Really?

[bonanova]

Is iⁱ real?

[DejMar]

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, e^xi =  cos(x) + i*sin(x), it can be shown that one of these multivalues is the real number of approximately 0.20787957635.

 

[ThunderCloud]

Spoiler

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