# What is i^i? 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 $$90\degree$$. The base number can be complex, not just real. So if the base number in this case is $$i$$, then it starts on the imaginary axis and rotates onto the real axis. $$ i^i = {\left( e^{\frac{i\pi}{2}}\right)}^i = e^{\frac{i^2\pi}{2}} = e^{-\frac{\pi}{2}} = \boxed{0.2087...} $$ Another method is by DeMoivre's Identity, $$e^{ix} = \cos(x) + i\sin(x)$$, so when $$x = \frac{\pi}{2}$$, the identity becomes $$e^\frac{i\pi}{2} = i$$. Then, the same algebra as previous follows to the same result. --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://nkintc.gitbook.io/brainless/steam/mathematics/complex-numbers/what-is-i-i.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.