Complex Numbers 2 Complex Numbers.

102
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Transcript of Complex Numbers 2 Complex Numbers.

Page 1: Complex Numbers 2  Complex Numbers.

Complex Numbers 2Complex Numbers 2

www.mathxtc.com

Page 2: Complex Numbers 2  Complex Numbers.

Complex NumbersComplex Numbers

Page 3: Complex Numbers 2  Complex Numbers.

Complex NumbersComplex Numbers

What is truth?What is truth?

Page 4: Complex Numbers 2  Complex Numbers.

Complex NumbersComplex NumbersWho uses themWho uses them

in real life?in real life?

Page 5: Complex Numbers 2  Complex Numbers.

Complex NumbersComplex NumbersWho uses themWho uses them

in real life?in real life?

Here’s a hint….Here’s a hint….

Page 6: Complex Numbers 2  Complex Numbers.

Complex NumbersComplex NumbersWho uses themWho uses them

in real life?in real life?

Here’s a hint….Here’s a hint….

Page 7: Complex Numbers 2  Complex Numbers.

Complex NumbersComplex NumbersWho uses themWho uses them

in real life?in real life?

The navigation system in the space The navigation system in the space shuttle depends on complex numbers!shuttle depends on complex numbers!

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Can you see a problem here?Can you see a problem here?

-2

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Who goes first?Who goes first?

-2

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Complex numbers do not have orderComplex numbers do not have order

-2

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What is a complex number?What is a complex number?

It is a tool to solve an equation.It is a tool to solve an equation.

Page 12: Complex Numbers 2  Complex Numbers.

What is a complex number?What is a complex number?

It is a tool to solve an equation.It is a tool to solve an equation. It has been used to solve equations for the It has been used to solve equations for the

last 200 years or so.last 200 years or so.

Page 13: Complex Numbers 2  Complex Numbers.

What is a complex number?What is a complex number?

It is a tool to solve an equation.It is a tool to solve an equation. It has been used to solve equations for the It has been used to solve equations for the

last 200 years or so.last 200 years or so. It is defined to be It is defined to be ii such that ;such that ;

2 1i

Page 14: Complex Numbers 2  Complex Numbers.

What is a complex number?What is a complex number?

It is a tool to solve an equation.It is a tool to solve an equation. It has been used to solve equations for the It has been used to solve equations for the

last 200 years or so.last 200 years or so. It is defined to be It is defined to be ii such that ;such that ;

Or in other words; Or in other words;

2 1i

1i

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ComplexComplex

i i is an imaginary is an imaginary numbernumber

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ComplexComplex

i i is an imaginary is an imaginary numbernumber

Or a complex numberOr a complex number

Page 17: Complex Numbers 2  Complex Numbers.

ComplexComplex

i i is an imaginary is an imaginary numbernumber

Or a complex numberOr a complex number Or an unreal numberOr an unreal number

Page 18: Complex Numbers 2  Complex Numbers.

Complex?Complex?

i i is an imaginary is an imaginary numbernumber

Or a complex numberOr a complex number Or an unreal numberOr an unreal number The terms are inter-The terms are inter-

changeablechangeable

complex

imaginary

unreal

Page 19: Complex Numbers 2  Complex Numbers.

Some observationsSome observations

In the beginning there were counting In the beginning there were counting numbersnumbers

1

2

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Some observationsSome observations

In the beginning there were counting In the beginning there were counting numbersnumbers

And then we needed integersAnd then we needed integers

1

2

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Some observationsSome observations

In the beginning there were counting In the beginning there were counting numbersnumbers

And then we needed integersAnd then we needed integers

1

2-1-3

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Some observationsSome observations

In the beginning there were counting In the beginning there were counting numbersnumbers

And then we needed integersAnd then we needed integers And rationalsAnd rationals 1

2-1-3

0.41

Page 23: Complex Numbers 2  Complex Numbers.

Some observationsSome observations

In the beginning there were counting In the beginning there were counting numbersnumbers

And then we needed integersAnd then we needed integers And rationalsAnd rationals And irrationalsAnd irrationals

1

2-1-3

0.41

2

Page 24: Complex Numbers 2  Complex Numbers.

Some observationsSome observations

In the beginning there were counting In the beginning there were counting numbersnumbers

And then we needed integersAnd then we needed integers And rationalsAnd rationals And irrationalsAnd irrationals And realsAnd reals

1

2-1-3

0.41

2

0

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So where do unreals fit in ?So where do unreals fit in ?

