3. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y
3
2
1
-4 -3 -2 -1 1 2 3 4 x
-1
-2
-3
4. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y
3
2
1
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1
-2
-3
5. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y (imaginary axis)
3
2
1
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1
-2
-3
6. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y (imaginary axis)
3
2
1
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1
A=2 -2
-3
7. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y (imaginary axis)
3
2
1
A
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1
A=2 -2
-3
8. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y (imaginary axis)
3
2
1
A
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1
A=2 -2
B = -3i -3
9. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y (imaginary axis)
3
2
1
A
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1
A=2 -2
B = -3i -3 B
10. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y (imaginary axis)
3
2
1
A
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1
A=2 -2
B = -3i -3 B
C = -2 + i
11. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y (imaginary axis)
3
2
C 1
A
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1
A=2 -2
B = -3i -3 B
C = -2 + i
12. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y (imaginary axis)
3
2
C 1
A
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1
A=2 -2
B = -3i -3 B
C = -2 + i
D=4-i
13. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y (imaginary axis)
3
2
C 1
A
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1 D
A=2 -2
B = -3i -3 B
C = -2 + i
D=4-i
14. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y (imaginary axis)
3
2
C 1
A
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1 D
A=2 -2
B = -3i -3 B
C = -2 + i
D=4-i
E=4+i
15. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y (imaginary axis)
3
2
C 1 E
A
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1 D
A=2 -2
B = -3i -3 B
C = -2 + i
D=4-i
E=4+i
16. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y (imaginary axis)
3 NOTE: Conjugates
2 are reflected in the
C 1 E real (x) axis
A
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1 D
A=2 -2
B = -3i -3 B
C = -2 + i
D=4-i
E=4+i
17. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y (imaginary axis)
3 NOTE: Conjugates
2 are reflected in the
C 1 E real (x) axis
A
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1 D
A=2 -2
B = -3i -3 B
C = -2 + i
D=4-i
E=4+i
Every complex number can be represented by a unique
point on the Argand Diagram.
19. Mod-Arg Form
Modulus
The modulus of a complex number is the
y length of the vector OZ
O x
20. Mod-Arg Form
Modulus
The modulus of a complex number is the
y length of the vector OZ
z = x + iy
O x
21. Mod-Arg Form
Modulus
The modulus of a complex number is the
y length of the vector OZ
z = x + iy
O x
22. Mod-Arg Form
Modulus
The modulus of a complex number is the
y length of the vector OZ
z = x + iy
y
O x x
23. Mod-Arg Form
Modulus
The modulus of a complex number is the
y length of the vector OZ
z = x + iy
r 2 x2 y2
y r x2 y2
O x x
24. Mod-Arg Form
Modulus
The modulus of a complex number is the
y length of the vector OZ
z = x + iy
r 2 x2 y2
r z
y r x2 y2
O x x
25. Mod-Arg Form
Modulus
The modulus of a complex number is the
y length of the vector OZ
z = x + iy
r 2 x2 y2
r z
y r x2 y2
z x2 y2
O x x
26. Mod-Arg Form
Modulus
The modulus of a complex number is the
y length of the vector OZ
z = x + iy
r 2 x2 y2
r z
y r x2 y2
z x2 y2
O x x
Argument
The argument of a complex number is
the angle the vector OZ makes with the
positive real (x) axis
27. Mod-Arg Form
Modulus
The modulus of a complex number is the
y length of the vector OZ
z = x + iy
r 2 x2 y2
r z
y r x2 y2
arg z z x2 y2
O x x
Argument
The argument of a complex number is
the angle the vector OZ makes with the
positive real (x) axis
28. Mod-Arg Form
Modulus
The modulus of a complex number is the
y length of the vector OZ
z = x + iy
r 2 x2 y2
r z
y r x2 y2
arg z z x2 y2
O x x
Argument
The argument of a complex number is
the angle the vector OZ makes with the
positive real (x) axis
1 y
arg z tan
x
29. Mod-Arg Form
Modulus
The modulus of a complex number is the
y length of the vector OZ
z = x + iy
r 2 x2 y2
r z
y r x2 y2
arg z z x2 y2
O x x
Argument
The argument of a complex number is
the angle the vector OZ makes with the
positive real (x) axis
y
1
arg z tan arg z
x
34. e.g . Find the modulus and argument of 4 4i
arg4 4i tan 4
4 4i 4 4
1
2 2
4
32 tan 1 1
4 2
35. e.g . Find the modulus and argument of 4 4i
arg4 4i tan 4
4 4i 4 4
1
2 2
4
32 tan 1 1
4 2
36. e.g . Find the modulus and argument of 4 4i
arg4 4i tan 4
4 4i 4 4
1
2 2
4
32 tan 1 1
4 2
4
37. e.g . Find the modulus and argument of 4 4i
arg4 4i tan 4
4 4i 4 4
1
2 2
4
32 tan 1 1
4 2
4
Every complex number can be written in terms of its modulus and
argument
38. e.g . Find the modulus and argument of 4 4i
arg4 4i tan 4
4 4i 4 4
1
2 2
4
32 tan 1 1
4 2
4
Every complex number can be written in terms of its modulus and
argument
z x iy
39. e.g . Find the modulus and argument of 4 4i
arg4 4i tan 4
4 4i 4 4
1
2 2
4
32 tan 1 1
4 2
4
Every complex number can be written in terms of its modulus and
argument
z x iy
r cos ir sin
40. e.g . Find the modulus and argument of 4 4i
arg4 4i tan 4
4 4i 4 4
1
2 2
4
32 tan 1 1
4 2
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
41. e.g . Find the modulus and argument of 4 4i
arg4 4i tan 4
4 4i 4 4
1
2 2
4
32 tan 1 1
4 2
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;
42. e.g . Find the modulus and argument of 4 4i
arg4 4i tan 4
4 4i 4 4
1
2 2
4
32 tan 1 1
4 2
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
43. e.g . Find the modulus and argument of 4 4i
arg4 4i tan 4
4 4i 4 4
1
2 2
4
32 tan 1 1
4 2
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
44. e.g . Find the modulus and argument of 4 4i
arg4 4i tan 4
4 4i 4 4
1
2 2
4
32 tan 1 1
4 2
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