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 } $

Video Tutorial on Dividing Complex Numbers


Example 2
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Practice Problems

Problem 1

Divide the complex numbers below: $ \frac{ 3 + 5i}{ 2 + 6i} $

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} $
Problem 2

Find the quotient $ \frac{6-2i}{5 + 7i} $

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.

Problem 3

Find the quotient

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

Problem 3

Find the quotient

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

A Parting Question

After looking at problem 4, do you think that all complex quotients of the form are equivalent to $$ -1$$?

The answer is yes

Any rational-expression in the form is equivalent to $$-1$$

(with the caveat that $$y-x \ne 0 $$)