What is i^i?

Answer geometrically by rotating an imaginary number or algebraically by an identity

This is the first homework question in my Signals and Systems class and it was a bit of a mind bender.

One of the rationalizations for this is that raising a number to a complex power is equivalent to a scaling (the real component) and rotation (the imaginary component). Using Euler's Identity, then exponentiation by the base imaginary unit is equivalent to rotation by 90°90\degree. The base number can be complex, not just real.

So if the base number in this case is ii, then it starts on the imaginary axis and rotates onto the real axis.

ii=(eiπ2)i=ei2π2=eπ2=0.2087...i^i = {\left( e^{\frac{i\pi}{2}}\right)}^i = e^{\frac{i^2\pi}{2}} = e^{-\frac{\pi}{2}} = \boxed{0.2087...}

Another method is by DeMoivre's Identity, eix=cos(x)+isin(x)e^{ix} = \cos(x) + i\sin(x), so when x=π2x = \frac{\pi}{2}, the identity becomes eiπ2=ie^\frac{i\pi}{2} = i. Then, the same algebra as previous follows to the same result.

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