Is $0$ an imaginary number? My question is due to an edit to the Wikipedia article:  Imaginary number.

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)?
The Wikipedia article cites a textbook that manages to confuse the issue further:


  Purely imaginary (complex) number : A complex number $z = x + iy$ is called a purely imaginary number iff $x=0$ i.e. $R(z) = 0$.
  
  Imaginary number : A complex number $z = x + iy$ is said to be an imaginary number if and only if $y \ne 0$ i.e., $I(z) \ne 0$.


This is a slightly different usage of the word "imaginary", meaning "non-real": among the complex numbers, those that aren't real we call imaginary, and a further subset of those (with real part $0$) are purely imaginary. Except that by this definition, $0$ is clearly purely imaginary but not imaginary!

Anyway, anybody can write a textbook, so I think that the real test is this: does $0$ have the properties we want a (purely) imaginary number to have?

I can't (and MSE can't) think of any useful properties of purely imaginary complex numbers $z$ apart from the characterization that $|e^{z}| = 1$. But $0$ clearly has this property, so we should consider it purely imaginary.

(On the other hand, $0$ has all of the properties a real number should have, being real; so it makes some amount of sense to also say that it's purely imaginary but not imaginary at the same time.)
A complex number z=a+ib where a and b are real numbers is called :
1- purely real , if b=0 ; e.g.- 56,78 ;
2- purely imaginary,  if a=0 ,e.g.- 2i, (5/2)i ;
3- imaginary,if b≠ 0 ,e.g.- 2+3i,1-i,5i ;
0 is purely imaginary and purely real but not imaginary.