We have always used them. 6 is not just 6 it is We have always used them. 6 is not just 6 it is 6 + 06 + 0i. i. Complex numbers incorporate all Complex numbers incorporate all numbers.numbers.

1

2-1-3

0.41

2

3 + 4i2i

0

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A number such as 3A number such as 3i i is a purely imaginary is a purely imaginary numbernumber

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A number such as 3A number such as 3i i is a purely imaginary is a purely imaginary numbernumber

A number such as 6 is a purely real numberA number such as 6 is a purely real number

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A number such as 3A number such as 3i i is a purely imaginary is a purely imaginary numbernumber

A number such as 6 is a purely real numberA number such as 6 is a purely real number 6 + 36 + 3ii is a complex number is a complex number

Page 29: Complex Numbers 2  Complex Numbers.

A number such as 3A number such as 3i i is a purely imaginary is a purely imaginary numbernumber

A number such as 6 is a purely real numberA number such as 6 is a purely real number 6 + 36 + 3ii is a complex number is a complex number x + iyx + iy is the general form of a complex is the general form of a complex

numbernumber

Page 30: Complex Numbers 2  Complex Numbers.

A number such as 3A number such as 3i i is a purely imaginary is a purely imaginary numbernumber

A number such as 6 is a purely real numberA number such as 6 is a purely real number 6 + 36 + 3ii is a complex number is a complex number x + iyx + iy is the general form of a complex is the general form of a complex

numbernumber If If x + iyx + iy = 6 – 4 = 6 – 4i i then then xx = 6 and = 6 and yy = = --44

Page 31: Complex Numbers 2  Complex Numbers.

A number such as 3A number such as 3i i is a purely imaginary is a purely imaginary numbernumber

A number such as 6 is a purely real numberA number such as 6 is a purely real number 6 + 36 + 3ii is a complex number is a complex number x + iyx + iy is the general form of a complex is the general form of a complex

numbernumber If If x + iyx + iy = 6 – 4 = 6 – 4i i then then xx = 6 and = 6 and yy = – = – 44 The ‘real part’ of 6 – 4The ‘real part’ of 6 – 4i i is 6is 6

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Worked ExamplesWorked Examples

1.1. SimplifySimplify 4

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Worked ExamplesWorked Examples

1.1. SimplifySimplify 4

2

4 4 1

4

2

i

i

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Worked ExamplesWorked Examples

1.1. SimplifySimplify

2.2. Evaluate Evaluate

4

2

4 4 1

4

2

i

i

3 4i i

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Worked ExamplesWorked Examples

1.1. SimplifySimplify

2.2. Evaluate Evaluate

4

2

4 4 1

4

2

i

i

3 4i i

23 4 12

12 1

12

i i i

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Worked ExamplesWorked Examples

3.3. SimplifySimplify 3 4i i

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Worked ExamplesWorked Examples

3.3. SimplifySimplify 3 4i i

3 4 7i i i

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Worked Examples

3. Simplify

4. Simplify

3 4i i

3 4 7i i i

3 7 4 6i i

Page 39: Complex Numbers 2  Complex Numbers.

Worked ExamplesWorked Examples

3.3. SimplifySimplify

4.4. SimplifySimplify

3 4i i

3 4 7i i i

3 7 4 6i i

3 7 4 6 13i i i

Page 40: Complex Numbers 2  Complex Numbers.

Worked ExamplesWorked Examples

3.3. SimplifySimplify

4.4. SimplifySimplify

5.5. SimplifySimplify

3 4i i

3 4 7i i i

3 7 4 6i i

3 7 4 6 13i i i

(3 7)(3 7)i i

Page 41: Complex Numbers 2  Complex Numbers.

Addition Subtraction Addition Subtraction MultiplicationMultiplication

3.3. SimplifySimplify

4.4. SimplifySimplify

5.5. SimplifySimplify

3 4i i

3 4 7i i i

3 7 4 6i i

3 7 4 6 13i i i

(3 7)(3 7)i i

2 2(3 7)(3 7) 3 7 9 49 58i i i

Page 42: Complex Numbers 2  Complex Numbers.

DivisionDivision

6.6. SimplifySimplify2

3 7i

Page 43: Complex Numbers 2  Complex Numbers.

DivisionDivision

6.6. SimplifySimplify

The trick is to make the denominator real:The trick is to make the denominator real:

2

3 7i

Page 44: Complex Numbers 2  Complex Numbers.

