GEOMETRY OF COMPLEX NUMBERS - Universitas...
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LOGO GEOMETRY OF COMPLEX NUMBERS Wuryansari Muharini Kusumawinahyu Dept. of Mathematics UB
Transcript of GEOMETRY OF COMPLEX NUMBERS - Universitas...
LOGO
GEOMETRY OF
COMPLEX NUMBERS
Wuryansari Muharini Kusumawinahyu
Dept. of Mathematics UB
Geometry of Complex Number
Rectangular / Cartesius
Coordinate / Argand Plane
– Point & Vector
– Addition & Subtraction
Coordinates
(Complex Plane)
Polar Coordinate
– Modulus & Argument
– Multiplication,
Division, Power &
Roots
Outline
Point & Vector in Complex Plane 1
Multiplication & Division 4
Polar Coordinate 3
Addition & Subtraction 2
Power & Roots 3
Complex Plane
Real Numbers
-5 -4 -3 -2 -1 0 1 2 3 4 5
x = -1.5
Real Line
Complex Numbers ?
Complex Numbers Complex Plane
-5 -4 -3 -2 -1 0 1 2 3 4 5
4
3
2
1
-1
-2
-3
-4
Re(z)
Im(z)
z = 2-3i
(z)iz)
biaz
ImRe(
iz 4 5z
iz 3iz 45
iz 43
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
5
4
3
2
1
-1
-2
-3
-4
-5
),( baz ),( dcw
wz
Vector & Addition ? wz
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
5
4
3
2
1
-1
-2
-3
-4
-5
),( baz ),( dcw wz
Subtraction
),( dcw
? wz )( wzwz
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
5
4
3
2
1
-1
-2
-3
-4
-5
biaz
Polar Coordinate
Re(z)
Im(z)
a
b
q
r
222 bar
22 bar
qcosra
qsinrb
Polar Form of Complex Numbers
q
rcis
ir
irrbiaz
)sin(cos
sincos
zzbar of modulus 22 Where
q
q
qtan
cos
sin
cos
sin
r
r
a
b
a
barctanq
q = argument of z
Multiplication & Division
)sin(cos
)sin(cos
iqdicw
irbiaz
)(
))sin()(cos(
q
rqcis
irqzw
)(
))sin()(cos(
q
cisq
r
iq
r
w
z
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
5
4
3
2
1
-1
-2
-3
-4
-5
qrcisz qcisw
)( q rqciszw
q
)( q cisq
r
w
z
MULTIPLICATION RULES
)( then
and
If
q
q
rqciszw
qcisdicw
rcisbiaz
The modulus of multiplication is
the multiplication of moduli
The argument of multiplication is
the sum of arguments
POWER
)(
then and if
212121
222111
cisrrzz
cisrzcisrz
)( So 321321321 qqq cisrrrzzz
)( 432143214321 qqqq cisrrrrzzzz
and
)( 321321321 nnn cisrrrrzzzz qqqq
)(
)(
then
if
terms terms
21
321
q
qqqq
q
ncisrzzz
cisrrrrzzz
rciszzzzz
n
nn
n
n
)( qncisrz nn
Generally
DE MOIVRE
THEOREM
(1667 – 1754)
Polar Form of Complex Numbers ….revisit
)360(
3600 ,
)sin(cos
0
00
krcis
rcis
irbiaz
q
q = principal argument of z
0 q
r
qrcisz
ROOTS
then if qrcisz ?1
nzzn
? and ? then Let 1
ppciszw n
)360(but
)(
0 krcisrciszw
ncispw
n
nn
knrpn 0360 and So q
1,,1,0,360
and 0
1
nkn
krrp nn
q
1,,1,0,360
z 0
n
nk
n
kcisrw n
q
Remark: once a complex number is represented
in the polar form, no difficulty to find its power
and roots as well.
