Associated Topics || Dr. Math Home || Search Dr. Math

### Dividing by Complex Numbers

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

Hi,

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.
Thanks.
```

```
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
number.

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
http://mathforum.org/dr.math/
```
Associated Topics:
High School Imaginary/Complex Numbers

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search