Dividing Complex
Numbers
Before we begin
Last time we learned about the “imaginary
numbers.”
 Numbers like 4i and -3i are called “pure
imaginary” because no real number has
been put with them.
 When you put a real number and an
imaginary number together, like -3 + 2i,
its called a “complex number.”

Conjugates of complex numbers
To get the conjugate of a complex number,
you change the sign in the middle.
The conjugate of 3 + i is 3 – i
The conjugate of -9 – 6i is -9 + 6i
Conjugates of Pure Imaginary
Numbers
What about the conjugate of 4i ?
4i written in “complex form” would be 0+4i
and its conjugate would be 0 – 4i
So to make it simple, the conjugate of 4i is
-4i, the conjugate of -9i is 9i , see?
Dividing Complex numbers
This is really easy, because you don’t really
do any dividing.
2 + 3i
2i
What makes this easy is that you’re looking
at the answer…its just not in simplest
form, yet
2 + 3i
2i
-2i
-2i
=
-4i – 6i
-4i
2
2
=
-4i + 6
4
=
3 – 2i
2

To simplify, just multiply the top and
bottom by the conjugate of the bottom.

Then simplify the whole fraction
Problem
-2 + 3i
5 + 2i
5 – 2i
5 – 2i
= -10 + 4i + 15i - 6i
25 – 4i 2
=
=

2
-10 + 19i - (-6)
25 – (-4)
-4 + 19i
29
Multiply top and bottom by conjugate of
bottom
Problem : Find the reciprocal of
5 i 3
Since the reciprocal of a number is just 1 over the
number, the reciprocal would be…
1
5 i 3
√5 + i √3
√5 + i √3
=
√5 + i √3
8
Now simplify by multiplying the top and bottom by
the conjugate of the bottom, just like before.
One more….
This is the easy one. If you have one term
with an i on top and one term with an i
on bottom, the i ‘s will cancel
3i
5i
=
3
5