# How to Divide Complex Numbers

To divide complex numbers. First, find the complex conjugate of the denominator, multiply the numerator and denominator by that conjugate and simplify

### Example

Let's divide the following 2 complex numbers

$\frac{ 5 + 2i}{ 7 + 4i}$ Step 1

Determine the conjugate of the denominator

The conjugate of $$(7 + 4i)$$ is $$(7 \red - 4i)$$

Step 2

Multiply the numerator and denominator by the conjugate.

$\big( \frac{ 5 + 2i}{ 7 + 4i} \big) \big( \frac{ 7 \red - 4i}{7 \red - 4i} \big)$ Step 3

Simplify. (Remember that $$i^2 = -1$$)

$\big( \frac{ 5 + 2i}{ 7 + 4i} \big) \big( \frac{ 7 \red - 4i}{7 \red - 4i} \big) \\ = \frac{ 35 + 14i -20i - 8\red{i^2 } }{ 49 \blue{-28i + 28i}-16 \red{i^2 }} \\ = \frac{ 35 + 14i -20i \red - 8 }{ 49 \blue{-28i + 28i} \red -16 } \\ = \frac{ 43 -6i }{ 65 }$

### Practice Problems

Step 1

Determine the conjugate of the denominator

The conjugate of $$2 + 6i$$ is $$(2 \red - 6i)$$

Step 2

Multiply the numerator and denominator by the conjugate.

$\big( \frac{ 3 + 5i}{ 2 + 6i} \big) \big( \frac { 2 \red - 6i}{ 2 \red - 6i} \big)$
Step 3

Simplify. (Remember that $$i^2 =-1$$)

$\big( \frac{ 3 + 5i}{ 2 + 6i} \big) \big( \frac { 2 \red - 6i}{ 2 \red - 6i} \big) \\ \frac{ 6 -18i +10i -30 \red{i^2} }{ 4 \blue{ -12i+12i} -36\red{i^2}} \\ \frac{ 6 -8i \red + 30 }{ 4 \red + 36}= \frac{ 36 -8i }{ 40 }= \frac{9 -2i}{10}$
Step 1

Determine the conjugate of the denominator

The conjugate of $$5 + 7i$$ is $$5 \red - 7i$$

Step 2

Multiply the numerator and denominator by the conjugate.

$\big( \frac{6-2i}{5 + 7i} \big) \big( \frac{5 \red- 7i}{5 \red- 7i} \big)$
Step 3

Simplify

$\big( \frac{6-2i}{5 + 7i} \big) \big( \frac{5 \red- 7i}{5 \red- 7i} \big) \\ \frac{ 30 -42i - 10i + 14\red{i^2}}{25 \blue{-35i +35i} -49\red{i^2} } \\ \frac{ 30 -52i \red - 14}{25 \red + 49 } = \frac{ 16 =52i}{ 74}$

#### What About Interesting Complex Quotients?

The next two problems will refer to the following two quotients

Problem 3

$\frac{3-2i}{3 + 2i}$

Problem 4

$\frac{3-2i}{2i -3 }$

Make a Prediction: Do you think that there will be anything special or interesting about either of the following quotients? Scroll down the page to see the answer.

Step 1

Determine the conjugate of the denominator

The conjugate of (3 + 2i) is (3 - 2i)

Step 2

Multiply the numerator and denominator by the conjugate

Step 3

Simplify

Step 1

Determine the conjugate of the denominator

The conjugate of (2i - 3) is (2i + 3)

Step 2

Multiply the numerator and denominator by the conjugate

Step 3

Simplify

Any rational-expression in the form is equivalent to $$-1$$
(with the caveat that $$y-x \ne 0$$)