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Dividing by Complex Numbers

Date: 05/04/2003 at 23:07:51
From: Ryan
Subject: Rational expressions with imaginary numbers


I'm difficulty with problems that have an imaginary number but don't 
cancel. Example: Divide each pair of complex numbers: (8+4i)/(1+2i)

Any help would be great. 

Date: 05/05/2003 at 05:02:42
From: Doctor Luis
Subject: Re: Rational expressions with imaginary numbers

Hi Ryan,

Good question. There's a trick for dividing by complex numbers, and to 
use it you need to understand something called the conjugate complex 

Essentially, the conjugate of a complex number is the number you get
when you replace (i) by (-i). For example, the conjugate of 1+3i is 
1+3(-i)=1-3i, and also, the conjugate of -3-2i is -3-2(-i)=-3+2i

Now, something funny happens when you multiply a complex number by its 
conjugate. The answer turns out to be a real number. I'll illustrate 
with 1+3i and its conjugate 1-3i

  (1+3i)*(1-3i) = 1^2 - (3i)^2 = 1 - (9 * (-1)) = 1 + 9 = 10

Here, I used the algebraic formula (a-b)(a+b) = a^2 - b^2 to multiply 
them. (Don't forget that i^2 = -1)

You should verify for yourself that for any complex number z=x+iy and 
its complex conjugate z'=x-iy, the product z*z' is a real number (and 
that it equals x^2+y^2).

Knowing this fact about complex numbers, to divide you simply multiply 
and divide by the conjugate of the denominator.

Here's how:

  20 + 30i     20 + 30i     -1 - 2i  
 --------- =  ---------- * ----------    (conjugate trick)
  -1 + 2i       -1 + 2i     -1 - 2i  

              (20 + 30i)(-1 - 2i)
           =  -------------------        (multiplying bottom)
                (-1)^2 + (2)^2

           = (40 - 70i)/5                (after multiplying top)

           = 8 - 14i                     (final answer)

That's all there is to it. You make the denominator into a number you 
can divide by (that is, a real number), using complex conjugates. With 
this background, you should be able to solve the division you asked 

Let us know if you have any more questions.

- Doctor Luis, The Math Forum 
Associated Topics:
High School Imaginary/Complex Numbers

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