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# Really?

Is ii real?

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Spoiler

Yes, it is real.
In complex number analysis, exponentiation with respect to the complex numbers is a multifunction. That is, in calculating the value of ii  it can be shown to have more than one value. Yet, unlike the inverse of the exponention a real number by an real number, which is also multivalued (e.g., square-root of 4 is {2, -2}), the magnitude of a number raised to a complex value is not always the same. Using de Moivre's theorem, It has shown that the multivalues of ii are equal to e(-π/2 + 2nπ) for any integer n, with the principal value being e-π/2 where n=0. As can be noted, each of these values are real numbers. Using Euler's formula, exi =  cos(x) + i*sin(x), it can be shown that one of these multivalues is the real number of approximately 0.20787957635.

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Spoiler

Yes. From Euler's formula, i = ei*Pi/2, so (ei*Pi/2)i = e-Pi/2, a real number.

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