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 . The base number can be complex, not just real.
So if the base number in this case is , then it starts on the imaginary axis and rotates onto the real axis.
Another method is by DeMoivre's Identity, , so when , the identity becomes . Then, the same algebra as previous follows to the same result.
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