Division of complex numbers calculator

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