Find the quotient of complex numbers calculator

1. How do you divide complex numbers?

To divide the complex numbers multiply the given complex number with the conjugate of the denominator on both numerator and denominator. Combine the Like terms and express the solution in the form of a+bi.

2. What is the formula for the division of complex numbers?

The formula for dividing complex numbers is

3. Where do I get a detailed explanation for the division of complex numbers?

You can get a detailed explanation for the division of complex numbers on our page.

4. What is the division of complex numbers (7+6i) and (2+3i)?

Step 1: Given expression

Step 2: Multiply with the complex conjugate of the denominator both numerator and denominator

Step 3: Simplifying the equation further we get the result as follows

Multiplying the expression (7+6i) and (2-3i) we get

= 7(2-3i)+6i(2-3i)

= 14-21i+12i-18i2

= 14-9i-18(-1)

=32-9i

Thus, the expression can be rewritten as

An easy to use calculator that divides two complex numbers.

Let w and z be two complex numbers such that w = a + ib and z = A + iB. The division of w by z is based on multiplying numerator and denominator by the complex conjugate of the denominator:

w / z = (a + ib) / (A + iB)

= (a + ib)(A - iB) / (A + iB)(A - iB)

= [ a A + b B + i(b A - a B) ] / [A 2 + B 2]

= ( a A + b B )/ [A 2 + B 2] + i (b A - a B) / [A 2 + B 2]

Divide Complex Numbers Calculator

Enter the real and imaginary parts (as an integer, a decimal or a fraction) of two complex numbers z and w and press "Divide".

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• Division 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.

Division of complex numbers calculator

learn how to divide complex numbers
show help ↓↓ examples ↓↓ tutorial ↓↓

Input first number :

Input second number:

Type r to input square roots $ \left( \color{blue}{\text{ r10 } = \sqrt{10}} ~ \right) $.

Examples:

$\dfrac{3-2i}{4+5i}$

$\dfrac{\frac{1}{2}-i}{2+\sqrt{2}i}$

EXAMPLES

Divide $ \left( 2 - 6i \right) $ by $ \left( 1 + i \right)$.

Divide $ \left( \dfrac{1}{2} - 2i \right) $ by $ \left( 2 - i \right)$.

Divide complex numbers $ \,\,\dfrac{ 2 - 3i}{ \sqrt{2} + i} $

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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