This is Real.pic.twitter.com/E5JQYz39dp
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Note though, when taking powers of complex numbers we have to be careful. Technically, if we don't restrict the range of θ to be [0,2π) then i^i takes on infinitely many values. All of which, however, are real.
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Proof that 𝑖ⁱ is real and equal to exp(-π/2). (Note: we are working in the principal branch of eⁱˣ)pic.twitter.com/OUo4TSJCDk
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The principal value has an argument in (-pi,pi].
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That's a matter of convention. It obviously is talking about the same region
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Actually, I didn't know that *all* the values of i^i were real, this is quite reassuring. It always annoyed me when ppl said that i^i is real because I would always think, 'Yeah, but that's for the one value you picked".
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Turns out it doesn't matter which one you pick, so there's a very real sense (pun accepted) in which i^i is real.
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Infinitely Real
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Question : Do we always need to use principal argument to find roots of complex number?
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