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Complex Numbers 2
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Complex Numbers
Complex Numbers
What is truth?
Complex Numbers
Who uses them
in real life?
Complex Numbers
Who uses them
in real life?
Here’s a hint….
Complex Numbers
Who uses them
in real life?
Here’s a hint….
Complex Numbers
Who uses them
in real life?
The navigation system in the space
shuttle depends on complex numbers!
Can you see a problem here?
Who goes first?
Complex numbers do not have order
What is a complex number?

It is a tool to solve an equation.
What is a complex number?
It is a tool to solve an equation.
 It has been used to solve equations for the
last 200 years or so.

What is a complex number?
It is a tool to solve an equation.
 It has been used to solve equations for the
last 200 years or so.
 It is defined to be i such that ;

i  1
2
What is a complex number?
It is a tool to solve an equation.
 It has been used to solve equations for the
last 200 years or so.
 It is defined to be i such that ;

i  1
2

Or in other words;
i  1
Complex

i is an imaginary
number
Complex


i is an imaginary
number
Or a complex number
Complex



i is an imaginary
number
Or a complex number
Or an unreal number
Complex?




i is an imaginary
number
Or a complex number
Or an unreal number
The terms are interchangeable
unreal
complex
imaginary
Some observations

In the beginning there were counting
numbers
1
2
Some observations
In the beginning there were counting
numbers
 And then we needed integers

1
2
Some observations
In the beginning there were counting
numbers
 And then we needed integers

1
-1
2
-3
Some observations
In the beginning there were counting
numbers
 And then we needed integers
 And rationals
1

0.41
-1
2
-3
Some observations
In the beginning there were counting
numbers
 And then we needed integers
 And rationals
1
0.41
 And irrationals

-1
2
2
-3
Some observations
In the beginning there were counting
numbers
 And then we needed integers
 And rationals
1
0.41
 And irrationals
2
-1
0
 And reals

-3
2

So where do unreals fit in ?
We have always used them. 6 is not just 6 it is
6 + 0i. Complex numbers incorporate all
numbers. 2i
3 + 4i
1
0.41
-1
2
0

2
-3

A number such as 3i is a purely imaginary
number
A number such as 3i is a purely imaginary
number
 A number such as 6 is a purely real number

A number such as 3i is a purely imaginary
number
 A number such as 6 is a purely real number
 6 + 3i is a complex number

A number such as 3i is a purely imaginary
number
 A number such as 6 is a purely real number
 6 + 3i is a complex number
 x + iy is the general form of a complex
number

A number such as 3i is a purely imaginary
number
 A number such as 6 is a purely real number
 6 + 3i is a complex number
 x + iy is the general form of a complex
number
 If x + iy = 6 – 4i then x = 6 and y = -4

A number such as 3i is a purely imaginary
number
 A number such as 6 is a purely real number
 6 + 3i is a complex number
 x + iy is the general form of a complex
number
 If x + iy = 6 – 4i then x = 6 and y = – 4
 The ‘real part’ of 6 – 4i is 6

Worked Examples
1.
Simplify
4
Worked Examples
1.
Simplify
4
4  4  1
 4  i2
 2i
Worked Examples
1.
Simplify
4
4  4  1
 4  i2
 2i
2.
Evaluate 3i  4i
Worked Examples
1.
Simplify
4
4  4  1
 4  i2
 2i
2.
Evaluate 3i  4i
3i  4i  12i 2
 12  1
 12
Worked Examples
3. Simplify 3i  4i
Worked Examples
3. Simplify 3i  4i
3i  4i  7i
Worked Examples
3. Simplify 3i  4i
3i  4i  7i
4. Simplify 3i  7  4i  6
Worked Examples
3. Simplify 3i  4i
3i  4i  7i
4. Simplify 3i  7  4i  6
3i  7  4i  6  i 13
Worked Examples
3. Simplify 3i  4i
3i  4i  7i
4. Simplify 3i  7  4i  6
3i  7  4i  6  i 13
5. Simplify (3i  7)(3i  7)
Addition Subtraction
Multiplication
3. Simplify 3i  4i
3i  4i  7i
4. Simplify 3i  7  4i  6
3i  7  4i  6  i 13
5. Simplify (3i  7)(3i  7)
(3i  7)(3i  7)   3i   7 2  9  49  58
2
Division
6. Simplify
2
3i  7
Division
6. Simplify
2
3i  7
The trick is to make the denominator real:
Division
6. Simplify
2
3i  7
The trick is to make the denominator real:
2
3i  7 2(3i  7)


