008 Complex Numbers

8/8/2019 008 Complex Numbers

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Complex NumbersComplex Numbers


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Consider the quadratic equation x 2 + 1= 0.

Solving for x , gives x 2 = 112 ! x

1! x

Complex Numbers


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Since there is not real number whoseSince there is not real number whosesquare issquare is - -1 , the equation has no1 , the equation has noreal solution. French mathematicianreal solution. French mathematician

Rene Descartes (1596Rene Descartes (1596 - -1650)1650)proposed thatproposed that i i be defined suchbe defined suchthat ,that ,

1!i


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1!i

Complex Numbers

12 !i Note that squaring both sides yields:therefore

and

so

and

iiiii !!! *1* 132

1)1(*)1(* 224 !!! iii

iiiii !!! *1*45

1*1* 2246 !!! iiii

A nd so on


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Finding Powers of Finding Powers of i i

T he successive powers of i rotate through the four values of i , -1, - i , and 1.

i n = i if

n = 1, 5 , 9 ,

i n = - 1 if

n = 2, 6 , 10,

i n = - i if

n = 3 , 7 , 11,

i n = 1 if n

= 4 , 8 , 12,


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then,1-I !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 theremainder as your

exponent instead.

Example: ?23 !i

3oremainder aith5423

!

.etcii -hichuseSo, 3 !

ii !23


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R eal NumbersImaginary Numbers

Real numbers and imaginarynumbers are subsets of the set of complex numbers.

Complex Numbers


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Imaginary UnitImaginary UnitUntil now , you have always been toldthat you cant take the square root of

a negative number.If you useimaginary units , you can!

The imaginary unit is .=I t is used to write the square root of a negative number.

1


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Property of the square rootProperty of the square rootof negative numbersof negative numbers

I f r is a positive real number , then

r ir ! r i

Examples:

!3 3i !4 !4i i2


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ExamplesExamples2

)3( 1. i22 )3(i!

)3*3(1!

)3(1!3!

26103 olve 2.2

! x363 2 ! x

122 ! x

122 ! x12i x s!

32i x s!


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Complex NumbersComplex NumbersA complex number has a real part & an imaginary part.

Standard form is: b ia

Real part Imaginary part

Example: 5+4iExample: 5+4i


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Adding and SubtractingAdding and Subtracting

To Add or Subt rac t C omplex Nu m b ers

1. Change all imaginary numbers to b i form .

2. A dd (or subtract ) the real parts of the complexnumbers .

3. A dd (or subtract ) the imaginary parts of the

complex numbers .4. Write the answer in the form a + b i .


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Adding and SubtractingAdding and Subtracting(add or subtract the real parts , then(add or subtract the real parts , thenadd or subtract the imaginary parts)add or subtract the imaginary parts)

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

i52!

Ex: )73()32( ii)73()32( ii!

i41!

Ex: )32()3(2 iii)32()23( iii!

i21!


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MultiplyingMultiplying

To Mu lti ply C omplex Nu m b ers


2. Multiply the complex numbers as you wouldmultiply polynomials .

3. Substitute 1 for each i 2.

4. Combine the real parts and the imaginary parts . Write the answer in a + b i form .


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MultiplyingMultiplyingTreat the is like variables , thenTreat the is like variables , then

change any that are not to thechange any that are not to thefirst powerfirst power

Ex: )3( ii2

3 ii!)1(3! i

i31!

Ex: )26)(32( ii2618412 iii!

)1(62212! i

62212! ii226!


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CAUTIO N!CAUTIO N!

?24 !

!! 2424 ii

!22 2i 22

824 {


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DividingDividing

To Divid e C omplex Nu m b ers


2. R ationalize the denominator by multiplying boththe numerator and the denominator by theconjugate of the denominator .

3. Substitute 1 for each i 2.


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DividingDividing

E xamples:

!i

i

3434

!i

i

i

i

3434

3434

!2316

3434i

ii

!916

924162

ii25

247 i

!

53

5!

53

5i

!

53

53

53

5i

i

i

!259

)53(5i

i!

59)5515 i

14)5515 i


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i

i

i

i E x

2121

*21

113 :

)21)(21()21)(113(

ii

ii!

2

2

4221221163iii

iii!

)1(41

)1(2253!

i

412253

!i

5

525 i!

55

525 i

!

i! 5


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EVALUATE theFOLLOWING

8

15

27

52

12

1.

2.

3. 2

4.5.

i

i

i

i

i


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C hange to biform 1. 36

2. 120

3. 48

4. 6005. 4 64


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1 . ( 2 8 ) (3 5 )

2 . (5 8 ) ( 4 2 ) (3 )

3 . 3 ( 2 5 )

4 . ( 4 3 )(2 5 )

45 .

23

6 .

i i

i i i

i i

i i

i

ii

i

P erform the followingoperations

008 Complex Numbers

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