1-מספרים מרוכבים - תיאוריה ותרגול
description
Transcript of 1-מספרים מרוכבים - תיאוריה ותרגול
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, "
1
. a, b z = a + ib
a z Re(z) , - b z Im(z).
2: , z - , z 2z a b= +.
z: , z - , z a ib= .
z
z , arg(z) .
: ?
( ) ( )cos 360 , sin 360a z k b z k = + = +
z , a ib= : , +
( )cos( 360 ) sin( 360 ) cos( 360 ) sin( 360 )z a ib z k z i k z k i k = + = + + + = + + +
tan, ? ba
, 180o tan . =
.
) )cos( 360 ) sin( 360 )z z k i k = + + + , , :
.360 - 0
1 1 1z a ib= 2 - + 2 2z a ib= 1 + 1 2 2, , ,a b a b R :
: . 1
1 2 1 2 1 2( ) ( )z z a a i b b+ = + + +
: . 2
1 2 1 2 1 2( ) ( )z z a a i b b = +
: . 3
z
0 a
b
z = a + ib
z
2 2
tan
z a bba
= +
= b
-
, "
2
1 2 1 2 1 2 1 2 2 1( ) ( )z z a a b b i a b a b = + +
1 : . 4 2, 0z z
1 2 2 1
1 2 3 1 2 3
1 2 2 1
1 2 3 1 2 3
1 2 3 1 2 1 3
1 2 1 2
1 2 1 2
1 1
2 2
2
1 2 1 2
1 2 1
1.2. ( ) ( )3.4. ( ) ( )5. ( )6. 2Re( )7. 2 Im( )
8.
9.
10.
11.
12.
13.
z z z zz z z z z zz z z zz z z z z zz z z z z z zz z zz z i z
z z z z
z z z z
z zz z
z z z
z z z z
z z z
+ = ++ + = + + = = + = + + = =
=
=
=
=
+ +
= 2
11
2 2
14.
15.
z
zzz z
z z
=
=
:
. 6 .1
. 7 .2
5 1 .3 , .
.
. C .4
1 (9 .5 2z z= :( n n nz z=.
1 (13 .6 2z z= :( n nnz z=.
: 7 .7
, n
1 1 22
2 2
z z zz z
=
-
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3
, .
. n
( ) ( ) ( )( ) ( ) ( )
6 5 2
4 6 3
4 2 3 1
4 3 2 2 2
i i ii i i
+ + +
+ .
( ) ( ) ( )( ) ( ) ( )
( ) ( ) ( )( ) ( ) ( )
( ) ( ) ( )( ) ( ) ( )
( ) ( )( ) ( ) ( )
6 5 26 5 2 6 5 2
4 6 3 4 6 3 4 6 3
6 5 2 2 2
4 6 3 3 3 2
16 1 4 9 1 14 2 3 1 4 2 3 1
4 3 2 2 24 3 2 2 2 1 16 9 4 4 4
17 13 2 17 2 17 2 17 2 178 2617 13 8 13 2 2 13 8 2 13 8 2 2
i i i i i ii i ii i i
+ + ++ + + + + += = =
+ + + + +
= = = =
: :
1 1 1 1(cos sin )z r i = 2 - + 2 2 2(cos sin )z r i = + ,
:
( )
( )
1 2 1 2 1 2 1 2
1 11 2 1 2 2
2 2
1. cos( ) sin( )
2. cos( ) sin( ) ( 0)
3. (cos sin )
360 3604. cos sin 0,1,2,..., 1
n
n n
z z r r i
z r i zz r
z r n i n
k kz r i k nn n
= + + +
= +
= +
+ + = + =
:
1 2 1 2arg( ) arg( ) arg( )z z z z = +
11 2
2
arg arg( ) arg( )z z zz
=
-
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4
1 : 4(cos 60 sin 60 )z i= 2 - + 5(cos 40 sin 40 )z i= : +
1 2 20(cos100 sin100 )z z i = 1 - +2
4 (cos 20 sin 20 )5
z iz
= +.
1 2z z= :21 16(cos120 sin120 )z i= +.
3 3 : 3z i= + .
3 3z i= : . +
( )
2 2
3 3
3 ( 3) 9 3 12
3tan 303
12 cos(30 360 ) sin(30 360 )
z i
r z
z k i k
= +
= = + = + =
= =
= + + +
:
( )( ) ( )( ) ( )
33
60
6 61
6 61
30 360 30 36012 cos( ) sin( ) 0,1,23 3
0 12 cos10 sin10
1 12 cos(10 120 ) sin(10 120 ) 12 cos130 sin130
2 12 cos(10 2 120 ) sin(10 2 120 ) 12 cos 250 sin 250
k kz i k
k z i
k z i i
k z i i
+ += + =
= = +
= = + + + = +
= = + + + = +
.1 6 :
): 1 )1 1 cos360 sin 360k i k= + .