DivisionDivision

6.6. SimplifySimplify

The trick is to make the denominator real:The trick is to make the denominator real:

2

3 7i

2 3 7 2(3 7)

3 7 3 7 58(3 7)

297 3

29

i i

i ii

i

Page 45: Complex Numbers 2  Complex Numbers.

Solving Quadratic FunctionsSolving Quadratic Functions

)(232

1166

2

166

2

52366

0136.7 2

Conjugatessolutionscomplexix

x

x

x

xxSolve

Page 46: Complex Numbers 2  Complex Numbers.

Powers of iPowers of i

7

6

5

4

3

2

i

i

i

i

i

i

ii

Page 47: Complex Numbers 2  Complex Numbers.

Powers of iPowers of i

7

6

5

4

3

22 11

i

i

i

i

i

i

ii

Page 48: Complex Numbers 2  Complex Numbers.

Powers of iPowers of i

7

6

5

4

3

22

111

1

11

i

i

i

i

iii

i

ii

Page 49: Complex Numbers 2  Complex Numbers.

Powers of iPowers of i

ii

i

ii

i

iii

i

ii

7

6

5

4

3

22

1

111

1

11

Page 50: Complex Numbers 2  Complex Numbers.

Powers of iPowers of i

1

1

161284

151173

141062

13951

iiii

iiiii

iiii

iiiii

Page 51: Complex Numbers 2  Complex Numbers.

Developing useful rulesDeveloping useful rules

22

2

22

2

2

))((

2

))((

2

2

)(

babia

biabiaz

bbia

biabiaz

bizz

azz

ConjugatebiazandbiazConsider

Page 52: Complex Numbers 2  Complex Numbers.

Developing useful rulesDeveloping useful rules

22

22

2

22

2

)(

)(

)(

)(

))((

)(

ba

babia

bia

bia

bia

bia

z

z

z

ba

biabiazz

ConjugatebiazandbiazConsider

Page 53: Complex Numbers 2  Complex Numbers.

Developing useful rulesDeveloping useful rules

21

21

21

.2

.1

zz

zz

diczandbiazConsider

Page 54: Complex Numbers 2  Complex Numbers.

Developing useful rulesDeveloping useful rules

2121

2121

21

.2

.1

zzzz

zzzz

diczandbiazConsider

Page 55: Complex Numbers 2  Complex Numbers.

Argand DiagramsArgand Diagrams

Jean Robert Argand was a Swiss amateur Jean Robert Argand was a Swiss amateur mathematician. He was an accountant mathematician. He was an accountant book-keeper.book-keeper.

Page 56: Complex Numbers 2  Complex Numbers.

Argand DiagramsArgand Diagrams

Jean Robert Argand was a Swiss amateur Jean Robert Argand was a Swiss amateur mathematician. He was an accountant mathematician. He was an accountant book-keeper.book-keeper.

He is remembered for 2 thingsHe is remembered for 2 things His ‘Argand Diagram’His ‘Argand Diagram’

Page 57: Complex Numbers 2  Complex Numbers.

Argand DiagramsArgand Diagrams

Jean Robert Argand was a Swiss amateur Jean Robert Argand was a Swiss amateur mathematician. He was an accountant mathematician. He was an accountant book-keeper.book-keeper.

He is remembered for 2 thingsHe is remembered for 2 things His ‘Argand Diagram’His ‘Argand Diagram’ His work on the ‘bell curve’His work on the ‘bell curve’

Page 58: Complex Numbers 2  Complex Numbers.

Argand DiagramsArgand Diagrams

Jean Robert Argand was a Swiss amateur Jean Robert Argand was a Swiss amateur mathematician. He was an accountant mathematician. He was an accountant book-keeper.book-keeper.

He is remembered for 2 thingsHe is remembered for 2 things His ‘Argand Diagram’His ‘Argand Diagram’ His work on the ‘bell curve’His work on the ‘bell curve’

Very little is known about Argand. No Very little is known about Argand. No likeness has survived.likeness has survived.

Page 59: Complex Numbers 2  Complex Numbers.

Argand DiagramsArgand Diagrams

x

y

1 2 3

1

2

3

2 + 32 + 3ii

Page 60: Complex Numbers 2  Complex Numbers.

Argand DiagramsArgand Diagrams

x

y

1 2 3

1

2

3

2 + 32 + 3ii

We can represent complex We can represent complex numbers as a point.numbers as a point.

Page 61: Complex Numbers 2  Complex Numbers.

