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Here is a funny exercise $$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$ (If you prove it don't publish it here please). Do you have similar examples?

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closed as primarily opinion-based by Najib Idrissi, Mike Miller, Yes, apnorton, user2345215 Jan 31 at 18:16

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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Maybe a moderator should put the zeta ones together since there are three already? –  anon Nov 3 '10 at 22:29
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Perhaps this should be a community wiki question. –  Nuno Nov 3 '10 at 22:31
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This is related. –  Guess who it is. Nov 3 '10 at 22:35
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I have tripped up many calculus students with this one: $log(1+2+3)=log1+log2+log3$. I am evil... –  user641 Dec 8 '12 at 1:23
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@SteveD If only we could find an odd example... –  peoplepower Jan 13 '13 at 0:31

66 Answers 66

If we define $P$ as the infinite lower triangular matrix where $P_{i,j} = \binom{i}{j}$ (we can call it the Pascal Matrix), then $$P^k_{i,j} = \binom{i}{j}k^{i-j}$$

where $P^k_{i,j}$ is the element of $P^k$ in the position $i,j.$

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Voronoi summation formula:

$\sum \limits_{n=1}^{\infty}d(n)(\frac{x}{n})^{1/2}\{Y_1(4\pi \sqrt{nx})+\frac{2}{\pi}K_1(4\pi \sqrt{nx})\}+x \log x +(2 \gamma-1)x +\frac{1}{4}=\sum \limits _{n\leq x}'d(n)$

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An awesome pattern.

    1 x 1 = 1
   11 x 11 = 121
  111 x 111 = 12321
 1111 x 1111 = 1234321
11111 x 11111 = 123454321
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I have another one, but I'm quite unwilling to post this here because it's MINE, I haven't found it anywhere, so don't steal this.

Let us take the four most important mathematical constants: The Euler number $e$, the Aurea Golden Ratio $\phi$, the Euler-Mascheroni constant $\gamma$ and finally $\pi$. Well we can see easily that

$$e\cdot\gamma\cdot\pi\cdot\phi \approx e + \gamma + \pi + \phi$$

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$$ \int_{-\infty}^{\infty}{\sin\left(x\right) \over x}\,{\rm d}x = \pi\int_{-1}^{1}\delta\left(k\right)\,{\rm d}k $$

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Here is an Asian kid paradox - perhaps they will understand (apologies for not being strictly mathematical). If $$Study=No \;Fail$$ and $$No \; Study=Fail$$ then $$Study+No\;Study=Fail+No\;Fail$$ $$\implies (1+No)Study=(1+No)Fail$$ Cancelling gives $$Study=Fail$$ Isn't that weird???

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$1+No$ is 0 in both sides. You can't cancel zero... –  CODE Jun 4 '13 at 16:47

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