# Complex Conjugates

How to rationalize the complex denominators by using the complex conjugates: definition, formula, examples, and their solutions.

## Definition, Formula

The conjugate of [*a* + *bi*] is [*a* - *bi*].

The conjugate of [*a* - *bi*] is [*a* + *bi*].

So [*a* + *bi*] and [*a* - *bi*]

are the conjugates of each other.

The product of these two conjugates is *a*^{2} + *b*^{2}.

This formula is used

to rationalize a complex denominator.

## Example 1: Simplify 2/(1 + 3*i*)

The conjugate of [1 + 3*i*] is [1 - 3*i*].

So, to rationalize the denominator,

multiply (1 - 3*i*)/(1 - 3*i*).

(1 + 3*i*)(1 - 3*i*) = 1^{2} + 3^{2}

The denominator's middle sign is (+), not (-).

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

1^{2} + 3^{2} = 1 + 9

1 + 9 = 10

Divide the numerator by 2

and reduce the denominator 10 to 5.

Then (given) = (1 - 3*i*)/5.

## Example 2: Simplify (1 - 7*i*)/(2 - *i*)

The conjugate of [2 - *i*] is [2 + *i*].

So, to rationalize the denominator,

multiply (2 + *i*)/(2 + *i*).

(2 - *i*)(2 + *i*) = 2^{2} + 1^{2}

The denominator's middle sign is (+), not (-).

(1 - 7*i*)(2 + *i*) = 2 + *i* - 14*i* - 7⋅(-1)

Multiplying complex numbers

2^{2} + 1^{2} = 4 + 1

+*i* - 14*i* = -13*i*

-7⋅(-1) = +7

4 + 1 = 5

2 + 7 = 9

So (given) = (9 - 13*i*)/5.

## Example 3: Multiplicative Inverse of 7 + 2*i*

The multiplicative inverse (= reciprocal) of [7 + 2*i*] is

1/(7 + 2*i*).

So it says to simplify 1/(7 + 2*i*).

The conjugate of [7 + 2*i*] is [7 - 2*i*].

So, to rationalize the denominator,

multiply (7 - 2*i*)/(7 - 2*i*).

(7 + 2*i*)(7 - 2*i*) = 7^{2} + 2^{2}

The denominator's middle sign is (+), not (-).

7^{2} + 2^{2} = 49 + 4

49 + 4 = 53

So (given) = (7 - 2*i*)/53.