04/10/23 23:56 15-4 - Complex Numbers 1

WARM-UP1. 4.

2. 5.

3.

25 5

3 3

5 27 3 10x x x

4 63 24x y

2

2x y

3

7 2

04/10/23 23:56 15-4 - Complex Numbers 2

COMPLEX NUMBERS

04/10/23 23:56 15-4 - Complex Numbers 3

IMAGINARY NUMBERSImaginary Numbers are numbers can be

written as a real number times “i.”

04/10/23 23:56 10.7 - Complex Numbers 4

IMAGINARY NUMBERSImaginary Numbers are numbers that can be

written as a real number times “i.”

Steps:1. View the radical without the negative and

simplify it2. After simplified form, attach an “i” onto the real

number3. Simplify further, if needed

04/10/23 23:56 10.7 - Complex Numbers 5

EXAMPLE 1SimplifyAfter simplified form, attach an “i” onto the real

number

4

4i 2i2i2i

04/10/23 23:56 10.7 - Complex Numbers 6

EXAMPLE 1

x = 0

Where does it cross the x-axis?

+2i

Simplify

04/10/23 23:56 10.7 - Complex Numbers 7

EXAMPLE 2

2i 2i

4

Simplify

04/10/23 23:56 10.7 - Complex Numbers 8

EXAMPLE 3

11i 11i

11

Simplify

04/10/23 23:56 10.7 - Complex Numbers 9

EXAMPLE 4

2 5i2 5i

20

Simplify

04/10/23 23:56 10.7 - Complex Numbers 10

YOUR TURN

5 3i 5 3i

75

Rules with Imaginaries:i1 = ii2 = –1

Answers CAN have an i but CAN NOT have two i’s

Two i’s make –1

04/10/23 23:56 10.7 - Complex Numbers 11

IMAGINARY NUMBERS

04/10/23 23:56 10.7 - Complex Numbers 12

WHY IS I2 EQUAL TO –1?

1i

1i 2 2

2 1i

How do we cancel the radicals?

Multiply

What is another way of writing this problem?

04/10/23 23:56 10.7 - Complex Numbers 13

EXAMPLE 5

3 3

2

3i

3 3i i

2i 2

31 3

Multiply

04/10/23 23:56 10.7 - Complex Numbers 14

EXAMPLE 6

16 16

24i

Multiply

04/10/23 23:56 10.7 - Complex Numbers 15

YOUR TURN

50 50

2

5 2i

Multiply

04/10/23 23:56 10.7 - Complex Numbers 16

EXAMPLE 7

6 6

6 6

6 6 6 6 36

6 6 6 6i i

22 6i

2

6 6

Multiply

04/10/23 23:56 10.7 - Complex Numbers 17

EXAMPLE 8

15 15

3 5

Multiply

04/10/23 23:56 10.7 - Complex Numbers 18

EXAMPLE 9

4i4i

8 2

Multiply

04/10/23 23:56 10.7 - Complex Numbers 19

YOUR TURN

28x 28x

34 16x x

Rationalize

04/10/23 23:56 10.7 - Complex Numbers 20

REVIEW1

2 6

1

2 6 1 6

2 6 6

6

2 6

6

12

6

12

Rationalize

04/10/23 23:56 10.7 - Complex Numbers 21

EXAMPLE 101

3i

23

i

i 1 3

i

3

i

3

i

1

3

i

i i

No i ’s AND radicals in the denominator

Rationalize

04/10/23 23:56 10.7 - Complex Numbers 22

EXAMPLE 111

6

1 6

6 6

i

i i

2

6

6

i

i

6

6

i

6

6

i

6

1 6

i

No i ’s AND radicals in the denominator

Rationalize

04/10/23 23:56 10.7 - Complex Numbers 23

EXAMPLE 1249

10

7 10

10

i

7 10

10

i

Rationalize

04/10/23 23:56 10.7 - Complex Numbers 24

YOUR TURN40

8

5i 5i

04/10/23 23:56 10.7 - Complex Numbers 25

COMPLEX NUMBERSComplex Numbers

a + b i

Conjugate is the complex number’s opposite sign Example: 2 + 3i ‘s conjugate is 2 – 3i

