Can a set containing $0$ be purely imaginary? A purely imaginary number is one which contains no non-zero real component. 
If I had a sequence of numbers, say $\{0+20i, 0-i, 0+0i\}$, could I call this purely imaginary? 
My issue here is that because $0+0i$ belongs to multiple sets, not just purely imaginary, is there not a valid case to say that the sequence isn't purely imaginary?
 A: 0 is both purely real and purely imaginary.  The given set is purely imaginary.  That's not a contradiction since "purely real" and "purely imaginary" are not fully incompatible.  Somewhat similarly baffling is that "all members of X are even integers" and "all members of X are odd integers" is not a contradiction.  It just means that X is an empty set.
A: A complex number is said to be purely imaginary if it's real part is zero.  Zero is purely imaginary, as it's real part is zero.  
A: As other answers say, zero is purely real as well as purely imaginary. Let me give an intuitive explanation based on the graphical representation of a complex number.
In the Argand plane, the points on the real axis represent complex numbers which are "purely real" as their imaginary part is zero. On the other hand, points on the imaginary axis denote complex numbers whose real part is zero and hence they are "purely imaginary". 
We know that, the real and imaginary axis meet at the origin which represents the complex number $0+0i$. As this point simultaneously lies on the real as well as the imaginary axis, we say that zero is both purely real and purely imaginary. 
