So far you have been able to extend the rules for real numbers to add, subtract, and multiply complex numbers, but what about dividing? Can you use what you know about real numbers to divide one complex number by another? In other words, if a problem looks like this:

$\frac { 3 + 2 i } { - 4 + 7 i }$

What needs to be done to get an answer in the form of a single complex number, a + bi?

Natalio had an idea. He said, “I'll bet we can use the conjugate!”

“How?” asked Ricki.

“Well, it's a fraction. Can't we multiply the numerator and denominator by the same number?” Natalio replied.

1. Natalio was not very clear in his explanation. Show Ricki what he meant they should do.

   Answer (a):

   $\frac{(3+2\textit{i})(-4-7\textit{i})}{(-4+7\textit{i})(-4-7\textit{i})}=\frac{2-29\textit{i}}{65}$

2. Using Natalio's ideas you probably still came up with a fraction in part (a), but the denominator should be a whole number. To write a complex number such as$\frac { c + d i } { m }$ in the form a + bi, just use the distributive property to rewrite the result as $\frac { c } { m } + \frac { d } { m } i$. Rewrite your result for part (a) in this form.

   Answer (b):

   $\frac{2}{65}-\frac{29}{65}\textit{i}$