this is a question of mine because of an edit to the Wikipedia article Imaginary number . the funny thing is that i couldn't find, in three of my old textbooks a clear definition of "imaginary number". (they were pretty good at defining "imaginary part", etc.) i understand that that zero lies on both the real and imaginary axis. is 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 an imaginary number had to square to be a real and negative number (not just non-positive). clearly we can (re)define a real number as a complex number with imaginary part 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 part zero, that would also include 0 among the pure imaginaries. what is the complete and formal definition of "imaginary number" (outside of the Wikipedia reference or anything derived from it)?