Had a lecture about complex numbers today, and I've been trying to figure the whole thing out and visualize it graphically. Please tell me if I am correct:
Real numbers are included in the complex numbers and can be thought of as vectors of varying length coinciding with the x-axis (ReZ-axis). So all real numbers are sort of one dimensional (along one line), while complex numbers have two dimensions - one real dimension and one imaginary.
This means that it is possible to get a real number even though you're adding, subracting, multiplying or dividing complex numbers that have both real and imaginary parts. But that will only happen when the imaginary coordinate is zero and the resulting vector coincides with the x-axis.