Argand DiagramsArgand Diagrams

x

y

1 2 3

1

2

3

Page 62: Complex Numbers 2  Complex Numbers.

Argand DiagramsArgand Diagrams

x

y

1 2 3

1

2

3

We can represent complex We can represent complex numbers as a vector.numbers as a vector.

O

A1 2z i OA

��������������

Page 63: Complex Numbers 2  Complex Numbers.

Argand DiagramsArgand Diagrams

x

y

1 2 3

1

2

3

1 2z i OA ��������������

2 2 3z i OB ��������������

O

A

B

Page 64: Complex Numbers 2  Complex Numbers.

Argand DiagramsArgand Diagrams

x

y

1 2 3

1

2

3

1 2z i OA ��������������

2 2 3z i OB ��������������

O

A

B 3 4 4z i OC ��������������

C

Page 65: Complex Numbers 2  Complex Numbers.

Argand DiagramsArgand Diagrams

x

y

1 2 3

1

2

3

1 2z i OA ��������������

2 2 3z i OB ��������������

O

A

B 3 4 4z i OC ��������������

C

1 2z z OA AC

OC

����������������������������

��������������

Page 66: Complex Numbers 2  Complex Numbers.

Argand DiagramsArgand Diagrams

x

y

1 2 3

1

2

3

1 2z i OA ��������������

2 2 3z i OB ��������������

O

A

B 3 4 4z i OC ��������������

C

?BA ��������������

Page 67: Complex Numbers 2  Complex Numbers.

Argand DiagramsArgand Diagrams

x

y

1 2 3

1

2

3

1 2z i OA ��������������

2 2 3z i OB ��������������

O

A

B 3 4 4z i OC ��������������

C

?BA ��������������

Page 68: Complex Numbers 2  Complex Numbers.

Argand DiagramsArgand Diagrams

x

y

1 2 3

1

2

3

1 2z i OA ��������������

2 2 3z i OB ��������������

O

A

B 3 4 4z i OC ��������������

C

OB BA OA ������������������������������������������

Page 69: Complex Numbers 2  Complex Numbers.

Argand DiagramsArgand Diagrams

x

y

1 2 3

1

2

3

1 2z i OA ��������������

2 2 3z i OB ��������������

O

A

B 3 4 4z i OC ��������������

C

OB BA OA

BA OA OB

������������������������������������������

������������������������������������������

Page 70: Complex Numbers 2  Complex Numbers.

Argand DiagramsArgand Diagrams

x

y

1 2 3

1

2

3

1 2z i OA ��������������

2 2 3z i OB ��������������

O

A

B 3 4 4z i OC ��������������

C

1 2

OB BA OA

BA OA OB

z z

������������������������������������������

������������������������������������������

Page 71: Complex Numbers 2  Complex Numbers.

De MoivreDe MoivreAbraham De Moivre was a Abraham De Moivre was a French Protestant who moved French Protestant who moved to England in search of to England in search of religious freedom.religious freedom.

He was most famous for his He was most famous for his work on probability and was work on probability and was an acquaintance of Isaac an acquaintance of Isaac Newton.Newton.

His theorem was possibly His theorem was possibly suggested to him by Newton.suggested to him by Newton.

Page 72: Complex Numbers 2  Complex Numbers.

cos sin cos sinn

i n i n

De Moivre’s TheoremDe Moivre’s Theorem

This remarkable formula works for This remarkable formula works for all values of all values of n.n.

Page 73: Complex Numbers 2  Complex Numbers.

Enter Leonhard Euler…..Enter Leonhard Euler…..

Page 74: Complex Numbers 2  Complex Numbers.

Euler who was the first to use Euler who was the first to use ii for complex for complex numbers had several great ideas. One of them numbers had several great ideas. One of them was thatwas thateeii = cos = cos + + ii sin sin

Here is an amazing proof….Here is an amazing proof….

Page 75: Complex Numbers 2  Complex Numbers.

1

Let sin

sin

y

y

Page 76: Complex Numbers 2  Complex Numbers.

1

2

Let sin

sin

1,now let so

1

y

y

dy y iz dy idzy

Page 77: Complex Numbers 2  Complex Numbers.

1

2

2

Let sin

sin

1,now let so

1

1

1

y

y

dy y iz dy idzy

idziz

Page 78: Complex Numbers 2  Complex Numbers.

1

2

2

2

Let sin

sin

1,now let so

1

1

1

1

1

y

y

dy y iz dy idzy

idziz

i dzz

Page 79: Complex Numbers 2  Complex Numbers.

