Question

Explain, in complete sentences and in your own words, the answers to the following questions relating to complex numbers.

What are the characteristics of a complex number? (1 point)
What is the relationship between a complex number and its conjugate? (2 point)
Describe the usefulness of the conjugate and its effect on other complex numbers. (2 points)

Answer 1

The complex number has two parts, real and imaginary. In a plane the real corresponds to x and the imaginary to y. The complex conjugate is simply a reflection across the x axis. For example:

Since this only changes the imaginary part (the "y" part) to negative it reflects it across the x axis.
The conjugate can be used to define division on the set of complex numbers. Namely:
\[\frac{A}{z}=\frac{A z^*}{z z^*}=\frac{Az^*}{r}=\frac{A}{r}(x-iy);\]
With:
\[ \rightarrow z=x+iy, z^*=x-iy, zz^*=x^2+y^2=r\]