Complex Numbers

Add and Subtract complex numbers

Multiply and divide complex numbers

Lets solve x 2 = -1

To solve these problem, French mathematician René Descartes

proposed that a number i be defined such that i 2 = -1. iSo 1

Simplify

No Decimals

28

Simplify

No Decimals

7272

74128

ii

Simplify

No Decimals

3

3

32

32

yi

y

Simplify

No Decimals

yyi

yi

y

24

32

32

3

3

Powers of i values go a in a circle

Power of i are a cyclical function

1

1

1

4

3

2

i

ii

i

i

16128

15117

14106

1395

1;1

;

1;1

1,

iii

iiiii

iii

iiii

Powers of i values go a in a circle

Find the value the expression

1

1

1

4

3

2

i

ii

i

i

158

29438 1

i

iii

Powers of i values go a in a circle

Find the value the expression

1

1

1

4

3

2

i

ii

i

i

1

1

2394158

29438

iii

iii

Adding

Pure imaginary numbers are like terms.

iii 17512

Multiply Imaginary Numbers

Remember i2 equals -1

6

)1(6

6

23

2

i

ii

Multiply Imaginary Numbers

Pull the i out first

212

212

ii

Multiply Imaginary Numbers

Pull the i out first

62

62)1(

3222

212

212

2

i

ii

Solve

Remember when you take the square root of a number you must remember its positive and negative answer.

iy

y

y

y

y

2

4

4

205

0205

2

2

2

Complex number

Have two parts: Real and Imaginary

bia alRe aginaryIm

When adding complex numbers,

add the real numbers together and then the imaginary numbers.

ii 4253

When adding complex numbers,

add the real numbers together and then the imaginary numbers.

i

i

ii

5

4523

4253

When subtracting complex numbers,

add the real numbers together and then the imaginary numbers.

ii 7364

When subtracting complex numbers,

add the real numbers together and then the imaginary numbers.

ii

ii

7364

7364

When subtracting complex numbers,

add the real numbers together and then the imaginary numbers.

i

ii

ii

1

7364

7364

Multiplying Complex numbers

Its time to remember how to FOIL again.

2315420

534

iii

ii

Multiplying Complex numbers

Its time to remember how to FOIL again.

1315420

315420

534

2

ii

iii

ii

Multiplying Complex numbers

Its time to remember how to FOIL again.

i

i

ii

iii

ii

1123

31120

1315420

315420

534

2

Divide Complex numbers

Complex Conjugate work as with real numbers conjugates. Important must

break into fractions

13

1510

149

15110

49

1015

23

23

23

5

23

5

2

2

ii

i

ii

i

i

i

i

i

i

i13

15

13

10

Homework

Page 274

# 19 – 41 odd,

49 – 55 odd

Homework

Page 274

# 18 – 40 even,

48 – 54 even