Warm-up:

Perform the indicated operation.

2 21. (3 4 7) (5 4 10)x x x x

2. (3 2)( 5)x x

Find the area and perimeter of the box.

3. Perimeter = ____

4. Area = ____2x+1

2x-3

Complex numbers

2

2

2

i

So here’s what the math people did: They used the letter “i” to represent the square root of (-1). “i” stands for “imaginary”

1i

So, does

1really exist?

Examples of how we use

1i

16

4 i 4i

16 1

Examples of how we use

1i

45

3 5 1

3 5 i

3 5i

45 1

then,1- If i

1*2 iii

i’s with exponents

Complex Numbers

A complex number has a real part & an imaginary part.

Standard form is:

bia

Real part Imaginary partExample: 5+4i

Adding and Subtracting complex numbersDo just like polynomial adding or subtracting

Ex: (3 2 ) (7 6 )i i (3 7) (2 6 )i i

10 8i Ex: (6 5 ) (1 2 )i i

(6 5 ) ( 1 2 )i i (6 1) ( 5 2 )i i

5 7i

MultiplyingDo just like polynomial multiplication but when you finish change i2 to -1

Ex: )3( ii 23 ii

)1(3 i

i31

Ex: )26)(32( ii 2618412 iii

)1(62212 i62212 i

i226

Conjugates: Two complex numbers of the form a + bi anda – bi are conjugates. The product is always a real number

Ex: (2 4 )(2 4 )i i 24 8 8 16i i i

4 16( 1)

20

Dividing Complex Numbers

Multiply the numerator and denominator by the conjugate of the denominator.

5 2 3 8

ii

(5 2 )(3 8 )(3 8 )(3 8 )

i ii i

2

2

15 40 6 169 24 24 64

i i ii i i

15 46 16( 1)9 64( 1)i

15 46 169 64i

1 4673

i

1 4673 73

i

5 1 i

5 52 2

i

6 32ii

3 32

i