The funny thing is, I couldn't find (in three of my old textbooks) a clear definition of an "imaginary number". (Though they were pretty good at defining "imaginary component", etc.)
I understand that the number zero lies on both the real and imaginary axes.
But is $\it 0$ both a real number and an imaginary number?
We know certainly, that there are complex numbers that are neither purely real, nor purely imaginary. But I've always previously considered, that a purely imaginary number had to have a square that is a real and negative number (not just non-positive).
Clearly we can (re)define a real number as a complex number with an imaginary component that is zero (meaning that $0$ is a real number), but if one were to define an imaginary number as a complex number with real component zero, then that would also include $0$ among the pure imaginaries.
What is the complete and formal definition of an "imaginary number" (outside of the Wikipedia reference or anything derived from it)?