There is an Algebraic way of looking at the complex number system, but for most people's purposes you have the concept down nicely.Right now, I am learning of the Complex Plane, and I believe that I am having difficulty truly understanding what this means. So far, this is how I visualized it:
a + bi
and these two parts of the complex number can be graphed on the complex plane, which is a variant of the cartesian plane with an x and y axis for real and imaginary numbers respectively.
If you add two real numbers A and B, you get a value that is itself real because it's like taking an actual length and adding on another actual length.
In this case, you are adding a real number a with an imaginary number bi, which can not be defined on a real number line. When you try to add these two guys together, it's quite unlike your first case because you cannot add these lengths together in a linear fashion. Therefore you have to represent imaginary numbers on a different line. your result for adding these two together is expressed as a + bi, which is the blue line.
Is this the correct way of thinking about complex numbers and their graphing on the complex plane? If I am of or if there's something more to this concept that I need to visualize or understand, I would really like to know.