1. Mod-Arg Form

2. Mod-Arg Form
y
z = x + iy
y
O
x
x
Modulus
The modulus of a complex number is the
length of the vector OZ

3. Mod-Arg Form
y
z = x + iy
r z
O
x
Modulus
The modulus of a complex number is the
length of the vector OZ
r 2  x2  y2
r  x2  y2
y
x

4. Mod-Arg Form
y
z = x + iy
r z
O
x
Modulus
The modulus of a complex number is the
length of the vector OZ
r 2  x2  y2
r  x2  y2
y
x
z  x2  y2

5. Mod-Arg Form
y
z = x + iy
r z
  arg z
O
x
Modulus
The modulus of a complex number is the
length of the vector OZ
r 2  x2  y2
r  x2  y2
y
x
z  x2  y2
Argument
The argument of a complex number is
the angle the vector OZ makes with the
positive real (x) axis

6. Mod-Arg Form
y
z = x + iy
r z
  arg z
O
x
Modulus
The modulus of a complex number is the
length of the vector OZ
r 2  x2  y2
r  x2  y2
y
x
z  x2  y2
Argument
The argument of a complex number is
the angle the vector OZ makes with the
positive real (x) axis
 y
arg z  tan  
 x
1
   arg z  

7. e.g . Find the modulus and argument of 4  4i

8. e.g . Find the modulus and argument of 4  4i
4  4i  4 2   4 
2
 32
4 2

9. e.g . Find the modulus and argument of 4  4i
4  4i  4   4 
2
 32
4 2
2
 4
arg4  4i   tan  
 4 
 tan 1  1
1

10. e.g . Find the modulus and argument of 4  4i
4  4i  4   4 
2
 32
4 2
2
 4
arg4  4i   tan  
 4 
 tan 1  1
1


4

11. e.g . Find the modulus and argument of 4  4i
4  4i  4   4 
2
 32
4 2
2
 4
arg4  4i   tan  
 4 
 tan 1  1
1


4
Every complex number can be written in terms of its modulus and
argument

12. e.g . Find the modulus and argument of 4  4i
4  4i  4   4 
2
 32
4 2
2
 4
arg4  4i   tan  
 4 
 tan 1  1
1


4
Every complex number can be written in terms of its modulus and
argument
z  x  iy
 r cos  ir sin 
 r cos  i sin  

13. e.g . Find the modulus and argument of 4  4i
4  4i  4   4 
2
2
 32
4 2
 4
arg4  4i   tan  
 4 
 tan 1  1
1


4
Every complex number can be written in terms of its modulus and
argument
z  x  iy
 r cos  ir sin 
 r cos  i sin  
The mod-arg form of z is;
z  r cos  i sin  
z  rcis
where; r  z
  arg z

14. e.g. i  4  4i

15. 
e.g. i  4  4i  4 2cis  


 4

16. 
e.g. i  4  4i  4 2cis  


 4
ii  3  i

17. 
e.g. i  4  4i  4 2cis  


 4
ii  3  i
z
 3
 4
2
2
 12

18. 
e.g. i  4  4i  4 2cis  


 4
ii  3  i
z
 3
 4
2
2
1
2
arg z  tan 1


6
1
3

19. 
e.g. i  4  4i  4 2cis  


 4
ii  3  i
z
 3
2
1
2
arg z  tan 1

 4
2
 3  i  2cis

6

6
1
3

20. 
e.g. i  4  4i  4 2cis  


 4
ii  3  i
z
 3
2
arg z  tan 1
1
2

 4
2
 3  i  2cis
 ii  Convert 6cis

6

6
to Cartsian form

6
1
3

21. 
e.g. i  4  4i  4 2cis  


 4
ii  3  i
z
 3
2
arg z  tan 1
1
2

 4
2
 3  i  2cis
 ii  Convert 6cis
6cis

6

6


6
to Cartsian form
 6(cos


 i sin )
6
6
6
1
3

22. 
e.g. i  4  4i  4 2cis  


 4
ii  3  i
z
 3
2
arg z  tan 1
1
2

 4
2
 3  i  2cis
 ii  Convert 6cis
6cis

6



6
to Cartsian form
 6(cos


 i sin )
6
6
6
3 1
 6(
 i)
2 2
 3 3  3i
6
1
3

23. Mod-Arg Relations

24. Mod-Arg Relations
1 z1 z2  z1 z2
arg z1 z 2   arg z1  arg z 2

25. Mod-Arg Relations
1 z1 z2  z1 z2
arg z1 z 2   arg z1  arg z 2
NOTE:
Multiplication
rotates z1 by
arg z2

26. Mod-Arg Relations
1 z1 z2  z1 z2
arg z1 z 2   arg z1  arg z 2
Proof: let z1  r1cis1 and z 2  r2 cis 2
z1 z 2  r1 cos1  i sin 1   r2 cos 2  i sin  2 
NOTE:
Multiplication
rotates z1 by
arg z2

27. Mod-Arg Relations
1 z1 z2  z1 z2
arg z1 z 2   arg z1  arg z 2
Proof: let z1  r1cis1 and z 2  r2 cis 2
NOTE:
Multiplication
rotates z1 by
arg z2
z1 z 2  r1 cos1  i sin 1   r2 cos 2  i sin  2 
 r1r2 cos1 cos 2  i sin 1 cos 2  i cos1 sin  2  sin 1 sin  2 
 r1r2 cos1 cos 2  sin 1 sin  2   isin 1 cos 2  cos1 sin  2 

