For questions concerning the creation and understanding of pictorial proofs.

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1
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4answers
96 views

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 ...
2
votes
4answers
182 views

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 ...
3
votes
0answers
110 views

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 $\sum_{n=1}^{\infty} \left(\frac 34 \right)^n$ ...
56
votes
2answers
1k views

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 ...
2
votes
1answer
84 views

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 ...
7
votes
2answers
197 views

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?
1
vote
1answer
49 views

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 ...
3
votes
0answers
215 views

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?
2
votes
1answer
97 views

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})+ ...
4
votes
0answers
722 views

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 ...
5
votes
2answers
526 views

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 ...
1
vote
2answers
143 views

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 ...
41
votes
2answers
986 views

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 ...
3
votes
4answers
261 views

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 ...