Complex Numbers. Aim: Identify parts of complex numbers. Imaginary NumbersReal Numbers.
Complex Numbers 2 Complex Numbers.
-
Upload
monica-spry -
Category
Documents
-
view
379 -
download
19
Transcript of Complex Numbers 2 Complex Numbers.
Complex Numbers 2Complex Numbers 2
www.mathxtc.com
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
������������������������������������������
������������������������������������������
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