Intermediate Algebra (11th edition)

OBJECTIVE 3 Add and subtract complex numbers.The commutative, asso-

ciative, and distributive properties for real numbers are also valid for complex

numbers. Thus, to add complex numbers, we add their real parts and add their

imaginary parts.

Adding Complex Numbers

Add.

(a)

Properties of real numbers

Add real parts. Add imaginary parts.

(b)

Associative property

= 1 + 4 i

= 34 + 3 + 1 - 624 + 32 + 1 - 12 + 34 i

14 + 2 i 2 + 13 - i 2 + 1 - 6 + 3 i 2

= 8 + 7 i

= 12 + 62 + 13 + 42 i

12 + 3 i 2 + 16 + 4 i 2

EXAMPLE 4

SECTION 8.7 Complex Numbers 477

NOW TRY
EXERCISE 4
Add.

(a)

(b)

• 16 - 4 i 2

15 - i 2 + 1 - 3 + 3 i 2

1 - 3 + 2 i 2 + 14 + 7 i 2

NOW TRY
EXERCISE 5
Subtract.

(a)

(b)

(c) 1 - 1 + 12 i 2 - 1 - 1 - i 2

15 - 2 i 2 - 19 - 7 i 2

17 + 10 i 2 - 13 + 5 i 2

Add real parts.

Add imaginary parts.

To subtract complex numbers, we subtract their real parts and subtract their

imaginary parts.

Subtracting Complex Numbers

Subtract.

(a)

Properties of real numbers

Subtract real parts. Subtract imaginary parts.

(b)

=- 1 + 3 i

= 17 - 82 + 3 - 3 - 1 - 624 i

17 - 3 i 2 - 18 - 6 i 2

= 3 + 3 i

= 16 - 32 + 15 - 22 i

16 + 5 i 2 - 13 + 2 i 2

EXAMPLE 5

(c)

=- 4 i NOW TRY

= 0 - 4 i

= 1 - 9 + 92 + 14 - 82 i

1 - 9 + 4 i 2 - 1 - 9 + 8 i 2

OBJECTIVE 4 Multiply complex numbers.

Multiplying Complex Numbers

Multiply.

(a)

Distributive property

Multiply.

Substitute for

Standard form

(b)

Use the FOIL method.

First Outer Inner Last

Multiply.

Combine imaginary terms;

Multiply.

= 22 + 14 i Combine real terms.

= 12 + 14 i+ 10

= 12 + 14 i- 101 - 12 i^2 =-1.

= 12 - 6 i+ 20 i- 10 i^2

= 3142 + 31 - 2 i 2 + 5 i 142 + 5 i 1 - 2 i 2

13 + 5 i 214 - 2 i 2

=- 12 + 8 i

= 8 i+ 121 - 12 - 1 i^2.

= 8 i+ 12 i^2

= 4 i 122 + 4 i 13 i 2

4 i 12 + 3 i 2

EXAMPLE 6

NOW TRY ANSWERS

4. (a) (b)


5. (a) (b)
   (c) 13 i

4 + 5 i - 4 + 5 i

1 + 9 i 8 - 2 i

NOW TRY