1. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
y (imaginary axis)
Diagram.
4
3
2
1
-4 -3 -2 -1
-1
-2
-3
1
2
3 4 x (real axis)

2. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
y (imaginary axis)
Diagram.
4
3
A=2
2
1
A
-4 -3 -2 -1
1 2 3 4 x (real axis)
-1
-2
-3

3. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
y (imaginary axis)
Diagram.
4
3
A=2
2
B = -3i
1
A
-4 -3 -2 -1
1 2 3 4 x (real axis)
-1
-2
-3 B

4. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
y (imaginary axis)
Diagram.
4
3
A=2
2
B = -3i
C = -2 + i
C 1
A
-4 -3 -2 -1
1 2 3 4 x (real axis)
-1
-2
-3 B

5. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
y (imaginary axis)
Diagram.
4
3
A=2
2
B = -3i
C = -2 + i
C 1
A
D=4-i
-4 -3 -2 -1
1 2 3 4 x (real axis)
-1
D
-2
-3 B

6. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
y (imaginary axis)
Diagram.
4
3
A=2
NOTE: Conjugates
2
B = -3i
are reflected in the
E
real (x) axis
C = -2 + i
C 1
A
D=4-i
-4 -3 -2 -1
1 2 3 4 x (real axis)
E=4+i
-1
D
-2
-3 B

7. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
y (imaginary axis)
Diagram.
4
3
A=2
NOTE: Conjugates
2
B = -3i
are reflected in the
E
real (x) axis
C = -2 + i
F
C 1
A
D=4-i
-4 -3 -2 -1
1 2 3 4 x (real axis)
E=4+i
-1
D
F = -(4 - i)
-2
= - 4 + i NOTE:
Opposites are -3 B
rotated 180°

8. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
y (imaginary axis)
Diagram.
4G
NOTE: Multiply
3
A=2
NOTE: Conjugates
by i results in
2
B = -3i 90° rotation
are reflected in the
E
real (x) axis
C = -2 + i
F
C 1
A
D=4-i
-4 -3 -2 -1
1 2 3 4 x (real axis)
E=4+i
-1
D
F = -(4 - i)
-2
= - 4 + i NOTE:
G = i(4 - i) Opposites are -3 B
rotated 180°
= 1+4i

9. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
y (imaginary axis)
Diagram.
4G
NOTE: Multiply
3
A=2
NOTE: Conjugates
by i results in
2
B = -3i 90° rotation
are reflected in the
E
real (x) axis
C = -2 + i
F
C 1
A
D=4-i
-4 -3 -2 -1
1 2 3 4 x (real axis)
E=4+i
-1
D
F = -(4 - i)
-2
= - 4 + i NOTE:
G = i(4 - i) Opposites are -3 B
rotated 180°
= 1+4i
Every complex number can be represented by a unique
point on the Argand Diagram.

10. Locus in Terms of Complex
Numbers
Horizontal and Vertical Lines

11. Locus in Terms of Complex
Numbers
Horizontal and Vertical Lines
y
c
x
Im z   c

12. Locus in Terms of Complex
Numbers
Horizontal and Vertical Lines
y
c
y
x
Im z   c
k
x
Re z   k

13. y
R
Circles
zz  R 2
R
R
R
x

14. y
R
Circles
zz  R 2
R
R
x
R
e.g.Find and describe the locus of points in the Argand diagram;
(i)  z  4  i  z  4  i   49

15. y
R
Circles
zz  R 2
R
R
x
R
e.g.Find and describe the locus of points in the Argand diagram;
(i)  z  4  i  z  4  i   49
z z  (4  i ) z  (4  i ) z  (4  i )(4  i )  49

