4.5 The complex ‘‘plane’’
In the last chapter 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.
The plot to the right shows several complex values. Despite appearances and some authors’ use of the terminology, that is not a complex plane. 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 the next chapter when we look at vector spaces.
Conjugation
Conjugation reflects a complex number across the horizontal Re(z) axis. If the number...
