Complex Numbers10.710.7

1. Write imaginary numbers using i.2. Perform arithmetic operations with complex numbers.3. Raise i to powers.

Imaginary unit:

Imaginary number: A number that can be expressed in the form bi, where b is a real number and i is the imaginary unit.

i3 i9 i52

1i12 i

16

21

32

1 16 4i 4i

1 21 21i

1 32 16 2i 4 2i

1

12

i

i

Complex number: A number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit.

i34 i57 i54

32

Examples:

realim

aginary

Complex Numbers: a + bComplex Numbers: a + bii

b = 0: Real numbers

a = 0: Imaginary numbers

real imaginary

Add Complex NumbersAdd Complex Numbers

ii 5435

1

12

i

i

ii 5435

Add the real parts – add the imaginary parts

i89

Subtract Complex NumbersSubtract Complex Numbers

ii 2138

1

12

i

i

ii 2138

i9

Slide 10- 9Copyright © 2011 Pearson Education, Inc.

Simplify. (4 + 7i) – (2 + i)

a) 2 + 7i2

b) 2 + 8i

c) 6 + 6i

d) 6 + 8i

Slide 10- 10Copyright © 2011 Pearson Education, Inc.

Simplify. (4 + 7i) – (2 + i)

a) 2 + 7i2

b) 2 + 8i

c) 6 + 6i

d) 6 + 8i

Multiply Complex NumbersMultiply Complex Numbers

ii 74

1

12

i

i

228i

128

28

Multiply Complex NumbersMultiply Complex Numbers

ii 857

1

12

i

i

25635 ii

15635 i

5635 i

i3556

standard a + bi form

Multiply Complex NumbersMultiply Complex Numbers

ii 425

1

12

i

i

228520 iii

22320 ii

12320 i

2320 i

i322

Multiply Complex NumbersMultiply Complex Numbers

235 i

1

12

i

i

29151525 iii

193025 i

93025 i

i3016

Rewrite & Foil

ii 3535

Slide 10- 15Copyright © 2011 Pearson Education, Inc.

Multiply. (4 + 7i)(2 + i)

a) 15 + 10i

b) 1 + 10i

c) 15 + 18i

d) 15 + 18i

Slide 10- 16Copyright © 2011 Pearson Education, Inc.

Multiply. (4 + 7i)(2 + i)

a) 15 + 10i

b) 1 + 10i

c) 15 + 18i

d) 15 + 18i

Divide Complex NumbersDivide Complex Numbers

i76

1

12

i

i

ii

i

76

276ii 17

6

i

76

i76i

76i

Binomial denominator conjugate

Divide Complex NumbersDivide Complex Numbers

i65

1

12

i

i

i

ii

6

66

5236

530ii

136

530

i

i375

3730 standard a + bi form

37530 i

Slide 10- 19Copyright © 2011 Pearson Education, Inc.

Write in standard form.

a)

b)

c)

d)

4

2 3

i

i

5 14

13 13

i

5 14

13 13

i

11 14

13 13

i

11 14

13 13

i

Slide 10- 20Copyright © 2011 Pearson Education, Inc.

Write in standard form.

a)

b)

c)

d)

4

2 3

i

i

5 14

13 13

i

5 14

13 13

i

11 14

13 13

i

11 14

13 13

i

Powers of i:Powers of i:1

12

i

i

i

111224 iii

iiiii 123

12 i

111

1

111

1

448

347

246

45

iii

iiiii

iii

iiiii

41i ii 104 ii 1

15i 334 ii ii 1

i

111224 iii

iiiii 123

12 i

Powers of i:Powers of i:

432 iiii

Simplify: