Math Calculators, Lessons and Formulas

It is time to solve your math problem

mathportal.org

Division of complex numbers calculator

google play badge app store badge

Use this online calculator to divide complex numbers.
The calculator shows a step-by-step, easy-to-understand solution on how the division was done.

Division of complex numbers calculator
learn how to divide complex numbers
help ↓↓ examples ↓↓ tutorial ↓↓
$\dfrac{3-2i}{4+5i}$
$\dfrac{\frac{1}{2}-i}{2+\sqrt{2}i}$
working...
EXAMPLES
example 1:ex 1:
Divide $ \left( 2 - 6i \right) $ by $ \left( 1 + i \right)$.
example 2:ex 2:
Divide $ \left( \dfrac{1}{2} - 2i \right) $ by $ \left( 2 - i \right)$.
example 3:ex 3:
Divide complex numbers $ \,\,\dfrac{ 2 - 3i}{ \sqrt{2} + i} $
Find more worked-out examples in the database of solved problems..
TUTORIAL

How to divide complex numbers?

This calculator uses multiplication by conjugate to divide complex numbers.

Example 1:

$$ \frac{ 4 + 2i }{1 + i} $$

We begin by multiplying numerator and denominator by complex conjugate of $ \color{purple}{1 + i} $.

$$ \frac{4 + 2i}{\color{purple}{1 + i}} \cdot \frac{\color{blue}{1 - i}}{\color{blue}{1 - i}} = \frac{(4+2i)(1-i)}{(1+i)(1-i)}$$

Then we expand and simplify both products. Keep in mind that $ i^2 = -1 $.

$$ \begin{aligned} \frac{(4+2i)(1-i)}{(1+i)(1-i)} &= \frac{4 - 4i + 2i - 2\color{blue}{i^2}}{1+i-i-i^2} = \\[ 1 em] &= \frac{4 - 2i - 2\color{blue}{(-1)}}{1-\color{purple}{i^2}} = \\[ 1 em] &= \frac{4 - 2i + 2)}{1-\color{purple}{(-1)}} = \\[ 1 em] &= \frac{6 - 2i)}{2} \end{aligned} $$

At the end we separate real and imaginary parts:

$$ \frac{6 - 2i}{2} = \frac{6}{2} - \frac{2}{2}i = 3 - i $$

Example 2:

Divide $ 10 - 25i $ by $ 5i $

Although the complex conjugate of $ 5i $ is $-5i$, we can simplify division process by multiplying numerator and denominator with $ - i $.

$$ \begin{aligned} \frac{10-25i}{5i} &= \frac{10-25i}{5i} \cdot \frac{-i}{-i} = \\[1 em] &= \frac{(10-25i)(-i)}{(5i)(-i)}= \\[ 1 em] &= \frac{-10i + 25i^2}{-5i^2} = \\[ 1 em] &= \frac{-10i - 25}{5} = \\[ 1 em] &= \frac{-25}{5} + \frac{-10}{5} i= \\[ 1 em] &= -5 - 2 i= \\[ 1 em] \end{aligned} $$

Example 3:

Divide $ 20 + 10i $ by $ 1 - 3i $

Solution

Search our database with more than 250 calculators
439 063 882 solved problems