28. Mod-Arg Relations
1 z1 z2  z1 z2
arg z1 z 2   arg z1  arg z 2
Proof: let z1  r1cis1 and z 2  r2 cis 2
NOTE:
Multiplication
rotates z1 by
arg z2
z1 z 2  r1 cos1  i sin 1   r2 cos 2  i sin  2 
 r1r2 cos1 cos 2  i sin 1 cos 2  i cos1 sin  2  sin 1 sin  2 
 r1r2 cos1 cos 2  sin 1 sin  2   isin 1 cos 2  cos1 sin  2 
 r1r2 cos1   2   i sin 1   2 

29. Mod-Arg Relations
1 z1 z2  z1 z2
arg z1 z 2   arg z1  arg z 2
Proof: let z1  r1cis1 and z 2  r2 cis 2
NOTE:
Multiplication
rotates z1 by
arg z2
z1 z 2  r1 cos1  i sin 1   r2 cos 2  i sin  2 
 r1r2 cos1 cos 2  i sin 1 cos 2  i cos1 sin  2  sin 1 sin  2 
 r1r2 cos1 cos 2  sin 1 sin  2   isin 1 cos 2  cos1 sin  2 
 r1r2 cos1   2   i sin 1   2 
 z1 z 2  r1r2
 z1 z 2

30. Mod-Arg Relations
1 z1 z2  z1 z2
arg z1 z 2   arg z1  arg z 2
Proof: let z1  r1cis1 and z 2  r2 cis 2
NOTE:
Multiplication
rotates z1 by
arg z2
z1 z 2  r1 cos1  i sin 1   r2 cos 2  i sin  2 
 r1r2 cos1 cos 2  i sin 1 cos 2  i cos1 sin  2  sin 1 sin  2 
 r1r2 cos1 cos 2  sin 1 sin  2   isin 1 cos 2  cos1 sin  2 
 r1r2 cos1   2   i sin 1   2 
 z1 z 2  r1r2
arg z1 z 2   1   2
 z1 z 2
 arg z1  arg z 2

31. Mod-Arg Relations
1 z1 z2  z1 z2
arg z1 z 2   arg z1  arg z 2
Proof: let z1  r1cis1 and z 2  r2 cis 2
NOTE:
Multiplication
rotates z1 by
arg z2
z1 z 2  r1 cos1  i sin 1   r2 cos 2  i sin  2 
 r1r2 cos1 cos 2  i sin 1 cos 2  i cos1 sin  2  sin 1 sin  2 
 r1r2 cos1 cos 2  sin 1 sin  2   isin 1 cos 2  cos1 sin  2 
 r1r2 cos1   2   i sin 1   2 
 z1 z 2  r1r2
arg z1 z 2   1   2
 z1 z 2
 arg z1  arg z 2
NOTE: it follows that;
z1 z 2 z3  z n  z1 z 2 z3  z n
arg z1 z 2 z3  z n   arg z1  arg z 2  arg z3    arg z n

32. z1
z1
2  
z2 z2
z 
arg 1   arg z1  arg z 2
 z2 

33. z1
z1
2  
z2 z2
z 
arg 1   arg z1  arg z 2
 z2 
NOTE: it follows that;
z1 z 2
z1 z 2

z3 z 4 z3 z 4
 z1 z 2 
arg

 z z   arg z1  arg z 2  arg z3  arg z 4
 3 4

34. z1
z1
2  
z2 z2
z 
arg 1   arg z1  arg z 2
 z2 
NOTE: it follows that;
z1 z 2
z1 z 2

z3 z 4 z3 z 4
 z1 z 2 
arg

 z z   arg z1  arg z 2  arg z3  arg z 4
 3 4
3 z  z
n
n
argz n   n arg z

35. e.g. Find the modulus and argument of z 
5  i  2  i 
3  2i

36. e.g. Find the modulus and argument of z 
2
2
52  12  2    1
z
32  2 2
5  i  2  i 
3  2i

37. e.g. Find the modulus and argument of z 
2
2
52  12  2    1
z
32  2 2
26 5

13
 10
5  i  2  i 
3  2i

38. 5  i  2  i 
e.g. Find the modulus and argument of z 
3  2i
2
2
2
2
5  1  2    1
1
2
1
z
2
2
arg z  tan 1    tan 1    tan 1  
 
 
 
3 2
5
  2
 3
26 5

13
 10

39. 5  i  2  i 
e.g. Find the modulus and argument of z 
3  2i
2
2
2
2
5  1  2    1
1
2
1
z
2
2
arg z  tan 1    tan 1    tan 1  
 
 
 
3 2
5
  2
 3
26 5

 1119   153 26  33 41
13
 175 48
 10

40. 5  i  2  i 
e.g. Find the modulus and argument of z 
3  2i
2
2
2
2
5  1  2    1
1
2
1
z
2
2
arg z  tan 1    tan 1    tan 1  
 
 
 
3 2
5
  2
 3
26 5

 1119   153 26  33 41
13
 175 48
 10
Patel: Exercise 4B; evens
Patel: Exercise 4C; 1 to 10 evens
Cambridge: Exercise 1D; 9, 11, 13, 14, 16, 20