Complex Numbers 2Complex Numbers 2

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Complex NumbersComplex Numbers

Complex NumbersComplex Numbers

What is truth?What is truth?

Complex NumbersComplex NumbersWho uses themWho uses them

in real life?in real life?

Complex NumbersComplex NumbersWho uses themWho uses them

in real life?in real life?

Here’s a hint….Here’s a hint….

Complex NumbersComplex NumbersWho uses themWho uses them

in real life?in real life?

Here’s a hint….Here’s a hint….

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!

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

-2

Who goes first?Who goes first?

-2

Complex numbers do not have orderComplex numbers do not have order

-2

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.

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.

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

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

ComplexComplex

i i is an imaginary is an imaginary numbernumber

ComplexComplex

i i is an imaginary is an imaginary numbernumber

Or a complex numberOr a complex number

ComplexComplex

i i is an imaginary is an imaginary numbernumber

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

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

Some observationsSome observations

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

1

2

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

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

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

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

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

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

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

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

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

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

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

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

Worked ExamplesWorked Examples

1.1. SimplifySimplify 4

Worked ExamplesWorked Examples

1.1. SimplifySimplify 4

2

4 4 1

4

2

i

i

Worked ExamplesWorked Examples

1.1. SimplifySimplify

2.2. Evaluate Evaluate

4

2

4 4 1

4

2

i

i

3 4i i

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

Worked ExamplesWorked Examples

3.3. SimplifySimplify 3 4i i

Worked ExamplesWorked Examples

3.3. SimplifySimplify 3 4i i

3 4 7i i i

Worked Examples

3. Simplify

4. Simplify

3 4i i

3 4 7i i i

3 7 4 6i i

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

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

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

DivisionDivision

6.6. SimplifySimplify2

3 7i

DivisionDivision

6.6. SimplifySimplify

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

2

3 7i

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

Solving Quadratic FunctionsSolving Quadratic Functions

)(232

1166

2

166

2

52366

0136.7 2

Conjugatessolutionscomplexix

x

x

x

xxSolve

Powers of iPowers of i

7

6

5

4

3

2

i

i

i

i

i

i

ii

Powers of iPowers of i

7

6

5

4

3

22 11

i

i

i

i

i

i

ii

Powers of iPowers of i

7

6

5

4

3

22

111

1

11

i

i

i

i

iii

i

ii

Powers of iPowers of i

ii

i

ii

i

iii

i

ii

7

6

5

4

3

22

1

111

1

11

Powers of iPowers of i

1

1

161284

151173

141062

13951

iiii

iiiii

iiii

iiiii

Developing useful rulesDeveloping useful rules

22

2

22

2

2

))((

2

))((

2

2

)(

babia

biabiaz

bbia

biabiaz

bizz

azz

ConjugatebiazandbiazConsider

Developing useful rulesDeveloping useful rules

22

22

2

22

2

)(

)(

)(

)(

))((

)(

ba

babia

bia

bia

bia

bia

z

z

z

ba

biabiazz

ConjugatebiazandbiazConsider

Developing useful rulesDeveloping useful rules

21

21

21

.2

.1

zz

zz

diczandbiazConsider

Developing useful rulesDeveloping useful rules

2121

2121

21

.2

.1

zzzz

zzzz

diczandbiazConsider

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.

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’

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’

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.

Argand DiagramsArgand Diagrams

x

y

1 2 3

1

2

3

2 + 32 + 3ii

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.

Argand DiagramsArgand Diagrams

x

y

1 2 3

1

2

3

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

��������������

Argand DiagramsArgand Diagrams

x

y

1 2 3

1

2

3

1 2z i OA ��������������

2 2 3z i OB ��������������

O

A

B

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

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

����������������������������

��������������

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 ��������������

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 ��������������

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

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

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

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

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

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

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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.

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.

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

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….

1

Let sin

sin

y

y

1

2

Let sin

sin

1,now let so

1

y

y

dy y iz dy idzy

1

2

2

Let sin

sin

1,now let so

1

1

1

y

y

dy y iz dy idzy

idziz

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

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

2

2

1

1

ln 1 [standard integral]

i dzz

i z z

2

2

1

1

ln 1 [standard integral]

sinnow

i dzz

i z z

yy iz z

i i

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

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

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

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

2ln 1 sin sin

ln cos sin

i i

i i

2

1

ln 1 sin sin

ln cos sin

ln cos sin

ln cos sin

i i

i i

i i

i i

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

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

1 cos sinln

cos sin cos sin

ii

i i

1 cos sinln

cos sin cos sin

ln cos sin

ii

i i

i i

1 cos sinln

cos sin cos sin

ln cos sin

cos sini

ii

i i

i i

e i

1 cos sinln

cos sin cos sin

ln cos sin

cos sini

ii

i i

i i

e i

One last amazing resultOne last amazing result

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

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??

now cos sin2 2

i i

2

now cos sin2 2

but cos sin

so cos sin2 2

i

i

i i

e i

e i i

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

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

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

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

The The EndEnd