3i  7 3i  7
58
(3i  7)

29
7  3i

29
Solving Quadratic Functions
7. Solve x  6 x  13  0
2
6  36  52
x
2
6   16
x
2
6  16  1
x
2
x  3  2i complex solutions (Conjugates)
Powers of i
ii
i 
2
i 
3
i 
4
i 
5
i 
6
i 
7
Powers of i
ii
i 
2
i 
3
i 
4
i 
5
i 
6
i 
7
  1
2
 1
Powers of i
ii
i 
2
  1
2
 1
i  1  i  i
3
i  1  1  1
4
i 
5
i 
6
i 
7
Powers of i
ii
i 
2
  1
2
 1
i  1  i  i
3
i  1  1  1
4
i i
5
i  1
6
i  i
7
Powers of i
i i i i i
1
5
9
13
i  i  i  i  1
2
6
10
14
i  i  i  i  i
3
7
11
15
i  i  i  i 1
4
8
12
16
Developing useful rules
Consider z  a  bi and z  a  bi (Conjugate)
z  z  2a
z  z  2bi
z  (a  bi )( a  bi )
2
 a  2bi  b
2
2
z  (a  bi )( a  bi )
2
 a  2abi  b
2
2
Developing useful rules
Consider z  a  bi and z  a  bi (Conjugate)
zz  (a  bi )( a  bi )
 a2  b2
 z
2
z (a  bi ) (a  bi )


z (a  bi ) (a  bi )
a 2  2abi  b 2

a2  b2
Developing useful rules
Consider z1  a  bi and z2  c  di
1. z1  z2  
2. z1 z2 
Developing useful rules
Consider z1  a  bi and z2  c  di
1. z1  z2   z1  z2
2. z1 z2  z1 z2 
Argand Diagrams
Jean Robert Argand was a Swiss amateur
mathematician. He was an accountant
book-keeper.
Argand Diagrams
Jean Robert Argand was a Swiss amateur
mathematician. He was an accountant
book-keeper.
He is remembered for 2 things
 His ‘Argand Diagram’
Argand Diagrams
Jean Robert Argand was a Swiss amateur
mathematician. He was an accountant
book-keeper.
He is remembered for 2 things
 His ‘Argand Diagram’
 His work on the ‘bell curve’
Argand Diagrams
Jean Robert Argand was a Swiss amateur
mathematician. He was an accountant
book-keeper.
He is remembered for 2 things
 His ‘Argand Diagram’
 His work on the ‘bell curve’

Very little is known about Argand. No
likeness has survived.
Argand Diagrams
y
2 + 3i
3
2
1
x
1
2
3
Argand Diagrams
y
2 + 3i
3
2
1
x
1
2
3
We can represent complex
numbers as a point.
Argand Diagrams
y
3
2
1
x
1
2
3
Argand Diagrams
y
3
2
z1  2  i  OA
1
O
A
x
1
2
3
We can represent complex
numbers as a vector.
Argand Diagrams
y
z2  2  3i  OB
B
3
2
1
O
A
1
2
z1  2  i  OA
x
3
Argand Diagrams
z2  2  3i  OB
y
B
3
C
z3  4  4i  OC
2
1
O
A
1
2
z1  2  i  OA
x
3
Argand Diagrams
z2  2  3i  OB
y
B
3
C
z3  4  4i  OC
2
1
O
A
1
2
z1  2  i  OA
x
3
z1  z2  OA  AC
 OC
Argand Diagrams
z2  2  3i  OB
y
B
3
C
z3  4  4i  OC
2
1
O
A
1
2
z1  2  i  OA
x
3
BA  ?
Argand Diagrams
z2  2  3i  OB
y
B
3
C
z3  4  4i  OC
2
1
O
A
1
2
z1  2  i  OA
x
3
BA  ?
Argand Diagrams
z2  2  3i  OB
y
B
3
C
z3  4  4i  OC
2
1
O
OB  BA  OA
A
1
2
z1  2  i  OA
x
3
Argand Diagrams
z2  2  3i  OB
y
B
3
C
z3  4  4i  OC
2
1
O
OB  BA  OA
BA  OA  OB
A
1
2
z1  2  i  OA
x
3
Argand Diagrams
z2  2  3i  OB
y
B
3
C
z3  4  4i  OC
2
1
O
OB  BA  OA
BA  OA  OB
 z1  z2
A
1
2
z1  2  i  OA
x
3
De Moivre
Abraham De Moivre was a
French Protestant who moved
to England in search of
religious freedom.
He was most famous for his
work on probability and was an
acquaintance of Isaac Newton.
His theorem was possibly
suggested to him by Newton.
De Moivre’s Theorem
 cos  i sin   cos n  i sin n
n
This remarkable formula works for
all values of n.
Enter Leonhard Euler…..
Euler who was the first to use i for complex
numbers had several great ideas. One of them
was that
ei = cos  + i sin 
Here is an amazing proof….
Let y  sin 
  sin 1 y
Let y  sin 
  sin y
1
 1
 
