P.O.D. Write as a common fraction in lowest terms.

83.0

6.7 & 6.8-The Imaginary Number i & Complex Numbers

The imaginary number “i”

Created to overcome the problem of square roots of negative numbers

The imaginary number “i”

Defined as

1i*Note* i2 = -1

The Imaginary number i can be used to write the square root of any negative number.

Property #1: If r is a positive real #, then

rir 55 iEx .

Property #2: By Property #1, it follows that

rri 2)(

555 22 iiEx

)(.

Using i to solve equations:

3x2+10= -26 3x2= -36

x=x=

Answer:

123212 ii

61362 i

Simplify: 123

Warning: The rule

does not apply when a and b are Both negative.

abba

123 ii You must first eliminate any negative radicands.

i2=-1

6

Rationalizing If a fraction has i in

the denominator, rationalize by multiplying top & bottom by i.

6.8- Complex Numbers

Complex number A number, a+bi,

where a and b are real numbers.

(a is the real part & bi is the imaginary part.)

Adding & Subtracting Complex #s:

)()( ii 2463 iAns 47:)()( ii 2463 iAns 81:

Multiplying Complex #s:

Use the Distributive Prop. or FOIL.

))(( ii 2543

2862015 iii )( 182615 i

i267

Conjugates:The numbers a+bi and a-bi are called conjugates.Ex. 3+4i and 3-4i are conjugates

Conjugates:The product of 2 conjugates is always a real #.Ex. (3+4i)(3-4i)

=9-16i2=9+16=25

*Conjugates are used to simplify the quotients w/ a complex # in the denominator:

i

i

i

i

i

i

32

32

32

5

32

5

i

i

i

i

i

i

32

32

32

5

32

5

2

2

94

315210

i

iii

)(

)(

194

1315210

ii

ii

13

17

13

7

13

177

P.O.D.

0324

:24 tt

numberscomplextheoverSolve

P.O.D.

)31(

1)(

iffindthenx

xxfIf