x2 t01 03 argand diagram (2013)
TRANSCRIPT
The Argand DiagramComplex numbers can be represented geometrically on an Argand Diagram.
1 2 3 4-1-2-3-4
123
-3-2-1
x
y4
(real axis)
(imaginary axis)
The Argand DiagramComplex numbers can be represented geometrically on an Argand Diagram.
1 2 3 4-1-2-3-4
123
-3-2-1
x
y4
(real axis)
(imaginary axis)
A = 2
A
The Argand DiagramComplex numbers can be represented geometrically on an Argand Diagram.
1 2 3 4-1-2-3-4
123
-3-2-1
x
y4
(real axis)
(imaginary axis)
A = 2B = -3i
A
B
The Argand DiagramComplex numbers can be represented geometrically on an Argand Diagram.
1 2 3 4-1-2-3-4
123
-3-2-1
x
y4
(real axis)
(imaginary axis)
A = 2B = -3iC = -2 + i
A
B
C
The Argand DiagramComplex numbers can be represented geometrically on an Argand Diagram.
1 2 3 4-1-2-3-4
123
-3-2-1
x
y4
(real axis)
(imaginary axis)
A = 2B = -3iC = -2 + iD = 4 - i A
B
C
D
The Argand DiagramComplex numbers can be represented geometrically on an Argand Diagram.
1 2 3 4-1-2-3-4
123
-3-2-1
x
y4
(real axis)
(imaginary axis)
A = 2B = -3iC = -2 + iD = 4 - iE = 4 + i
A
B
C
D
E
NOTE: Conjugates are reflected in the real (x) axis
The Argand DiagramComplex numbers can be represented geometrically on an Argand Diagram.
1 2 3 4-1-2-3-4
123
-3-2-1
x
y4
(real axis)
(imaginary axis)
A = 2B = -3iC = -2 + iD = 4 - iE = 4 + i
A
B
C
D
E
NOTE: Conjugates are reflected in the real (x) axis
F = -(4 - i)= - 4 + i
F
NOTE: Opposites are rotated 180°
The Argand DiagramComplex numbers can be represented geometrically on an Argand Diagram.
1 2 3 4-1-2-3-4
123
-3-2-1
x
y4
(real axis)
(imaginary axis)
A = 2B = -3iC = -2 + iD = 4 - iE = 4 + i
A
B
C
D
E
NOTE: Conjugates are reflected in the real (x) axis
F = -(4 - i)= - 4 + i
F
NOTE: Opposites are rotated 180°
G = i(4 - i)= 1+4i
GNOTE: Multiply by i results in 90° rotation
The Argand DiagramComplex numbers can be represented geometrically on an Argand Diagram.
1 2 3 4-1-2-3-4
123
-3-2-1
x
y4
(real axis)
(imaginary axis)
A = 2B = -3iC = -2 + iD = 4 - iE = 4 + i
A
B
C
D
E
NOTE: Conjugates are reflected in the real (x) axis
Every complex number can be represented by a unique point on the Argand Diagram.
