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

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.

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.

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

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

"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)

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

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.

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

Help! What do I do wrong:

  e^(πi) = -1 <=>

  e^(2πi) = 1 <=>

  ln(e^(2πi)) = ln(1) <=>

  2πi = 0 <=>

  i = 0

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

ln(e^(2πi)) is not 2πi

http://www.wolframalpha.com/input/?i=ln(e^(2%CF%80i))

It's similar mistake to claiming that sqrt(square(-2)) is -2

Logarithms are tricky with complex numbers because rotation wraps around if you go 360 degrees (2 pi i).