through this article on geometric series on Wikipedia and found this diagrammatic representation of an infinite geometric series with a said common factor of (1/2) but shouldn't the common ...
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How to prove an equilateral triangle ABC having first brocard point with angles 30, 30, 30
Can you please help me prove an equilateral triangle having first brocard point with angles 30, 30, 30, I tried to draw the shape and to write that Cot 30= cot
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An intuitive approach to a Lemma related to subgroups of quotient groups.
Consider the following Lemma:
Let $K$ be a normal subgroup of $G$ and $T$ be a subgroup of $G/K$,then there exists a subgroup $H$ of $G$ such that $T=H/K$. (Of course, it is automatically implied ...
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Are pictures legitimate as a proof in mathematics?
While I'm studying Topology (teaching it myself with videos and books) I've seen some 'proofs' with pictorial approach and solution, I haven't seen it before. So is it legitimate?
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Is there a geometric proof of $\frac1r = \frac{1}{h_a} + \frac{1}{h_b} + \frac{1}{h_c}$ in a given triangle?
Can anyone provide a geometric proof of $\frac1r = \frac{1}{h_a} + \frac{1}{h_b} + \frac{1}{h_c}$ in a given triangle, if there is one? Here, $h_i$ refers to the distance from the side $i$ to the ...
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Is there anything missing in this proof?
I came with this geometry problem and numerous lengthy solutions were proposed, so I thought there must be something missing in my solution.
The problem: Given that $\angle CAB=3x$, $\angle BCA=x$, ...
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Is there a dissection proof of the Pythagorean Theorem, such that a single cut makes the needed pieces from the squares?
Can you cut the squares of a right triangle with one cut (cutting all three pieces at the same time) so that it’s possible to arrange them and prove Pythagoras’ theorem?
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Is there a sufficiently reachable plausibility argument that $\pi$ is irrational?
I was teaching someone earlier today (precisely, a twelve-year-old) and we came upon a problem on circles. Little did I know in what direction it would lead. I was able to give a quick plausibility ...
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Quadrilateral in a Parallelogram - Interesting Proofs!
Here's an interesting problem, and result, that I wish to share with the math community here at Math SE. I think I've found a proof without words...
I came across this problem sometime back, and here'...
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Geometric proof of a trig identity on $\cos t \cos u\cos v$
Consider the following trigonometric identity, valid for any set of angles $u,v,t$:
$$\cos t⋅\cos u⋅\cos v =\frac14\left[\cos(t + u + v)+\cos(t + u - v)+\cos(u+v-t)+\cos(v + t - u)\right]$$
This ...
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Proof without words: $1+8\times\text{triangular number}$ is an odd perfect square
A recent question asked how to show that $8T_n+1$ is a perfect square if $T_n$ is a triangular number. This follows immediately from $T_n=\frac12 n(n+1)\implies 8T_n+1=4n^2+4n+1=(2n+1)^2$.
Can this ...
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Proof of four triangles with equal area.
I have been working on proofs without words but I have a bit problem about Steven Snovers proof. It proves that these four triangles have equal area. Proof comes from rotation but I couldnt get it.
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Proof without words for Stirling formula
It it easy to prove $$x^n=\sum_{i=0}^{n}S(n,i)x_{(i)}$$
by induction or recursive formula...
But is there some proof without word for it? or at least a direct proof that concern with partitions of ...
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What is the explanation for this visual proof of the sum of squares?
Supposedly the following proves the sum of the first-$n$-squares formula given the sum of the first $n$ numbers formula, but I don't understand it.
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To prove $\frac{n!}{p!q!}$. [closed]
The number of permutaion of n objects, where p and are of one kind,q are of second kind,are of different kind is $\frac{n!}{p!q!}$
How can we proove above theorem $\frac{n!}{p!q!}$.I tried to prove ...
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Examples of 3d visual proofs
I am looking for examples of three dimensional constructible proofs. By this I mean activities such as steps in proving $1^2+2^2+\cdots+n^2=n(n+1)(2n+1)/6$. In this construction the identity is proven ...