Remember: NO IMAGINARY NUMBERS in the denominator

SIMPLIFY RADICALS FIRST then OPERATE

Real Number

Imaginary Number

Add

04/10/23 23:56 10.7 - Complex Numbers 26

REVIEW 2 5 3 4x x

5 x5 x

Add

04/10/23 23:56 10.7 - Complex Numbers 27

EXAMPLE 13 2 5 3 4i i

5 i5 i

Subtract

04/10/23 23:56 10.7 - Complex Numbers 28

EXAMPLE 14 7 3 5 2i i

2 5i2 5i

7 3 5 2i i

Subtract

04/10/23 23:56 10.7 - Complex Numbers 29

YOUR TURN 2 4 2i i

3i 3i

MultiplyFOIL (First Outer Inner Last)

24 2 2x x x

04/10/23 23:56 10.7 - Complex Numbers 30

REVIEW 22 x

2 2x x

44 2x4 2 2x x 2 4 4x x

FOIL (First Outer Inner Last)Multiply

04/10/23 23:56 10.7 - Complex Numbers 31

EXAMPLE 15 22 i

24 2 2i i i 2 2i i

44 2i4 2 2i i 24 4i i

4 4 1i 3 4i3 4i

Multiply

04/10/23 23:56 10.7 - Complex Numbers 32

EXAMPLE 16 2 3 6i i

15 16i15 16i

212 2 18 3( )i i i

12 16 3( 1)i

12 16 3i

Multiply

04/10/23 23:56 10.7 - Complex Numbers 33

YOUR TURN 9 2 9 2i i

8585

Rationalize

What is the problem with the bottom?

04/10/23 23:56 10.7 - Complex Numbers 34

REVIEW1

4 3

1

4 3

4 3

4 3

We try to get rid of the radicals on the

bottom so the bottom will be even

With the CONJUGATE

4 3

16 3

4 3

13

4 3

13 13

or

Rationalize

What is the problem with the bottom?

04/10/23 23:56 10.7 - Complex Numbers 35

EXAMPLE 171

4 3i

1

4 3i

4 3

4 3

i

i

We try to get rid of the radicals on the

bottom so the bottom will be even

With the CONJUGATE

2

4 3

16 3 3 3

i

i i i

4 3

19

4 3

19 19

i

or

i

Rationalize

04/10/23 23:56 10.7 - Complex Numbers 36

EXAMPLE 181

2 3i

2 3

13

i

Rationalize

04/10/23 23:56 10.7 - Complex Numbers 37

EXAMPLE 192

1

i

i

1 3

2

i

Rationalize

04/10/23 23:56 10.7 - Complex Numbers 38

YOUR TURN3 2

3 2

i

i

5 12

13

i

04/10/23 23:56 COMPLEX NUMBERS 39

IMAGINARY POWERSSteps:1. Look at the exponent of the

imaginary number and divide by 4 2. View only the remainder or decimal3. Convert the problem with the number

REMEMBER: No i’s in the denominator

04/10/23 23:56 COMPLEX NUMBERS 40

IMAGINARY NUMBERS

Imaginary Imaginary ExponentExponent

RemainderRemainder DecimalDecimal Imaginary Imaginary NumberNumber

i1 R1 0.25 i

i2 R2 0.50 –1

i3 R3 0.75 –i

i4 R0 0.00 1

04/10/23 23:56 COMPLEX NUMBERS 41

IMAGINARY NUMBERS40 i 1

11 i i

22 i -1

33 i i

0 0.0R

1 0.25R

2 0.50R

3 0.75R

Simplify

04/10/23 23:56 10.7 - Complex Numbers 42

EXAMPLE 207i

7i 7 4 1 3

1.75

r

Imaginary Imaginary ExponentExponent

RemainderRemainder DecimalDecimal Imaginary Imaginary NumberNumber

i1 R1 0.25 i

i2 R2 0.50 –1

i3 R3 0.75 –i

i4 R0 0.00 1

i

Simplify

04/10/23 23:56 10.7 - Complex Numbers 43

EXAMPLE 2125i

i

24i i

25

46 1r

1r i

Simplify

04/10/23 23:56 10.7 - Complex Numbers 44

YOUR TURN589,276,538i

1

Simplify

i25 = ii36 = 1

04/10/23 23:56 10.7 - Complex Numbers 45

EXAMPLE 2225 36i i

i

Simplify

04/10/23 23:56 10.7 - Complex Numbers 46

YOUR TURN21 30i i

1 i

Simplify

04/10/23 23:56 10.7 - Complex Numbers 47

EXAMPLE 24 7

2i

128i