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I played with imaginary numbers while in college.
I've made some mind-stretching "discoveries" over the years.
I'll post some of them here, a few at a time.
If any of you have found such "facts", I've love to see them.
Consider a line with slope $i.$
. . It would have the form: $y \:=\:ix + b$
Since $i = \text{-}\frac{1}{i}$ ($i$ is its own negative reciprocal),
. . the line $y \:=\: ix + b$ is perpendicular to itself.
Consider the circle $x^2+y^2 \:=\:\text{-}1$.
. . It has center $(0,0)$ and radius $i$.
Its area is: $\pi(i^2) \:=\:\text{-}\pi$, a real number.
Surprisingly, $i^i$ is a real number: .$i^i \:=\:e^{\text{-}\frac{\pi}{2}}$