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Geometric proof of $QM \ge AM$
Prove by geometric reasoning that:
$$\sqrt{\frac{a^2 + b^2}{2}} \ge \frac{a + b}{2}$$
The proof should be different than one well known from Wikipedia:
DISCLAIMER: I think I devised such proof (...
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Using a visual "proof" to show that $\sum_{n=1}^{\infty} \left(\frac 34 \right)^n =1$
The "proof without words" that $\sum_{n=1}^{\infty} \left(\frac 12 \right)^n =1$ is fairly well known:
But why can't we apply the exact same logic to $\sum_{n=1}^{\infty} \left(\frac 34 \right)^n$ or ...
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Visual proof of $\sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}$?
In his gorgeous paper "How to compute $\sum \frac{1}{n^2}$ by solving triangles", Mikael Passare offers this idea for proving $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$:
Proof of equality ...
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On equilateral triangles
I came across the following geometry problem.
In the exterior of a triangle $ABC$ three equilateral triangles $ABC' , BCA'$ and $CAB'$ are constructed. Prove that the centroids of these triangles ...
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Explain this calculus proof without words
This demonstrates that $\int_0^1 t^{p/q} + t^{q/p} dt = 1$. Could you please explain how the proof without words shows that?
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I can't figure how to prove this coordinate algebraic geometry proof.
Consider the points $A$ and $C$ are on $y=x^p$. We are told that point $A$ has coordinates $( a, b )$, where $0<a<1$ and point $C$ has coordinates $( c, d )$, where
$1<c$. Give a ...
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Looking for proof-without-words of Bezout's identity
I'm looking for a "proof-without-words" of Bezout's identity (for integers).
Does anyone know of one?
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Does $\operatorname{MSE}(\hat{\theta}) = \operatorname{Var}(\theta)+ \left(\operatorname{Bias}(\hat{\theta},\theta)\right)^2$?
We know that
$\operatorname{MSE}(\hat{\theta})=\operatorname{E}\left[(\hat{\theta}-\theta)^2\right]$ and
$\operatorname{MSE}(\hat{\theta})=\operatorname{Var}(\hat{\theta})+ \left(\operatorname{Bias}(\...
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Why matrix representation of convolution cannot explain the convolution theorem?
A record saying that Convolution Theorem is trivial since it is identical to the statement that convolution, as Toeplitz operator, has fourier eigenbasis and, therefore, is diagonal in it, has ...
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Intuitive/Visual proof that $(1+2+\cdots+n)^2=1^3+2^3+\cdots+n^3$ [duplicate]
$$(1+2+\cdots+n)^2=1^3+2^3+\cdots+n^3$$
I noticed this only because $\displaystyle \sum_{i=1}^n i = \frac{n(n+1)}{2}$ and $\displaystyle \sum_{i=1}^n i^3 = \frac{n^2(n+1)^2}{4}$.
But the two things ...
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If $Ax=b$, for $b\ne 0$, has more than one solution, then $Ax=0$ does as well. T or F. Prove this.
I get that this is true, because there's one free variable, so no matter what the augmented matrix is, there always will be an infinite amount of solutions. Right? But how to I explain this as a proof?...
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Geometric interpretation for sum of fourth powers
Summing the first $n$ first powers of natural numbers:
$$\sum_{k=1}^nk=\frac12n(n+1)$$
and there is a geometric proof involving two copies of a 2D representation of $(1+2+\cdots+n)$ that form a $n\...
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Proof-without-words for $\bar a\times (\bar b\times\bar c)=\bar b (\bar a\cdot\bar c)-\bar c (\bar a\cdot \bar b)$ or some visual-biased explanation?
Griffiths' Introduction to Electromagnetism -book has equations called 20.10 below.
I have proved this equation d) pretty much on the first mathematics -course I had but I have not yet understood a ...
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