dy, now let y  iz so dy  idz
2
 1 y
Let y  sin 
  sin 1 y
 1
 
dy,now let y  iz so dy  idz
 1  y2

1

idz
2
 1   iz 
Let y  sin 
  sin y
1
 1
 
dy, now let y  iz so dy  idz
2
 1 y

1

idz
2
 1   iz 
 1
 i
dz
 1 z2
Let y  sin 
  sin y
1
 1
 
dy, now let y  iz so dy  idz
2
 1 y

1

idz
2
 1   iz 
 1
 i
dz
 1 z2
 1
  i
dz
 1  z2


  i ln 1  z  z [standard integral]
2
 1
  i
dz
 1  z2


  i ln 1  z  z [standard integral]
2
y sin 
now y  iz  z  
i
i
 1
  i
dz
 1  z2


  i ln 1  z  z [standard integral]
2
y sin 
now y  iz  z  
i
i
2


sin

sin



  i ln  1  




i 
i 



 1
  i
dz
 1  z2


  i ln 1  z 2  z [standard integral]
y sin 
now y  iz  z  
i
i
2


sin

sin



  i ln  1  




i 
i 



 i ln

1  sin   i sin 
2

 1
  i
dz
 1  z2


  i ln 1  z 2  z [standard integral]
y sin 
now y  iz  z  
i
i
2


sin

sin



  i ln  1  




i 
i 



 i ln

1  sin   i sin 
2

 1
  i
dz
 1  z2


  i ln 1  z 2  z [standard integral]
y sin 
now y  iz  z  
i
i
2


sin

sin



  i ln  1  




i 
i 



 i ln

1  sin   i sin 
2

  i ln

1  sin   i sin 
2
  i ln  cos  i sin  

  i ln

1  sin   i sin 
2
  i ln  cos   i sin  
i   ln  cos   i sin  
i  ln  cos   i sin  
1

  i ln

1  sin   i sin 
2

  i ln  cos  i sin  
i   ln  cos  i sin  
i  ln  cos  i sin  

i  ln 
 cos

i  ln 
 cos
1
1


 i sin  
1
cos  i sin  


 i sin  cos  i sin  
  i ln

1  sin   i sin 
2

  i ln  cos  i sin  
i   ln  cos  i sin  
i  ln  cos  i sin  

i  ln 
 cos

i  ln 
 cos
1
1


 i sin  
1
cos  i sin  


 i sin  cos  i sin  
1
cos   i sin  

i  ln 


 cos  i sin  cos  i sin  
1
cos  i sin  

i  ln 


 cos  i sin  cos  i sin  
i  ln  cos  i sin  
1
cos   i sin  

i  ln 


 cos   i sin  cos   i sin  
i  ln  cos   i sin  
ei  cos   i sin 
1
cos   i sin  

i  ln 


 cos   i sin  cos   i sin  
i  ln  cos   i sin  
ei  cos   i sin 
One last amazing result
Have you ever thought about ii ?
One last amazing result
What if I told you that ii is a real
number?


now i  cos  i sin
2
2




now i  cos  i sin
2
2
i
but e  cos  i sin 
i

so e  cos  i sin  i
2
2
2




now i  cos  i sin
2
2
but ei  cos  i sin 
i

so e  cos  i sin  i
2
2

2
i
 i2 
i
e

i
 
 




now i  cos  i sin
2
2
but ei  cos  i sin 
i

so e  cos  i sin  i
2
2
2
i

 i2 
i
e   i
 
e
i2

2


 ii
e 2  ii
ii = 0.20787957635076190855
3
3
now i  cos
 i sin
2
2
but ei  cos  i sin 
so e

e

i
3
i
2
e
i2
5
2
i

i
 i

3
2
e
3
3
 cos
 i sin
i
2
2
3
2
 ii
 ii
ii = 111.31777848985622603
So ii is an infinite number of
real numbers
The
End