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We know that $i^i$ is real. But how to explain it geometrically maybe in terms of rotation. like we can explain geometrically multiplication of two complex numbers and so on. Can someone show me a little bit about geometric interpretation of $i^i$ and tell me if what I think below is true?

Note :(Some informations): It is equal to $e^{i \cdot \log(i)}$, but $\log$ is only well-defined up to adding integer multiples of $2 \pi i$. Thus in the correct setting $i^i$ is each of the numbers $e^{-\frac{(4n+1)\pi}{2}}$, $n$ an integer. Each of those are equally valid, thus finding a geometric interpretation would by a bit silly.

Thank you for any help

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Your note is precisely the point. There is a geometric meaning for addition and multiplication and real powers, but imaginary exponents don't have any because they must be defined in terms of $\exp$ and $\ln$ or something equivalent. And there is no geometric interpretation of $\exp$.

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    $\begingroup$ @zeraouliarafik: Err please read carefully, I mentioned your note in my answer! $\endgroup$ – user21820 Jun 30 '15 at 0:53
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I don't have one, but maybe this helps.

I'd like to call $z^i$ the 'semiprocal' of $z$, because the $i^{\text{th}}$ power brings us 'halfway' to the reciprocal of $z$, being $\left(z^i\right)^i = z^{-1} = \frac1z$

Likewise, $i^i$ brings us halfway to the reciprocal of $i$, being $\left(i^i\right)^i = i^{-1} = \frac1i = -i$

Knowing that $i^i$ is 'halfway' between $i$ and $-i$, it is not that surprising that the answer is purely real.

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