Is Zero an Imaginary Number?

If you're thinking that this title sounds familiar, it's because you're thinking of my famous post "Is Zero a Square number?". You can think of this post as the sequel. I can foresee a whole franchise of posts of this nature in my future.

To answer today's question, we first need to decide how we are going to define "imaginary number".
No prizes for guessing which website I'll be going to for that!

"An imaginary number is a number whose square is less than or equal to zero" (Wikipedia)

Oh. Well, I guess we're done here. Zero is an imaginary number, as zero squared is zero, which is obviously less than or equal to zero. Job done.

Oh come on, you can't possibly be satisfied with that!

What is the rationale behind including zero? If you don't include zero, what things are affected? Would it really make a difference? I need to know WHY! Darn you Wikipedia!!!

I need another definition!

So I will check out my usual second port of call,  Wolfram Math World:

"A (purely) imaginary number can be written as a real number multiplied by the "imaginary unit" i (equal to the square root ), i.e., in the form ."

OK, so that's a little bit better. Again, zero would fit this definition because you could write it as 0i (or 0j if you prefer) and zero is certainly a real number.

But I'm still not happy. It just doesn't feel right. 

Let's think graphically for a moment. On an Argand diagram, real numbers are numbers along the real axis (the horizontal axis) and imaginary numbers are along the imaginary axis (the vertical axis). The number 0 + 0i has coordinates (0,0) and is hence on the origin, both on the real and the imaginary axis. So this means it is either real and imaginary, or neither. We know zero is real, so the best option would be to take it as both.

The evidence is stacking up on one side: zero is an imaginary number. But I'm still not convinced.

In fact, the only thing that I even find slightly convincing is the argument put to me by one of my A level students. He said to me: zero can't be an imaginary number, because all imaginary numbers have an argument, but zero's argument would be undefined.

I really liked this. What is the angle between the positive real axis and the line connecting (0, 0) to (0, 0)? It could be anything, so it's undefined. Can an imaginary number have an undefined argument? A quick Google suggests actually yes, it can. Foiled again.

Fine, I give up. Zero is an imaginary number. It is also a real number. I can definitely see why including zero would be useful in terms of subspaces etc as it allows the imaginary numbers to have an additive identity.

So, as my students would say, "Allow it".

Emma x x x