Lesson 5.4 --- Dividing Complex Numbers Using Complex Conjugates

When dividing a complex number, you start off by writing the a+bi/x+yi equation down on your paper.

After that, you multiply the a+bi/x+yi equation by the complex conjugate of the denominator over itself. That way, you would be able to change the equations form, while at the same time, using the same equation. So it would be something like a+bi/x+yi x x-yi/x-yi

Use foil to distribute the numbers and then simplify it.

Finally, you write it in it's standard a+bi form

5+3i

1-2i
Write down the equation

5+3i

1-2i

1+2i

1+2i
Multiply the equation by the complex conjugate of the denominator over itself

(5+3i)(1+2i)

(1-2i)(1+2i)
Optional Step: Write down the new equation

5+3i+10i+6i²

1+2i-2i-4i²
Distribute using FOIL

5+13i+6i²

1-4i²
Combine the like terms

5+13i+6(-1)

1-4(-1)
=

5+13i-6

1+4
=

-1+13i

5
Simplify

-1

5
+

13

5
Write it in standard form

Here are some practice problems

Solve these problems by dividing using complex conjugates.

8

1+i

2i

1-i

3+i

3-i

-7+6i

9-4i

2+5i

5+2i