X2 T01 03 argand diagram (2011)
-
Upload
nigel-simmons -
Category
Technology
-
view
834 -
download
0
Transcript of X2 T01 03 argand diagram (2011)
The Argand Diagram
The Argand DiagramComplex numbers can be represented geometrically on an Argand Diagram.
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
y
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
y
(real 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
y
(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
y
(real axis)
(imaginary axis)
A = 2
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
y
(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
y
(real axis)
(imaginary axis)
A = 2B = -3i
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
y
(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
y
(real axis)
(imaginary axis)
A = 2B = -3iC = -2 + i
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
y
(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
y
(real axis)
(imaginary axis)
A = 2B = -3iC = -2 + iD = 4 - 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
y
(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
y
(real axis)
(imaginary axis)
A = 2B = -3iC = -2 + iD = 4 - iE = 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
y
(real axis)
(imaginary axis)
A = 2B = -3iC = -2 + iD = 4 - iE = 4 + i
A
B
C
D
E
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
y
(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
y
(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.
Mod-Arg Form
y
xO
Mod-Arg FormModulusThe modulus of a complex number is the length of the vector OZy
xO
Mod-Arg Form
y
xO
ModulusThe modulus of a complex number is the length of the vector OZz = x + iy
Mod-Arg Form
y
xO
ModulusThe modulus of a complex number is the length of the vector OZz = x + iy
Mod-Arg Form
y
xO
ModulusThe modulus of a complex number is the length of the vector OZz = x + iy
x
y
Mod-Arg Form
y
xO
ModulusThe modulus of a complex number is the length of the vector OZz = x + iy
x
y 22
222
yxr
yxr
Mod-Arg Form
y
xO
ModulusThe modulus of a complex number is the length of the vector OZz = x + iy
x
y 22
222
yxr
yxr
zr
Mod-Arg Form
y
xO
ModulusThe modulus of a complex number is the length of the vector OZz = x + iy
x
y 22
222
yxr
yxr
zr
22 yxz
Mod-Arg Form
y
xO
ModulusThe modulus of a complex number is the length of the vector OZz = x + iy
x
y 22
222
yxr
yxr
zr
22 yxz
ArgumentThe argument of a complex number is the angle the vector OZ makes with the positive real (x) axis
Mod-Arg Form
y
xO
ModulusThe modulus of a complex number is the length of the vector OZz = x + iy
x
y 22
222
yxr
yxr
zr
22 yxz
ArgumentThe argument of a complex number is the angle the vector OZ makes with the positive real (x) axis
zarg
Mod-Arg Form
y
xO
ModulusThe modulus of a complex number is the length of the vector OZz = x + iy
x
y 22
222
yxr
yxr
zr
22 yxz
ArgumentThe argument of a complex number is the angle the vector OZ makes with the positive real (x) axis
zarg
xyz 1tanarg
Mod-Arg Form
y
xO
ModulusThe modulus of a complex number is the length of the vector OZz = x + iy
x
y 22
222
yxr
yxr
zr
22 yxz
ArgumentThe argument of a complex number is the angle the vector OZ makes with the positive real (x) axis
zarg
xyz 1tanarg zarg
ige 44ofargument andmodulus theFind ..
22 4444 i
ige 44ofargument andmodulus theFind ..
22 4444 i
ige 44ofargument andmodulus theFind ..
2432
22 4444 i
ige 44ofargument andmodulus theFind ..
44tan44arg 1i
2432
22 4444 i
ige 44ofargument andmodulus theFind ..
44tan44arg 1i
2432
1tan 1
22 4444 i
ige 44ofargument andmodulus theFind ..
44tan44arg 1i
2432
1tan 1
22 4444 i
ige 44ofargument andmodulus theFind ..
44tan44arg 1i
2432
1tan 1
4
22 4444 i
ige 44ofargument andmodulus theFind ..
44tan44arg 1i
2432
1tan 1
4
Every complex number can be written in terms of its modulus and argument
22 4444 i
iyxz
ige 44ofargument andmodulus theFind ..
44tan44arg 1i
2432
1tan 1
4
Every complex number can be written in terms of its modulus and argument
22 4444 i
iyxz
ige 44ofargument andmodulus theFind ..
44tan44arg 1i
2432
1tan 1
4
Every complex number can be written in terms of its modulus and argument
sincos irr
22 4444 i
iyxz
ige 44ofargument andmodulus theFind ..
44tan44arg 1i
2432
1tan 1
4
Every complex number can be written in terms of its modulus and argument
sincos irr sincos ir
22 4444 i
iyxz
ige 44ofargument andmodulus theFind ..
44tan44arg 1i
2432
1tan 1
4
Every complex number can be written in terms of its modulus and argument
sincos irr sincos ir
The mod-arg form of z is;
22 4444 i
iyxz
ige 44ofargument andmodulus theFind ..
44tan44arg 1i
2432
1tan 1
4
Every complex number can be written in terms of its modulus and argument
sincos irr sincos ir
The mod-arg form of z is; sincos irz
22 4444 i
iyxz
ige 44ofargument andmodulus theFind ..
44tan44arg 1i
2432
1tan 1
4
Every complex number can be written in terms of its modulus and argument
sincos irr sincos ir
The mod-arg form of z is; sincos irz
rcisz
22 4444 i
iyxz
zzrarg
where;
ige 44ofargument andmodulus theFind ..
44tan44arg 1i
2432
1tan 1
4
Every complex number can be written in terms of its modulus and argument
sincos irr sincos ir
The mod-arg form of z is; sincos irz
rcisz
iie.g. 44
iie.g. 44
424 cis
iie.g. 44
424 cis
iii 3
iie.g. 44
424 cis
iii 3
24
13 22
z
iie.g. 44
424 cis
iii 3
24
13 22
z3
1tanarg 1z
iie.g. 44
424 cis
iii 3
24
13 22
z3
1tanarg 1z
6
iie.g. 44
424 cis
iii 3
24
13 22
z3
1tanarg 1z
6
623 cisi
iie.g. 44
424 cis
iii 3
24
13 22
z3
1tanarg 1z
6
623 cisi
Exercise 4B; evens