

How to explain Euler's identity using triangles and spirals - skybrian
http://docs.google.com/present/view?id=dgbfz8sm_16f84msqd2

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jonsen
The most pleasant and most informative set of slides I've seen for a long
time. Well done!

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dmolina
I adore explanations like that! I was very good in math in school, but I
always prefer this kind of explanations, that give a more intuitive vision
(and it is a lot better to explain to other people).

Thanks.

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gregfjohnson
I do appreciate this nice geometrical explanation, but the explanation of
Euler's identity via Taylor series expansion has always seemed intuitive and
reasonable to me.

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nlindig
definitely! e^(i*pi) = cos(pi) + i sin(pi) = -1.

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Qz
It's pretty awesome, but I can't help the feeling that the formula has lost a
little magic to me now...

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eru
"Poets say science takes away from the beauty of the stars - mere globs of gas
atoms. I, too, can see the stars on a desert night, and feel them. But do I
see less or more?" (Richard P. Feynman)

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Qz
Actually you're right, magic was the wrong word. Maybe it's less _mystery_ and
more _magic_.

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jonsen
Help! What do I do wrong:

    
    
      e^(πi) = -1 <=>
    
      e^(2πi) = 1 <=>
    
      ln(e^(2πi)) = ln(1) <=>
    
      2πi = 0 <=>
    
      i = 0

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lotharbot
The complex log function is a "branching function" [1] with an infinite number
of branches. For comparison, the square root function has two branches.

sqrt(x^2) has two values, +x and -x.

ln(e^x) has an infinite number of values, x + 2nπi (for any integer n).

2πi = 0 + 2nπi for some n. (You can repeat this for, say, e^4πi,
e^989781123972πi, e^-091784πi, etc. Those are all equal to 1.)

[1] <http://en.wikipedia.org/wiki/Complex_logarithm>

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vixen99
I enjoy and appreciate explanations like this but why oh why does it have to
be '<fucking> with me'? I am hoping no explanation for my comment is needed.

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eru
It's just the way the xkcd comic was written.

