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 –

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

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