So @kazumiochin and @udernation asked about imaginary numbers. As a BS in Mathematics, I am stupendously-qualified to carry forth on the subject.
Thread:

Let's talk about square roots. If @*@=#, where @ and # are both numbers, then we can also write @^2=#, or "at squared equals pound". Consequently, a 'square root' of # is @.

Why did I say *a* square root? Because if @ is a square root of #, then so is -@.
2*2=4, and -2*-2=4 also, since the two negative numbers make a positive one when multiplied together.

This works dandy as long as # is positive. If # is negative, though, we have a problem. As noted above, two negative numbers multiplied together make a positive number. And two positive numbers also make a positive number. This poses the following dilemma:

In order to find a square root of a negative number, we need a number that, when multiplied by itself, creates a negative number. But a negative number or a positive number won't do the trick. We need a number that is neither positive nor negative.

... what? ... ok. 0. 0 isn't positive or negative. But 0*0=0, so that won't help us with our problem. Gotta keep thinking...

OK. What if we just pretended there was a number that would do what we want? We'll just make it up. Also, since sqrt(-#)=sqrt(-1 * #)=sqrt(-1) * sqrt(#), if we know the square root of -1, we can find the square root for any negative number.

Since we're all collectively playing pretend for this number, we'll call it an 'imaginary' number, and we'll use the letter i to stand in for it. So i*i=-1. And the square root of -# is @*i.

Uh oh. New problem: How do we do math with i? Is @*i bigger or smaller than @? What does @*i + @ equal? If I was plotting points on a graph, where does @*i go?

We know that i isn't positive or negative, so @*i also can't be positive or negative. And the only other number we know that isn't positive or negative is 0, but that's not very helpful here, cuz i is definitely not 0. Huh. What do what do.

Guys, guys, what if we just did a little more pretending? Any number multiplied by i also becomes imaginary, and all those imaginary numbers exist in their own little space called the Complex Plane.

Think of the Complex Plane as a kind of shadow dimension. Things work more or less the same in the real world as they do in the shadow world, but we don't get a lot of crossover between them (get out of here, Yugi!).

And like the relationship between light, objects and shadows, we can assign a shadow (an imaginary part) to all of our real numbers: @ becomes @ + 0i.
@*i + @ becomes 0+@*i + @+0i. If we let the bodies play with bodies and the shadows play with shadows, that simplifies to @+@*i

When we multiply, wires get crossed a little bit. @*(1+i) becomes @+@*i. And i*(1+i) = i-1. But it's not too different from regular algebra.

As for graphing, the Complex Plane extends along a third axis, independent of the x-axis and y-axis, with unit vector i.

Why bother with all this imaginary nonsense?
Well, mathematicians are just behind physicists as relentless fans of fictional stuff (looking at you, String Theory). And pretending is fun. But i also lets us do a lot of cool things.

The only one I know about off-hand is that because of i, we get the fundamental theorem of algebra: that any polynomial equation has a number of roots equal to the largest exponent.

f(x)=(x-2)*(x+2) when graphed gives us a parabola that passes through the x-axis at x=2 and x=-2. These are the 'roots' of the equation. What about f(x)=x*x? It should also have two roots, but 0 is only one of them. Our other root is 0i.

Another fun fact: i*i=-1, i*i*i=-i, and i*i*i*i=1. That is, the powers of i cycle through 1, i, -1, and -i.