Complex Conjugates

Complex Conjugates

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

Definition, Formula

(a + bi)(a - bi) = a^2 - (b^2)(-1) = a^2 + b^2

Let's see the product of the complex conjuates.

Product of a sum and a difference

Powers of i

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

The product of the complex conjuates
makes a real number.

There are no i because of the squares.

Example 1

Simplify the given expression. 2 / (1 + 3i)

The product of the complex conjuates
makes a real number.

This property can be used
when rationalizing complex denominators.

To rationalize 1 + 3i,
multiply its conjugate (1 - 3i)
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 b2 term.

Example 2

Simplify the given expression. (1 - 7i) / (2 - i)

To rationalize 2 - i,
multiply its conjugate (2 + i)
to both of the numerator and the denominator.

Multiplying complex numbers

Example 3

Find the multiplicative inverse of the given expression. 7 + 2i

The multiplicative inverse (or the reciprocal)
of 7 + 2i is 1 / (7 + 2i).

So simplify 1 / (7 + 2i).

To rationalize 7 + 2i,
multiply its conjugate (7 - 2i)
to both of the numerator and the denominator.