The document describes how complex numbers can be represented geometrically on an Argand diagram, which uses a horizontal x-axis for the real part and a vertical y-axis for the imaginary part. It provides examples of various complex numbers plotted as points on the diagram and notes that conjugates are reflections across the x-axis, while opposites involve a 180° rotation. The document then discusses how loci of complex functions can be represented as horizontal and vertical lines, circles, or other shapes on the Argand diagram. It provides two examples, showing that the locus of (i) (z + 4 + i)(z + 4 - i) = 49 is a circle with center (-4, -1) and radius 7 units

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