Add, subtract, multiply, or divide the complex numbers

Question

(7+2i)/(3-2i)

Answer 1

Danielle,

 

The key to these problems is to simplify the denominator so that it does not include any complex numbers. The standard way to do this is to multiply the numerator and denominator by the complex conjugate of the denominator.

 

You may not know what the complex conjugate is. I'll illustrate by examples: the conjugate of 3 + 2i is 3 - 2i; of 1 + 2i is 1 - 2i; of 4 - 6i is 4 + 6i. I think you probably get it. Try Googling "complex conjugates Khan Academy," if you need more help.

 

What is the conjugate of 3 - 2i? When you find that, multiplying the top and bottom by that quantity is the way to go. 

 

Bryce

Answer 2

Looks like this one's division.  (Although we don't actually do any division to solve the problem).

In order to make this into a+bi form we want to eliminate the imaginary part from the denominator.  We can do this by multiplying both the top and bottom of the fraction by the conjugate of the complex number that's in the denominator.  The conjugate simply changes the sign between the real and imaginary part, so the conjugate for this problem is 3+2i.

After multiplying out the top and bottom (FOIL or otherwise), we should be left with a regular old integer in the denominator (remember that i^2 = -1). 

From there it's a matter of teacher preference on how to leave your answer.  Most would likely prefer to split the fraction up into two pieces and put the i to the side in order to make it look like a+bi form again.