# Complex Conjugates

How to use the complex conjugates to divide complex numbers: definition, formula, examples, and their solutions.

## Definition, Formula

Let's see the product of the complex conjuates.

Product of a sum and a difference

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

The product of the complex conjuates

makes a real number.

There are no *i* because of the squares.

## Example 1

The product of the complex conjuates

makes a real number.

This property can be used

when rationalizing complex denominators.

To rationalize 1 + 3*i*,

multiply its conjugate (1 - 3*i*)

to both of the numerator and the denominator.

This part is similar to

rationalizing a radical denominator.

Rationalizing a denominator

The only difference is the (+) sign of the *b*^{2} term.

## Example 2

To rationalize 2 - *i*,

multiply its conjugate (2 + *i*)

to both of the numerator and the denominator.

## Example 3

The multiplicative inverse (or the reciprocal)

of 7 + 2*i* is 1 / (7 + 2*i*).

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

To rationalize 7 + 2*i*,

multiply its conjugate (7 - 2*i*)

to both of the numerator and the denominator.