4.5 The complex “plane”
In section 3.9, we discussed the algebraic properties of C, the complex numbers. We return to them again here to look at their geometry. For any point (a, b) in the real plane, consider the corresponding complex number a + bi.
In the graph of the complex numbers, the horizontal axis is the real part of the complex variable z, and the vertical axis is the imaginary part. These replace the x and y axes, respectively. complex number complex number$geometry number$complex C`bold plane$complex complex$plane
The plot in Figure 4.23 shows several complex values. That is not a complex plane despite appearances and some authors’ terminology. A plane has two dimensions. We visualized C, which is one-dimensional, in the two-dimensional real plane. We return to these issues about dimensions with respect to a field in section 5.2 when we look at vector...