1

2

2

2

Let sin

sin

1,now let so

1

1

1

1

1

y

y

dy y iz dy idzy

idziz

i dzz

Page 80: Complex Numbers 2  Complex Numbers.

2

2

1

1

ln 1 [standard integral]

i dzz

i z z

Page 81: Complex Numbers 2  Complex Numbers.

2

2

1

1

ln 1 [standard integral]

sinnow

i dzz

i z z

yy iz z

i i

Page 82: Complex Numbers 2  Complex Numbers.

2

2

2

1

1

ln 1 [standard integral]

sinnow

sin sinln 1

i dzz

i z z

yy iz z

i i

ii i

Page 83: Complex Numbers 2  Complex Numbers.

2

2

2

2

1

1

ln 1 [standard integral]

sinnow

sin sinln 1

ln 1 sin sin

i dzz

i z z

yy iz z

i i

ii i

i i

Page 84: Complex Numbers 2  Complex Numbers.

2

2

2

2

1

1

ln 1 [standard integral]

sinnow

sin sinln 1

ln 1 sin sin

i dzz

i z z

yy iz z

i i

ii i

i i

Page 85: Complex Numbers 2  Complex Numbers.

2

2

2

2

1

1

ln 1 [standard integral]

sinnow

sin sinln 1

ln 1 sin sin

i dzz

i z z

yy iz z

i i

ii i

i i

Page 86: Complex Numbers 2  Complex Numbers.

2ln 1 sin sin

ln cos sin

i i

i i

Page 87: Complex Numbers 2  Complex Numbers.

2

1

ln 1 sin sin

ln cos sin

ln cos sin

ln cos sin

i i

i i

i i

i i

Page 88: Complex Numbers 2  Complex Numbers.

2

1

ln 1 sin sin

ln cos sin

ln cos sin

ln cos sin

1ln

cos sin

1 cos sinln

cos sin cos sin

i i

i i

i i

i i

ii

ii

i i

Page 89: Complex Numbers 2  Complex Numbers.

2

1

ln 1 sin sin

ln cos sin

ln cos sin

ln cos sin

1ln

cos sin

1 cos sinln

cos sin cos sin

i i

i i

i i

i i

ii

ii

i i

Page 90: Complex Numbers 2  Complex Numbers.

1 cos sinln

cos sin cos sin

ii

i i

Page 91: Complex Numbers 2  Complex Numbers.

1 cos sinln

cos sin cos sin

ln cos sin

ii

i i

i i

Page 92: Complex Numbers 2  Complex Numbers.

1 cos sinln

cos sin cos sin

ln cos sin

cos sini

ii

i i

i i

e i

Page 93: Complex Numbers 2  Complex Numbers.

1 cos sinln

cos sin cos sin

ln cos sin

cos sini

ii

i i

i i

e i

Page 94: Complex Numbers 2  Complex Numbers.

One last amazing resultOne last amazing result

Have you ever thought about Have you ever thought about iii i ??

Page 95: Complex Numbers 2  Complex Numbers.

One last amazing resultOne last amazing result

What if I told you that What if I told you that iii i is a is a real real numbernumber??

Page 96: Complex Numbers 2  Complex Numbers.

now cos sin2 2

i i

Page 97: Complex Numbers 2  Complex Numbers.

2

now cos sin2 2

but cos sin

so cos sin2 2

i

i

i i

e i

e i i

Page 98: Complex Numbers 2  Complex Numbers.

2

2

now cos sin2 2

but cos sin

so cos sin2 2

i

i

ii i

i i

e i

e i i

e i

Page 99: Complex Numbers 2  Complex Numbers.

2

2

2

2

2

now cos sin2 2

but cos sin

so cos sin2 2

i

i

ii i

i i

i

i i

e i

e i i

e i

e i

e i

iiii = 0.20787957635076190855= 0.20787957635076190855

Page 100: Complex Numbers 2  Complex Numbers.

2

5

2

3

2

3

2

3

2

3 3now cos sin

2 2

but cos sin

3 3so cos sin

2 2

i

i

ii i

i i

i

i i

e i

e i i

e i

e i

e i

iiii = 111.31777848985622603= 111.31777848985622603

Page 101: Complex Numbers 2  Complex Numbers.

So So iii i is an infinite number of is an infinite number of real numbersreal numbers

Page 102: Complex Numbers 2  Complex Numbers.

The The EndEnd