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.

Complex numbers division calculator

input two complex number and tap calculate button
help ↓↓ examples ↓↓ tutorial ↓↓

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First number :

Z₁ =

Second number:

Z₂ =

Examples:

\frac{3 - 2 i}{4 + 5 i}

\frac{\frac{1}{2} - i}{2 + 2 i}

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Examples

ex 1:

Divide

(2 - 6 i)

(1 + i)

ex 2:

Divide

(\frac{1}{2} - 2 i)

(2 - i)

ex 3:

Divide complex numbers

\frac{2 - 3 i}{2 + i}

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Find more worked-out examples in our database of solved problems.

TUTORIAL

How to divide complex numbers?

This calculator uses multiplication by conjugate to divide complex numbers.

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

We begin by multiplying numerator and denominator by complex conjugate of $1 + i$ .

\frac{4 + 2 i}{1 + i} \cdot \frac{1 - i}{1 - i} = \frac{( 4 + 2 i ) ( 1 - i )}{( 1 + i ) ( 1 - i )}

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

\frac{( 4 + 2 i ) ( 1 - i )}{( 1 + i ) ( 1 - i )} = \frac{4 - 4 i + 2 i - 2 i ^{2}}{1 + i - i - i ^{2}} = = \frac{4 - 2 i - 2 ( - 1 )}{1 - i ^{2}} = = \frac{4 - 2 i + 2 )}{1 - ( - 1 )} = = \frac{6 - 2 i )}{2}

At the end we separate real and imaginary parts:

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

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

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

\frac{10 - 25 i}{5 i} = \frac{10 - 25 i}{5 i} \cdot \frac{- i}{- i} = = \frac{( 10 - 25 i ) ( - i )}{( 5 i ) ( - i )} = = \frac{- 10 i + 25 i ^{2}}{- 5 i ^{2}} = = \frac{- 10 i - 25}{5} = = \frac{- 25}{5} + \frac{- 10}{5} i = = - 5 - 2 i =

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

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