 # Complex Number Division Formula

A Complex number is in the form of a+ib, where a and b are real numbers the ‘i’ is called the imaginary unit. The imaginary number, i, has the property, such as

$$\begin{array}{l}i^{2}\end{array}$$
=
$$\begin{array}{l}-1\end{array}$$
.

To find the division of any complex number use below-given formula.

Let two complex numbers are a+ib, c+id, then the division formula is,

$\LARGE \frac{a+ib}{c+id}=\frac{ac+bd}{c^{2}+d^{2}}+\frac{bc-ad}{c^{2}+d^{2}}i$

#### Solved Examples

Question 1: Divide the complex roots.
$$\begin{array}{l}\large \frac{7 – 6i}{2 – 3i}\end{array}$$

Step 1 –

$$\begin{array}{l}\large \frac{7-6i}{2-3i}\times \frac{2+3i}{2+3i}\end{array}$$

Step 2 –

$$\begin{array}{l}\large \frac{14+21i-12i-18i^{2}}{4+6i-6i-9i^{2}}\end{array}$$

Step 3 –

$$\begin{array}{l}\large \frac{14+21i-12i-18(-1)}{4+6i-6i-9(-1)}\end{array}$$

=

$$\begin{array}{l}\large \frac{14+21i-12i+18}{4+6i-6i+9}\end{array}$$

Step 4 –

$$\begin{array}{l}\large \frac{32+9i}{13}\end{array}$$

Step 5 –

$$\begin{array}{l}\large \frac{32}{13}+\frac{9}{13}i\end{array}$$