Daily Check:

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

Homework Review

CCGPS Analytic GeometryDay 32 (9-20-13)

UNIT QUESTION: In what ways can algebraic methods be used in problem solving?Standard: MCC9-12.N.RN.1-3, N.CN.1-3, A.APR.1

Today’s Question:How do we take the square root of negative numbers?Standard: MCC9-12..N.CN.1-3

2

2

2

i1i

• You can't take the square root of a negative number, right?

• When we were young and still in Math I, no numbers that, when multiplied by themselves, gave us a negative answer. 

• Squaring a negative number always gives you a positive.   (-1)² = 1. (-2)² = 4 (-3)² = 9

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 16 1

4 i 4i

81 81 1

9 i 9i

Examples of how we use

1i

45 45 1

3 3 5 1

3 5 1

3 5 i

3 5i

2 2 2 5 5 1

200 200 1

2 5 2 1

10 2 i

10 2i

1.3Powers of i and

Complex Operations

Since -1, theni

12 i

ii 3

14 i

ii 5

16 i

ii 7

18 i

*For larger exponents, divide the exponent by 4, then use the remainder

as your exponent instead.

Example: ?23 i3 ofremainder a with 5

423

.etcii - which use So, 3

ii 23

!Try These131. i272. i

543. i

724. i

$25,000 Pyramid

$25,000 Pyramid

2i-1 -i 1 -1

i -1 -i

-i 1

i

3i 4i 14i

45i 54i 63i

71i 92i

141i

$25,000 Pyramid

2i-1 -i 1 -i

-i -1 i

-1 -i

i

3i 4i 23i

35i 66i 77i

82i 95i

173i

Complex Numbers

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

Standard form is:

bia

Real part Imaginary partExample: 5+4i

The Complex Plane

Imaginary Axis

Real Axis

Graphing in the complex plane

i34 .

i52 .i22 .

i34

.

Adding and SubtractingAdd or subtract the real parts, and then, add or subtract the imaginary parts.

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 3i

1) (9-4i)-(-2+3i)

11 7i

Your Turn!

2) 9-(10+2i)-5i

1 7i

4 3 4 33) (11i 2 ) (2 6 )i i i

9 8i

Your Turn!

MultiplyingTreat the i’s like variables, then change any that are not to the first power

Ex: )3( ii 23 ii

)1(3 i

i31

Ex: )26)(32( ii 2618412 iii

)1(62212 i62212 i

i226

1) ( 3 )(8 5 )i i

29 7i

Your Turn!

2) (4 3 )(4 3 )i i

25

3) 2 (1 4 )i i

8 2i

Your Turn!

4) (3 2 )( 5 9 )i i

33 17i

Conjugates: Two complex numbers of the form a + bi anda – bi are complex 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

Conjugates: Two complex numbers of the form a + bi anda – bi are complex 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

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

Dividing Complex Numbers

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

Simplify completely.

5 2 3 81) *3 8 3 8

i ii i

(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

Writing in Standard Form

52) 1 i

5 52 2

i

Your Turn!

8 33) 1 2

ii

2 195 5

i

6 34) 2ii

3 32

i

Your Turn!

5 65) 3ii

523

i

Assignment

Complex Numbers Practice WS