x2 t01 03 argand diagram (2012)
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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
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