Question

Describe the usefulness of the conjugate and its effect on other complex numbers. :)

Answer 1

Well, let's give an example. Here's a complex number:
3+2i

To get the complex conjugate of that number, we just change the sign of the imaginary part.
3-2i
Now, what is cool about the complex conjugate? Well, let's see what happens when we multiply those two things.
(3+2i)(3-2i)
FOIL it out.
9 -6i +6i -4i^2
Hey, sweet! The two middle terms cancel out, so I'm left with
9-4i^2
and what do we know about i^2? Well, i^2=-1, so I can substitute and get
9-4(-1) = 9+4 = 13

Wait, what just happened? Well, we multiplied two complex numbers together, and since the middle terms canceled, we were left with just a normal non-complex number.

Answer 2

That's the cool thing about the complex conjugate. Whenever I multiply a complex number by its conjugate, the result is a real number.