10.7 Complex Numbers

Objective 1

Simplify numbers of the form where b > 0.

,b

Slide 10.7- 2

Imaginary Unit i

The imaginary unit i is defined as

That is, i is the principal square root of –1.

21, where 1.i i

Slide 10.7- 3

Simplify numbers of the form where b > 0.

,b

For any positive real number b, .b i b

b

Slide 10.7- 4

Simplify numbers of the form where b > 0.

,b

It is easy to mistake for with the i under the radical. For this reason,

we usually write as as in the definition of

2i 2i

2i 2,i .b

Write each number as a product of a real number and i.

25 25i 5i

81 81i 9i

7

44 44i 4 11i 2 11i

7i

Slide 10.7- 5

CLASSROOM EXAMPLE 1 Simplifying Square Roots of Negative Numbers

Solution:

Multiply.

6 5 6 5i i

2 6 5i

( 1) 30

8 6

30

8 6i i 2 8 6i 2 48i2 16 3i

4 3

5 7 5 7i

35i

Slide 10.7- 6

CLASSROOM EXAMPLE 2 Multiplying Square Roots of Negative Numbers

Solution:

16 25 16 25i i 4 5i i

220i

20 1

20

Divide.

80

5

80

5

i

i

80

5

16

4

40

10

40

10

i

40

10i

4i

2i

Slide 10.7- 7

CLASSROOM EXAMPLE 3 Dividing Square Roots of Negative Numbers

Solution:

Objective 2

Recognize subsets of the complex numbers.

Slide 10.7- 8

Complex Number

If a and b are real numbers, then any number of the form a + bi is called a complex number. In the complex number a + bi, the number a is called the real part and b is called the imaginary part.

Slide 10.7- 9

Recognize subsets of the complex numbers.

For a complex number a + bi, if b = 0, then a + bi = a, which is a real number.

Thus, the set of real numbers is a subset of the set of complex numbers.

If a = 0 and b ≠ 0, the complex number is said to be a pure imaginary number.

For example, 3i is a pure imaginary number. A number such as 7 + 2i is a nonreal complex number.

A complex number written in the form a + bi is in standard form.

Slide 10.7- 10

Recognize subsets of the complex numbers.

The relationships among the various sets of numbers.

Slide 10.7- 11

Recognize subsets of the complex numbers.

Objective 3

Add and subtract complex numbers.

Slide 10.7- 12

Add.

( 1 8 ) (9 3 )i i ( 1 9) ( 8 3)i

8 11i

( 3 2 ) (1 3 ) ( 7 5 )i i i

[ 3 1 ( 7)] [2 ( 3) ( 5)]i

9 6i

Slide 10.7- 13

CLASSROOM EXAMPLE 4 Adding Complex Numbers

Solution:

Subtract.

( 1 2 ) (4 )i i ( 1 4) (2 1)i 5 i

(8 5 ) (12 3 )i i (8 12) [ 5 ( 3)]i

4 2i (8 12) ( 5 3)i

Slide 10.7- 14

CLASSROOM EXAMPLE 5 Subtracting Complex Numbers

Solution:

( 10 6 ) ( 10 10 )i i [ 10 ( 10)] (6 10)i

0 4i 4i

Objective 4

Multiply complex numbers.

Slide 10.7- 15

Multiply.

6 (4 3 )i i 6 (4) 6 (3 )i i i 224 18i i

24 18( 1)i

18 24i

Slide 10.7- 16

CLASSROOM EXAMPLE 6 Multiplying Complex Numbers

Solution:

(6 4 )(2 4 )i i 6(2) 6(4 ) ( 4 )(2) ( 4 )(4 )First Outer Inner Last

i i i i

212 24 8 16i i i

12 16 6 )11 (i

12 16 16i

28 16i

Slide 10.7- 17

CLASSROOM EXAMPLE 6 Multiplying Complex Numbers (cont’d)

Multiply.

Solution:

(3 2 )(3 4 )i i 3(3) 3(4 ) (2 )(3) (2 )(4 )First Outer Inner Last

i i i i

29 12 6 8i i i

9 18 8 )1(i

9 18 8i

1 18i

Slide 10.7- 18

CLASSROOM EXAMPLE 6 Multiplying Complex Numbers (cont’d)

Multiply.

Solution:

The product of a complex number and its conjugate is always a real number.

(a + bi)(a – bi) = a2 – b2( –1) = a2 + b2

Slide 10.7- 19

Multiply complex numbers.

Objective 5

Divide complex numbers.

Slide 10.7- 20

Find the quotient.

23

3

i

i

(23 )(3 )

(3 )(3 )

i i

i i

2

69 23 3 1

3 1

i i

70 20

10

i

10(7 2 )

10

i 7 2i

Slide 10.7- 21

CLASSROOM EXAMPLE 7 Dividing Complex Numbers

Solution:

5 i

i

(5 )( )

( )

i i

ii

2

2

5i i

i

5 ( 1)

( 1)

i

5 1

1

i 1 5i

Slide 10.7- 22

CLASSROOM EXAMPLE 7 Dividing Complex Numbers (cont’d)

Find the quotient.

Solution:

Objective 6

Find powers of i.

Slide 10.7- 23

Because i2 = –1, we can find greater powers of i, as shown below.

i3 = i · i2 = i · ( –1) = –i

i4 = i2 · i2 = ( –1) · ( –1) = 1

i5 = i · i4 = i · 1 = i

i6 = i2 · i4 = ( –1) · (1) = –1

i7 = i3 · i4 = (i) · (1) = –I

i8 = i4 · i4 = 1 · 1 = 1

Slide 10.7- 24

Find powers of i.

Find each power of i.

28i 74i 7 11

19i 16 3i i 44 3i i 41 ( ) ii

9i 9

1

i

8

1

i i

24

1

i i

2

1

1 i

1

i

1( )

( )

i

i i

2

i

i

( 1)

i

1i

i

Slide 10.7- 25

CLASSROOM EXAMPLE 8 Simplifying Powers of i

Solution:

22i 22

1

i

20 2

1

i i

54

1

( 1)i

5

1

1 ( 1)

1

11