F = -(4 - i)= - 4 + i
F
NOTE: Opposites are rotated 180°
NOTE: Multiply by i results in 90° rotation
G = i(4 - i)= 1+4i
G
Locus in Terms of Complex Numbers
Horizontal and Vertical Lines
Locus in Terms of Complex Numbers
Horizontal and Vertical Lines
y
x
c
cz Im
Locus in Terms of Complex Numbers
Horizontal and Vertical Lines
y
x
y
x
c
cz Im
k
kz Re
Circlesy
x
R
R
R
R2Rzz
Circlesy
x
R
R
R
R2Rzz
e.g.Find and describe the locus of points in the Argand diagram; (i) 4 4 49z i z i
Circlesy
x
R
R
R
R2Rzz
e.g.Find and describe the locus of points in the Argand diagram; (i) 4 4 49z i z i (4 ) (4 ) (4 )(4 ) 49zz i z i z i i
Circlesy
x
R
R
R
R2Rzz
e.g.Find and describe the locus of points in the Argand diagram; (i) 4 4 49z i z i (4 ) (4 ) (4 )(4 ) 49zz i z i z i i
2 2
4( ) (4 )(4 ) 49
4( ) (4 )(4 ) 49
4( ) (4 )(4 ) 49
8 2 16 1 49
zz z z iz iz i i
z z z z iz iz i i
z z z z iz iz i i
x y x y
Circlesy
x
R
R
R
R2Rzz
e.g.Find and describe the locus of points in the Argand diagram; (i) 4 4 49z i z i (4 ) (4 ) (4 )(4 ) 49zz i z i z i i
2 28 16 2 1 49x x y y
2 2
4( ) (4 )(4 ) 49
4( ) (4 )(4 ) 49
4( ) (4 )(4 ) 49
8 2 16 1 49
zz z z iz iz i i
z z z z iz iz i i
z z z z iz iz i i
x y x y
Circlesy
x
R
R
R
R2Rzz
e.g.Find and describe the locus of points in the Argand diagram; (i) 4 4 49z i z i (4 ) (4 ) (4 )(4 ) 49zz i z i z i i
2 24 1 49x y
2 2
4( ) (4 )(4 ) 49
4( ) (4 )(4 ) 49
4( ) (4 )(4 ) 49
8 2 16 1 49
zz z z iz iz i i
z z z z iz iz i i
z z z z iz iz i i
x y x y
2 28 16 2 1 49x x y y
Circlesy
x
R
R
R
R2Rzz
e.g.Find and describe the locus of points in the Argand diagram; (i) 4 4 49z i z i (4 ) (4 ) (4 )(4 ) 49zz i z i z i i
14:centre ,units7:radius
Locus is a circle
2 2
4( ) (4 )(4 ) 49
4( ) (4 )(4 ) 49
4( ) (4 )(4 ) 49
8 2 16 1 49
zz z z iz iz i i
z z z z iz iz i i
z z z z iz iz i i
x y x y
2 24 1 49x y
2 28 16 2 1 49x x y y
Circlesy
x
R
R
R
R2Rzz
e.g.Find and describe the locus of points in the Argand diagram; (i) 4 4 49z i z i (4 ) (4 ) (4 )(4 ) 49zz i z i z i i
14:centre ,units7:radius
Locus is a circle
2Rzz Locus is a circle
centre ωradius R
2 2
4( ) (4 )(4 ) 49
4( ) (4 )(4 ) 49
4( ) (4 )(4 ) 49
8 2 16 1 49
zz z z iz iz i i
z z z z iz iz i i
z z z z iz iz i i
x y x y
2 24 1 49x y
2 28 16 2 1 49x x y y
1 1( ) 1iiz z
1 1( ) 1iiz z
z z zz
1 1( ) 1iiz z
z z zz 2 22x x y
1 1( ) 1iiz z
z z zz 2 22x x y
2 2
2 2
2 0( 1) 1x x yx y
1 1( ) 1iiz z
z z zz 2 22x x y
2 2
2 2
2 0( 1) 1x x yx y
centre: 1 0,radius: 1 unit
Locus is a circle
1 1( ) 1iiz z
z z zz 2 22x x y
2 2
2 2
2 0( 1) 1x x yx y
centre: 1 0,radius: 1 unit
Locus is a circle
y
x1 2
1 1( ) 1iiz z
z z zz 2 22x x y
2 2
2 2
2 0( 1) 1x x yx y
centre: 1 0,radius: 1 unit
Locus is a circle
excluding the point (0,0)
y
x1 2
Cambridge: Exercise 1C; 1 ace, 2 bdf, 3, 4 ace, 5 bdfh, 6, 10 to 15
1 1( ) 1iiz z
z z zz 2 22x x y
2 2
2 2
2 0( 1) 1x x yx y
centre: 1 0,radius: 1 unit
Locus is a circle
excluding the point (0,0)
y
x1 2