16. y
R
Circles
zz  R 2
R
R
x
R
e.g.Find and describe the locus of points in the Argand diagram;
(i)  z  4  i  z  4  i   49
z z  (4  i ) z  (4  i ) z  (4  i )(4  i )  49
z z  4( z  z )  iz  iz  (4  i )(4  i )  49
z z  4( z  z )  iz  iz  (4  i )(4  i )  49


z z  4( z  z )  iz  iz  (4  i )(4  i )  49
x 2  y 2  8 x  2 y  16  1  49

17. y
R
Circles
zz  R 2
R
R
x
R
e.g.Find and describe the locus of points in the Argand diagram;
(i)  z  4  i  z  4  i   49
z z  (4  i ) z  (4  i ) z  (4  i )(4  i )  49
z z  4( z  z )  iz  iz  (4  i )(4  i )  49
z z  4( z  z )  iz  iz  (4  i )(4  i )  49


z z  4( z  z )  iz  iz  (4  i )(4  i )  49
x 2  y 2  8 x  2 y  16  1  49
x 2  8 x  16  y 2  2 y  1  49

18. y
R
Circles
zz  R 2
R
R
x
R
e.g.Find and describe the locus of points in the Argand diagram;
(i)  z  4  i  z  4  i   49
z z  (4  i ) z  (4  i ) z  (4  i )(4  i )  49
x 2  8 x  16  y 2  2 y  1  49
z z  4( z  z )  iz  iz  (4  i )(4  i )  49
 x  4    y  1
z z  4( z  z )  iz  iz  (4  i )(4  i )  49


z z  4( z  z )  iz  iz  (4  i )(4  i )  49
x 2  y 2  8 x  2 y  16  1  49
2
2
 49

19. y
R
Circles
zz  R 2
R
R
x
R
e.g.Find and describe the locus of points in the Argand diagram;
(i)  z  4  i  z  4  i   49
z z  (4  i ) z  (4  i ) z  (4  i )(4  i )  49
x 2  8 x  16  y 2  2 y  1  49
z z  4( z  z )  iz  iz  (4  i )(4  i )  49
 x  4    y  1
z z  4( z  z )  iz  iz  (4  i )(4  i )  49


z z  4( z  z )  iz  iz  (4  i )(4  i )  49
x 2  y 2  8 x  2 y  16  1  49
2
2
Locus is a circle
centre :  4 ,  1
radius : 7 units
 49

20. y
R
Circles
 z    z     R 2
zz  R 2
R
R
x
Locus is a circle
centre ω
radius R
R
e.g.Find and describe the locus of points in the Argand diagram;
(i)  z  4  i  z  4  i   49
z z  (4  i ) z  (4  i ) z  (4  i )(4  i )  49
x 2  8 x  16  y 2  2 y  1  49
z z  4( z  z )  iz  iz  (4  i )(4  i )  49
 x  4    y  1
z z  4( z  z )  iz  iz  (4  i )(4  i )  49


z z  4( z  z )  iz  iz  (4  i )(4  i )  49
x 2  y 2  8 x  2 y  16  1  49
2
2
Locus is a circle
centre :  4 ,  1
radius : 7 units
 49

21. 1 1
(ii )   1
z z

22. 1 1
(ii )   1
z z
z  z  zz

23. 1 1
(ii )   1
z z
z  z  zz
2x  x 2  y 2

24. 1 1
(ii )   1
z z
z  z  zz
2x  x 2  y 2
x2  2 x  y 2  0
( x  1) 2  y 2  1

25. 1 1
(ii )   1
z z
z  z  zz
2x  x 2  y 2
x2  2 x  y 2  0
( x  1) 2  y 2  1
Locus is a circle
centre: 1,0 
radius: 1 unit

26. 1 1
(ii )   1
z z
z  z  zz
Locus is a circle
centre: 1,0 
radius: 1 unit
2x  x 2  y 2
x2  2 x  y 2  0
y
( x  1) 2  y 2  1
1
2 x

27. 1 1
(ii )   1
z z
z  z  zz
Locus is a circle
centre: 1,0 
radius: 1 unit
2x  x 2  y 2
x2  2 x  y 2  0
y
( x  1) 2  y 2  1
excluding the point (0,0)
1
2 x

28. 1 1
(ii )   1
z z
z  z  zz
Locus is a circle
centre: 1,0 
radius: 1 unit
2x  x 2  y 2
x2  2 x  y 2  0
y
( x  1) 2  y 2  1
excluding the point (0,0)
1
2 x
Cambridge: Exercise 1C; 1 ace, 2 bdf, 3, 4 ace, 5 bdfh, 6,
10 to 15