Divide Complex Numbers

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

Formula

Formula

(a + bi)(a - bi)

= a2 - (bi)2
Product of a Sum and a Difference

= a2 - b2⋅i2
Power of a Product

= a2 - b2⋅(-1)
Power of i

= a2 + b2

So (a + bi)(a - bi) = a2 + b2.

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].)

(1 + 3i)(1 - 3i)
= 12 + 32
= 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)
= 22 + 12
= 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.