Is this a pure imaginary number or real number? Is $\dfrac{0}{2yi}$ a pure imaginary number or a real number?
I'm debating, $0$ is a real number but if you divide by $i$, it's imaginary.
 A: You told us nothing about $y$ but, assuming that $y$ is a non-zero complex number, then $\dfrac0{2yi}=0$, which is both a real number and a pure imaginary number. It's actually the only complex number with both properties.
A: If $y\neq 0$, in the complex plane, by definition, $0=0+0i$. Since the imaginary and real parts are 0, 0 is purely real and imaginary. However, it's also member of the complex numbers. 
A: We have that 
$$\frac{0}{2yi}=0$$
which is an integer, a rational, a real and a complex number.
Notably it indicates the neutral element with respect to addition. 
A: Well, provided that $y \neq 0$, you are multiplying by $\frac{1}{2y}$.  (Of course, if $y = 0$ then it is undefined.)  So, I will talk of multiplication rather than division.  I am guessing that $y$ is intended to be real but that does not actually affect the answer. 
A complex number is real if the imaginary component is zero.  Conversely, it is imaginary if the real component is zero. Most complex numbers e.g. $1 + i$ are neither. $0$ is special in that it is both.  
Generally multiplying by $i$ will flip real numbers to imaginary and vice versa.  Since $0 \times i = 0$ multiplying by $i$ does change it, it goes from both to both.  
Something a little like this occurs with the more familiar real numbers.  Multiplying by $-1$ flips positive and negative.  It leaves $0$ alone, so is $0$ positive or negative?  The usual answer is that $0$ is neither but considering it as both could work.  The Bourbaki (Wikipedia) school proposed this.  I found a reference thanks to posting a question here: Bourbaki and zero (this site).
