Complex Number Calculator
Worksheet on Dividing Complex Numbers
Multiplying Complex Number$$(3 + 2i)(4 + 2i)$$
It's All about complex conjugates and multiplication
To divide complex numbers. First, find the complex conjugate of the denominator, multiply the numerator and denominator by that conjugate and simplify.
Example 1
Let's divide the following 2 complex numbers
$ \frac{5 + 2i}{7 + 4i} $
Step 1Determine the conjugate of the denominator
The conjugate of $$ (7 + 4i)$$ is $$ (7 \red - 4i)$$.
Step 2Multiply the numerator and denominator by the conjugate.
$ \big( \frac{ 5 + 2i}{ 7 + 4i} \big) \big( \frac{ 7 \red - 4i}{7 \red - 4i} \big) $
Step 3Simplify
Remember$$ (i^2 = -1) $$
$ \big( \frac{ 5 + 2i}{ 7 + 4i} \big) \big( \frac{ 7 \red - 4i}{7 \red - 4i} \big) \\ \boxed{ \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{ 35 + 14i -20i \red - 8 }{ 49 \blue{-28i + 28i} +16 }\\\frac{ 43 -6i }{ 65 }$
Note: The reason that we use the complex conjugate of the denominator is so that the $$ i $$ term in the denominator "cancels", which is what happens above with the i terms highlighted in blue $$ \blue{-28i + 28i} $$.
Video Tutorial on Dividing Complex Numbers
Practice Problems
Problem 1.1
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
Multiplythe 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.
$\big( \frac{ 3 + 5i}{ 2 + 6i} \big) \big( \frac { 2 \red - 6i}{ 2 \red - 6i} \big) \\\frac{ 6 -18i +10i -30 \red{i^2} }{ 4 \blue{ -12i+12i} -36\red{i^2}} \text{ } _{ \small{ \red { [1] }}}\\\frac{ 6 -8i \red + 30 }{ 4 \red + 36}= \frac{ 36 -8i }{ 40 }\\\boxed{ \frac{9 -2i}{10}}$
$$ \red { [1]} $$ Remember $$ i^2 = -1 $$
Problem 1.2
Find the following 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
Multiplythe 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} } \text{ } _{\small{ \red { [1] }}}\\\frac{ 30 -52i \red - 14}{25 \red + 49 } = \frac{ 16 - 52i}{ 74} $
$$ \red { [1]} $$ Remember $$ i^2 = -1 $$
Problem 1.3
Find the following quotient
$ \frac{3-2i}{3+2i} $
Step 1
Determine the conjugate of the denominator
The conjugate of $$ 3 + 2i $$ is $$ (3 \red -2i) $$.
Step 2
Multiplythe numerator and denominator by the conjugate.
$ \big( \frac{ 3 -2i}{ 3 + 2i} \big) \big( \frac { 3 \red - 2i}{ 3 \red - 2i} \big) $
Step 3
Simplify.
$\big( \frac{ 3 -2i}{ 3 + 2i} \big) \big( \frac { 3 \red - 2i}{ 3 \red - 2i} \big) \\ \frac{ 9 \blue{ -6i -6i } + 4 \red{i^2 } }{ 9 \blue{ -6i +6i } - 4 \red{i^2 }} \text{ } _{ \small{ \red { [1] }}}\\\frac{ 9 \blue{ -12i } -4 }{ 9 + 4 } \\\frac{ 5 -12i }{ 13 } \\$
$$ \red { [1]} $$ Remember $$ i^2 = -1 $$
More Like Problem 1.3...
Problem 1.3.1
Make a Prediction
Look carefully at the problems 1.5 and 1.6 below.
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(from our free downloadable worksheet).
Problem 1.4
$\frac{ \red 3 - \blue{ 2i}}{\blue{ 2i} - \red { 3} }$
Problem 1.5
$\frac{\red 4 - \blue{ 5i}}{\blue{ 5i } - \red{ 4 }}$
Problem 1.4
Find the following quotient
$ \frac{3-2i}{2i-3} $
Step 1
Determine the conjugate of the denominator
The conjugate of $$ 2i - 3 $$ is $$ (2i \red + 3) $$.
Step 2
Multiplythe numerator and denominator by the conjugate.
$ \big( \frac{ 3 -2i}{ 2i -3 } \big) \big( \frac { 2i \red + 3 }{ 2i \red + 3 } \big) $
Step 3
Simplify.
$\big( \frac{ 3 -2i}{ 2i -3 } \big) \big( \frac { 2i \red + 3 }{ 2i \red + 3 } \big) \\\frac{ \blue{6i } + 9 - 4 \red{i^2 } \blue{ -6i } }{ 4 \red{i^2 } + \blue{6i } - \blue{6i } - 9 } \text{ } _{ \small{ \red { [1] }}}\\\frac{ 9 + 4 }{ -4 - 9 } \\\boxed{-1}$
$$ \red { [1]} $$ Remember $$ i^2 = -1 $$
Problem 1.5
Find the following quotient
$ \frac{4 - 5i}{5i - 4} $
Step 1
Determine the conjugate of the denominator
The conjugate of $$ 5i - 4 $$ is $$ (5i \red + 4 ) $$.
Step 2
Multiplythe numerator and denominator by the conjugate.
$ \big( \frac{ 4 -5i}{ 5i -4 } \big) \big( \frac { 5i \red + 4 }{ 5i \red + 4 } \big) $
Step 3
Simplify.
$\big( \frac{ 4 -5i}{ 5i -4 } \big) \big( \frac { 5i \red + 4 }{ 5i \red + 4 } \big) \\\frac{\blue{20i} + 16 -25\red{i^2} -\blue{20i}}{ 25\red{i^2} + \blue{20i} - \blue{20i} -16} \text{ } _{ \small{ \red { [1] }}}\\\frac{ 16 + 25 }{ -25 - 16 } \\\frac{ 41 }{ -41 } \\\boxed{-1}$
$$ \red { [1]} $$ Remember $$ i^2 = -1 $$
How did we get the same result?
After looking at problems 1.5 and 1.6 , do you think that all complex quotients of the form
$ \frac{ \red a - \blue{ bi}}{\blue{ bi} - \red { a} } $
are equivalent to $$ -1$$? ( taken from our free downloadable worksheet )
The answer is yes
Any rational-expression in the form $$ \frac{y-x}{x-y} $$ is equivalent to $$-1$$.
(with the caveat that $$y-x \ne 0 $$).
Complex Number Calculator
Worksheet on Dividing Complex Numbers
Multiplying Complex Number$$(3 + 2i)(4 + 2i)$$
