Why is $i^i$ real? [duplicate]

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Possible Duplicate:
How to raise a complex number to the power of another complex number?

My calculator (as well as WolframAlpha) gives me the approximation:

$$0.2078795763507619085469...$$

But I don't understand how exponentiating two purely imaginary constructs yields a real (albeit irrational) number. When I do $i^{i+1}$ it gives me an imaginary number as well as $(i+1)^i$. So why does $i^i$ fall into that precise point where it is real and no longer imaginary? What is happening? I understand that exponentiation is not repeated multiplication, and it wouldn't make sense to multiply $i$ by itself $i$ times (because it would only yield $i$, $-i$, $1$, or $-1$). So what are we doing behind the scenes to get such a number?

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1 Answer

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Using Euler's formula:

$$ i = e^{i\pi / 2} $$

So:

$$ i^i = (e^{i\pi / 2})^i = e^{i^2\pi/2} = e^{-\pi/2} = 0.207... $$

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