Can you represent complex numbers using a single number?
Well, it depends on what you mean. The complex numbers form a field (a field is basically a set where addition, subtraction, multiplication, and division are possible), so they deserve to be called “numbers” as much as real numbers or rational numbers do; often in algebraic geometry or complex differential geometry you hear people refer to the “complex line” which really means what you would ordinarily call the complex plane. The word “line” is simply used to refer to an object which is in some sense a one-dimensional object over the complex numbers (which corresponds to a two-dimensional object over the real numbers). That being said, there is a certain precise sense in which it is not possible to geometrically visualize the complex numbers as sitting on a line (an object with dimension one over the real numbers). Part of the structure of the real line is the order structure of the real numbers; for the real numbers, it is possible to say that, for example, 10 < 15 and -32.1 < -29.8.
