Warm Up Warm Up 1. 3 12 6

22.

11

Simplify

SAT Review question

Dana walks from home to school at a rate of 5 mph. It takes her 2 hourslonger to walk home from school than it did to walk to school. If her totalwalking time to and from school was 8 hours, what was Dana’s rate ofspeed walking home from school?

a) 3 b) 4 c) 5 d) 8 e) 15

Solve 2

2

3. 2 1 17

4. 7 10 1

x

x

Section 5.4 Complex Numbers Section 5.4 Complex Numbers Essential QuestionEssential Question

• What is an imaginary number?

Imaginary UnitImaginary Unit• Until now, you have always been told

that you can’t take the square root of a negative number. If you use imaginary units, you can!

• The imaginary unit is i.• i= • It is used to write the square root of

a negative number.

1

Property of the square root Property of the square root of negative numbersof negative numbers

• If r is a positive real number, then

r ri

Examples:

3 3i 4 4i i2

then,1- If i

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

4

23

.etcii - which use So, 3

ii 23

ExamplesExamples2)3( 1. i

22 )3(i)3*3(1

)3(13

26103 Solve 2. 2 x

363 2 x122 x

122 x

12ix 32ix

Complex NumbersComplex Numbers• A complex number has a real part &

an imaginary part.• Standard form is:

bia

Real part Imaginary part

Example: 5+4iExample: 5+4i

Adding and Subtracting complex numbersAdding and Subtracting complex numbers1. add or subtract the real parts1. add or subtract the real parts

2. add or subtract the imaginary parts2. add or subtract the imaginary parts

Ex: )33()21( ii )32()31( ii

i52

Ex: ( 2 8) (5 50)

You try!You try!

(13 2 ) ( 5 6 )

(8 18) (4 3 2)

i i

i

MultiplyingMultiplying

1. Treat the i’s like a variable1. Treat the i’s like a variable2. Change any that are not to the 2. Change any that are not to the

first powerfirst powerEx: 5 * 10

5 * 10i i2 50i

1(5 2)

5 2

Ex: )26)(32( ii 2618412 iii

)1(62212 i62212 i

i226

You try!You try!

(6 2 )(2 3 )

8 (9 4 )

i i

i i

ConjugatesConjugates• The conjugate of a complex number

has the same real part and the opposite imaginary part

• Ex. Find the conjugate of 5 + 3i5 – 3i

Ex. Find the conjugate of 3 – 2i 3 + 2i

Imaginary numbers in the Imaginary numbers in the denominatordenominator

• i’s cannot be in the denominator (like radicals)

• To get rid of the i’s, multiply numerator and denominator by the conjugate

• If there is only an imaginary part in the denominator, multiply by the same imaginary number

ExampleExample14

2i

14 2*

2 2

i

i i

2

28

4

i

i

28

4

i

7i

3 11:

1 2

iEx

i

)21)(21(

)21)(113(

ii

ii

2

2

4221

221163

iii

iii

)1(41

)1(2253

i

41

2253

i

5

525 i

5

5

5

25 i

i 5

3 11 1 2*

1 2 1 2

i i

i i

You try!You try!

5

5

1

i

i

Assignment

Pg. 277: #17-21(odd), Pg. 277: #17-21(odd), 29-33(odd), 37-41(odd), 29-33(odd), 37-41(odd), 47-51(odd), 57-61(odd), 47-51(odd), 57-61(odd),

65-69(odd), 9265-69(odd), 92

AssessmentAssessment• Concept circles