x2 t01 03 argand diagram (2013)
TRANSCRIPT
![Page 1: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/1.jpg)
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)
![Page 2: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/2.jpg)
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
![Page 3: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/3.jpg)
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
![Page 4: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/4.jpg)
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
![Page 5: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/5.jpg)
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
![Page 6: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/6.jpg)
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
![Page 7: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/7.jpg)
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°
![Page 8: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/8.jpg)
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
![Page 9: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/9.jpg)
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
![Page 10: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/10.jpg)
Locus in Terms of Complex Numbers
Horizontal and Vertical Lines
![Page 11: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/11.jpg)
Locus in Terms of Complex Numbers
Horizontal and Vertical Lines
y
x
c
cz Im
![Page 12: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/12.jpg)
Locus in Terms of Complex Numbers
Horizontal and Vertical Lines
y
x
y
x
c
cz Im
k
kz Re
![Page 13: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/13.jpg)
Circlesy
x
R
R
R
R2Rzz
![Page 14: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/14.jpg)
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
![Page 15: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/15.jpg)
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
![Page 16: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/16.jpg)
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
![Page 17: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/17.jpg)
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
![Page 18: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/18.jpg)
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
![Page 19: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/19.jpg)
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
![Page 20: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/20.jpg)
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
![Page 21: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/21.jpg)
1 1( ) 1iiz z
![Page 22: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/22.jpg)
1 1( ) 1iiz z
z z zz
![Page 23: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/23.jpg)
1 1( ) 1iiz z
z z zz 2 22x x y
![Page 24: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/24.jpg)
1 1( ) 1iiz z
z z zz 2 22x x y
2 2
2 2
2 0( 1) 1x x yx y
![Page 25: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/25.jpg)
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
![Page 26: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/26.jpg)
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
![Page 27: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/27.jpg)
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
![Page 28: X2 t01 03 argand diagram (2013)](https://reader034.vdocument.in/reader034/viewer/2022052412/557e77d1d8b42a4d108b4b6d/html5/thumbnails/28.jpg)
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