But it drives home the reason why I parted ways with advanced math. There just came a point when the math no longer resonated with the natural world that I was comfortable living in.
Imaginary (and complex) numbers were the first real schism where my mind repelled the very idea.
Nonetheless, decades later, I would be talking with a friend who is much smarter than me. We were talking about the lossy nature of converting audio from the time domain to the frequency domain and back. I understood that phase information was lost in the translation (I also thought it was curious that the human brain seemed to be phase-agnostic and so could not perceive the difference when audio was round-tripped through the frequency domain).
"Oh," he said, "when you do a Fast Fourier transform to move audio into the frequency domain the phase information is represented by the imaginary part of a complex number."
Wait, what? So complex numbers have a real-world analog in constructing an FFT — a "real" algorithm that does real work?
I see now that the really smart people just understand the natural world much better than I do — or perhaps see it differently than I do.