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Division of complex numbers calculator

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.

Divide complex numbers: $$\dfrac{20+10i}{1-3i}$$

solution

$$\dfrac{20+10i}{1-3i} = -1+7i$$

explanation

Step 1: Determine the conjugate of the denominator. ( to find the conjugate just change the sign of the imaginary part ).

In this example, the conjugate of $\color{orangered}{ 1-3i }\,$ is $\color{blue}{ 1+3i }$.

Step 2: Multiply both the numerator and denominator by the conjugate:

\begin{aligned} \frac{ 20+10i }{ 1-3i } &= \frac{ 20+10i }{ 1-3i } \cdot \frac{ \color{blue}{ 1+3i } }{ \color{blue}{ 1+3i } } = \\[1 em] &= \frac{ \left( 20+10i \right) \cdot \left( 1+3i \right) }{ \left( 1-3i \right) \cdot \left( 1+3i \right) } \end{aligned}

Step 3: Simplify numerator and denominator (use $\color{blue}{i^2 = -1}$)

\begin{aligned} \left( 20+10i \right) \cdot \left( 1+3i \right) &= 20 \cdot 1 + 20 \cdot \left(3 \,i \right) + \left( 10 \,i \right) \cdot \left(1 \right) + \left( 10 \,i \right) \cdot \left(3 \,i \right) = \\[1 em] &= 20 + 60 \, i + 10 \, i + 30 \color{blue}{(-1)} = \\[1 em] &= -10+70i\end{aligned} \begin{aligned} \left( 1-3i \right) \cdot \left( 1+3i \right) &= 1 \cdot 1 + 1 \cdot \left(3 \,i \right) + \left( -3 \,i \right) \cdot \left(1 \right) + \left( -3 \,i \right) \cdot \left(3 \,i \right) = \\[1 em] &= 1 + 3 \, i -3 \, i -9 \color{blue}{(-1)} = \\[1 em] &= 10\end{aligned}

Step 4: Separate real and imaginary parts:

$$\frac{ 20+10i }{ 1-3i } = \frac{ -10+70i }{ 10 } = \frac{ -10 }{ 10 } + \frac{ 70 }{ 10 } i= -1+7i$$

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Script name : complex-numbers-division-calculator

Form values: 20 , 1 , 10 , 1 , 2 , 3 , g , Divide 20+10i by 1-3i , , Divide 20+10i by 1-3i

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Division of complex numbers calculator
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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

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