The fact that the complex numbers are two dimensional is part of what makes them useful in electrical engineering, but, at the same time, it makes it impossible to represent mathematical relationships graphically the way we can with Real Numbers.
In a standard x and y graph, numbers for x are fed into an equation (or formula), a calculation is made and an answer for a corresponding y value is computed. These pairs of (real number) values are then graphed on the x–y axes for a two dimensional visual representation of the relationship between x and y.
This works because each number itself is only one dimensional. The two dimensional graph shows how the value of each y (or vertical) coordinate depends on each x (or horizontal) value. In this case, each number is one dimensional, so showing them together requires only two dimensions.
In the case of Complex Analysis, each x value is two dimensional and each y value is also two dimensional – this is FOUR dimensions, which is one more that most humans can comprehend.
Although time is often considered to be a fourth dimension, I think that it is more accurate to say that we experience a fourth spatial dimension over time. In his book Flatland, the author Edwin Abbott describes how a two dimensional creature would experience the third spatial dimension over time.
If you’re not sure how this works, that last link is worth a look, because it can help us to conceptualize how we might experience a fourth spatial dimension over time.
In the next post I’ll look at what this means for solving equations graphically, and analytic geometry in general.
This post from last week touched on these ideas.