Transcript Document

Argand diagrams: modulus,
argument and polar form
IB HL
Adrian Sparrow
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The Argand plane
The plane of complex numbers is known as the Argand plane.
The y-axis is the imaginary part and the x-axis is the real part.
Examples
z1  3  2i
z2  2  i
z1  z2
Draw a sketch to show
each of these:
z1  z2
2z2 
z2
z1


2z2
z1
*
z1  z2  1 3i


z1
*
2z2   4  2i
z1
*
 3 2i

Modulus
The modulus, z, is the length the linez.
Examples
z1  3  2i
z2  2  i
Find,
z1
z2
z1 and z2
z1  32  22  13
z2  22 12  5
Find,
z1  z 2
10
2z 2 
2 5
z1 
*

13


Argument
The argument of a complex number is the angle,
, made between
the line and the positive axis.
Note that:      
Considerz1  4  4i
4
t an 
4


argz1 
4


4
Consider z2  3  3i
3
t an

3



4

3
argz2    
4 4


Argument - examples 1
z1  4
z 3  3
arg z1  0
arg z3  


z 2  5i
argz2 
z4  4i

2
argz4  



2
Argument - examples 2
1. Find the argument of the 2. Find the argument of the
complex numberw =1 3i.
complex numberz = 3  i.
2


1
1

3
2
 1
1
tan  
or sin  
2
3



argz 
3
1
  or cos  
tan
1
2


6
3



argw  

3
Polar and Euler form
Cartesian form: z = x + yi
Polar form: z = z cos  isin 
The modulus of
the complex
number.

z  z cos  isin  
can be z  z cis
The Euler form of
z  z cis is given as
z  ze
The argument of the
complex number.

i
We will derive the Euler form of
a complex number in a later
lessons.
Cartesian to polar form
Write the complex number
z = 3+i 3 in polar form.
Draw a sketch on your Argand
diagram
2 3
3



2

3  3
 12  2 3
3
  
z  2 3cis 
 6 
3  1 
sin  
 
2 3  2 

1. z  2  2i



 3 
z  2 2cis 
 4 
2. z  2  i 2 3
Make a right angled triangle and


find the hypotenuse, modulus, and
the angle, argument.
2
Questions - write each of the
following in polar form.

6
 
z  4cis 
3 
3. z  1 3i

4. z  4  4i

 2 
z  2cis 
 3 
  
z  4 2cis 
 4 
Polar to cartesian form
Write the polar complex number Questions - write each of the
  
following in cartesian form.
z = 2cis-  in cartesian form.
 6 

1. z = 6cis
Draw a sketch on your Argand
z  3 i 3 3
3
diagram.
 
3
1
2


6

Work backward using the angle .
6
 1
sin 

6 2

opp
sin
hyp
  
2. z =10cis 
 4 
z 5 2 i 5 2
 5 
3. z = 4cis
 
 6 
z  2 3  2i
2 

4. z =16cis 
z  8  i 8 5
 
opp1,adj  3
z  3 i
 3 
 