Beginning Algebra, 11th Edition

(c)

Multiply.

Add. NOW TRY

OBJECTIVE 4 Divide complex numbers.The quotient of two complex num-

bers, such as

is expressed in standard form by changing the denominator into a real number.

As seen in Example 3(c),the product is 5, a real number. This

suggests multiplying the numerator and denominator of the given quotient by

as follows.

Multiply.

Combine like terms.

Factor out 5.

= 2 - 3 i Divide out the common factor.

=

512 - 3 i 2

5

=

10 - 15 i

5

=

8 - 16 i+i- 21 - 12

1 - 41 - 12

=

8 - 16 i+i- 2 i^2

1 - 4 i^2

1 - 2 i

= 1 - 2 i= 1

8 +i

1 + 2 i

#^1 -^2 i

1 - 2 i

8 +i

1 + 2 i

1 - 2 i

11 + 2 i 211 - 2 i 2

8 +i

1 + 2 i

,

= 5

= 1 + 4

= 1 - 41 - 12 i^2 =- 1

= 1 - 4 i^21 xy 22 =x^2 y^2

= 12 - 12 i 22 1 x+y 21 x-y 2 =x^2 - y^2

11 + 2 i 211 - 2 i 2

SECTION 9.4 Complex Numbers 577

NOW TRY
EXERCISE 3
Find each product.

(a)

(b)

(c) 15 - 7 i 215 + 7 i 2

12 - 4 i 213 + 2 i 2

8 i 11 - 3 i 2

NOW TRY ANSWERS

3. (a) (b)
   (c) 74

24 + 8 i 14 - 8 i

Use parentheses

around to avoid

errors.

• 1

Factor first.

Then divide out

the common

factor.

The complex numbers and are conjugates.That is, the conjugateof

the complex number is Multiplying the complex number by

its conjugate a-bigives the real number a^2 +b^2.

abi abi. a+bi

1 + 2 i 1 - 2 i

Product of Conjugates

That is, the product of a complex number and its conjugate is the sum of the

squares of the real and imaginary parts.

1 abi 21 abi 2 a^2 b^2

To divide complex numbers, multiply both the numerator and denominator by the

conjugate of the denominator. We used a similar method to rationalize some radical

expressions in Chapter 8.