Understanding Complex Numbers: A Visual Explanation

[QuarkCharmer]

Complex Numbers, I just don't understand what they are supposed to represent. I understand how they are used, to some degree in solving equations etc. but the definitions leave me clueless. The idea that they "extend the idea of the one-dimensional number line to the two-dimensional complex plane" sort of makes sense. If each number consists of a real part and an imaginary part (a+bi etc.), then starting from a number-line representing the real number a, you can extend a y-axis to represent i. But what does that really mean?? What would be a good visual explanation of this concept?

 

[symbolipoint]

I have been struggling with the same difficulty for several years. Finally, there was a YouTube video a few months ago which gave a reasonably logical demonstration that complex numbers represent rotation. Think of a number represented in just one direction on a number line. Multiplication by i^2 is rotation by 180 degrees, and multiplication instead by just i is rotation by just 90 degrees.