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Transcript 
Complex Numbers 2
The Argand Diagram
Representing Complex Numbers
Real numbers are usually represented as positions on a
horizontal number line.
Real
-3
-2
-1
0
1
2
3
4
5
Addition, subtraction, multiplication and division with real
numbers takes place on this number line.
The Argand Diagram
Complex numbers also have an imaginary part so
another dimension needs to be added to the number
line
Imaginary
7
6
5
4
3
2
1
-8 -7 -6 -5 -4 -3 -2 -1
-2
-3
-4
-5
-6
-7
Real
1
2
3
4
5
6
7
8
Complex numbers can be
represented on the Argand
diagram by straight lines.
Putting complex numbers on
an Argand diagram often
helps give a feel for a
problem.
Some examples
u  62j
v  27j
w  5  4 j
z  7  6 j
Imaginary
7
6
5
4
w
3
u
2
1
Real
-8
-7
-6
-5
-4
-3
-2
-1
1
2
-2
-3
-4
-5
z
-6
-7
v
3
4
5
6
7
8
Complex numbers and their
conjugates
Imaginary
w  5  4 j w*  5  4 j
z  6  3 j z*  6  3 j
7
6
5
w
4
z*
3
2
1
Real
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
-2
z
-3
-4
-5
-6
-7
w*
6
7
8
Addition
Imaginary
w  54j
7
6
w
5
4
3
z  6  3 j
2
1
Real
z
-8
-7
-6
-5
-4
-3
-2
-1
-2
-3
w z
-4
-5
-6
-7
1
2
3
4
5
6
7
8
Subtraction
Imaginary
u  23j
7
v  63j
6
5
4
u v
3
2
1
Real
-8
-7
-6
-5
-4
-3
-2
-1
-2
-3
-4
-5
-6
-7
1
2
3
4
5
6
7
8
The modulus of a complex
number
Imaginary
x  yj  x  y
2
x + yj
y
Real
O
x
2
The argument of a complex
number
Imaginary
z  23j
w  3  5 j
7
6
5
4
z=2 + 3j
3
between -180o and 180o
2
1
-8
-7
-6
-5
-4
-3
-2
-1
1
-2
α
-3
-4
-5
w=-3 - 5j
θ
-6
-7
Real
2
3
4
5
6
7
8
arg( z )  arctan( 32 )
 56
arctan( 53 )  59
arg( w)  121
Radians
  180
c
180  c
60 

3
3
c
5  180 5
150 

6
6
Loci using complex numbers
z  64j
zw
w  1 2 j
7
6
5
4
3
2
1
1
2
3
4
5
6
7
8
The distance to a point
z  (3  2 j )
z  (4  7 j )
z 47j
Imaginary
z  (3  2 j)  4
7
6
5
4
3
2
z  (3  2 j)  4
1
Real
-8
-7
-6
-5
-4
-3
-2
-1
z  (3  2 j)  4
-2
z  (3  2 j)  4
-5
-3
-4
-6
-7
1
2
3
4
5
4
6
7
8
Loci using arguments
arg( z ) 
Im

arg( z  j ) 
4
Re
0  arg( z  j ) 


Im
4
Re
Im
4
Re
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