# Divide Complex Numbers

How to divide complex numbers (rationalize the complex denominator): 2 examples and their solutions.

## Formula

### Formula

(a + bi)(a - bi)

= a^{2} - (bi)^{2}

Product of a Sum and a Difference

= a^{2} - b^{2}⋅i^{2}

Power of a Product

= a^{2} - b^{2}⋅(-1)

Power of i

= a^{2} + b^{2}

So (a + bi)(a - bi) = a^{2} + b^{2}.

This formula is used

to rationalize the denominator a + bi or a - bi

when dividing complex numbers.

## Example 1

### Example

### Solution

The denominator, 1 + 3i, is a complex number.

To rationalize the denominator 1 + 3i,

multiply, the conjugate of 1 + 3i, 1 - 3i

to both of the numerator and the denominator.

Rationalize Denominator

(The conjugate of [a + b] is [a - b].

The conjugate of [a - b] is [a + b].)

2(1 - 3i) = 2 - 6i

Multiply a Monomial and a Polynomial

(1 + 3i)(1 - 3i)

= 1^{2} + 3^{2}

= 1 + 9

So

[2/(1 + 3i)]⋅[(1 - 3i)/(1 - 3i)]

= (2 - 6i)/(1 + 9).

1 + 9 = 10

Divide the numerator and the denominator

by 2.

So (1 - 3i)/5 is the answer.

## Example 2

### Example

### Solution

The denominator, 2 - i, is a complex number.

To rationalize the denominator 2 - i,

multiply, the conjugate of 2 - i, 2 + i

to both of the numerator and the denominator.

(The conjugate of [a + b] is [a - b].

The conjugate of [a - b] is [a + b].)

(1 - 7i)(2 + i)

= 2 + i - 14i + 7

Multiply Complex Numbers

(2 - i)(2 + i)

= 2^{2} + 1^{2}

= 4 + 1

So

[(1 - 7i)/(2 - i)]⋅[(2 + i)/(2 + i)]

= (2 + i - 14i + 7)/(4 + 1).

2 + 7 = 9

+i - 14i = -13i

Add and Subtract Complex Numbers

4 + 1 = 5

So (9 - 13i)/5 is the answer.