6 6
0
1
2
3
4
360 3601 1 cos sin cos60 sin 60 , 0,1,2,3,4,56 6
0 cos0 sin 0 1 0 1
1 31 cos60 sin 602 2
1 32 cos120 sin1202 2
3 cos180 sin180 1 0 1
4 cos 240 sin 240
k i k k i k k
k z i i
k z i i
k z i i
k z i i
k z i
= + = + =
= = + = + =
= = + = +
= = + = +
= = + = + =
= = + =
5
1 32 2
1 35 cos300 sin 3002 2
i
k z i i
= = + =
-
, "
5
:
.0 - z n .1
)1 - z n .2 1)n z+
:
0 . 1 02 1cos135 sin135 , 4 2 4 2z i z i= + = : . 3 1
2
zzz
=.
:1
1z :
( ) ( )1 1
2 2
1 2
3 1
2
3
4 2 4 2
4 2 4 2 32 32 64 8
4 2tan 1 360 45 3154 2
8 315 , 135
8 315 8 (315 ( 135 )) 8 450 8 (450 360 ) 8 90( 135 )
90 3608 2 (30 120 )3k
z i z rcis
r
z cis z cisz cisz cis cis cis cisz cis
kz cis cis k
k
= =
= + = + = =
= = = =
= =
= = = = = =
+= = +
( )
0
1
2
0 1 2
3 10 2 30 2 32 2
3 11 2 (30 120 ) 2 150 2 32 2
2 2 (30 240 ) 2 270 2 0 2
2 30 3 , 2 150 3 , 2 270 2
z cis i i
k z cis cis i i
k z cis cis i i
z cis i z cis i z cis i
= = = + = +
= = + = = + = +
= = + = = =
= = + = = + = =
: . 2
)2. 1)z z= +
6. 4 24 16 64 0i i+ + + =z z z
0483. 24 =+ zz -
nz. z= n1 - .
-
, "
6
:2
z: . a ib= a, + b R :
( )
2
2
2 2
2 2
( 1)
( 1)
( 1)( 1) 12 2 2 1
z z
a ib a ib
a ib a ib a ib a abi a abi b ib a iba ib a a b ib abi
= +
+ = + +
= + + + + = + + + + + + +
= + + + +
:
2 2
2 2 2 2 2 2
2 1 (2 2 )
2 1 1 0 1 02 2 3 2 0 (3 2 ) 0
a ib a a b b ab i
a a a b a a b a a bb b ab b ab b a
= + + + +
= + + + + = + + =
= + + = + =
0b: =:
2: 1 0a a+ + . . =a R 0 b . =
3: 2 0a+ = :
32
a = : .
22
2
2
1 2
3 3 1 02 2
9 3 1 04 2
7 74 2
3 7 3 7,2 2 2 2
b
b
b b
z i z i
+ + =
+ =
= =
= + =
: .
6 4 2 4 2 2 2 4
2 4
2 2
1 24 4
0 4 16 64 ( 4) 16 ( 4) ( 4)( 16 )( 4)( 16 ) 0
4 0 42 , 2
16 0 16
i i z z i z z z iz z i
z zz i z iz i z i
= + + + = + + + = + +
+ + =
+ = = = =
+ = =
z z z
-
, "
7
4
4 4
3
4
5
16 16 270 16 270
16 270
270 360 270 36016 270 16 2 2 (67.5 90 ) , 0,1,2,34 4 4
0 2 67.5
1 2 (67.5 90 ) 2 157.5
2 2 (157.5 90 ) 2 247.5
i i cis cis
z cis
kz cis cis cis k cis k k
k z cisk z cis cisk z cis cis
= =
=
+= = = + = + =
= =
= = + =
= = + =
6
1 2 3 4 5 5
3 2 (247.5 90 ) 2 337.5
2 , 2 , 2 67.5 , 2 157.5 , 2 247.5 , 2 337.5
k z cis cis
z i z i z cis z cis z cis z cis
= = + =
= = = = = =
.
4 2
4 2 2
3 48 048 163
z z
z z z
+ =
= =
z: z rcis= :( )z rcis = 16 . 16 180cis =
: "
4 2
4 2 2
4 2
164 16 180 ( 2 ) 16 (180 2 )4 16 (180 2 )
z zr cis cis r cis r cisr cis r cis
=
= =
=
:
4 2
4 2
4 16 (180 2 )
164 180 2 360
r cis r cis
r rk
=
=
= +
2: 0r = :
0,1: 0z . 2 0 , =
0r: :
: r2 -
-
, "
8
4 2 2
2
3
4
16 164 180 2 360 6 180 360
4180 360 30 60 , 0,1,...,5
63 10 4 30 4(cos30 sin 30 ) 4( ) 2 3 2
2 21 4 90 4
32 4 150 4(cos150 sin150 ) 4(
r r rk k
rk k k
k z cis i i i
k z cis i
k z cis i
= = = + = +
= +
= = + =
= = = + = + = +
= = =
= = = + =
5
6
7
1) 2 3 22 2
3 13 4 210 4(cos 210 sin 210 ) 4( ) 2 3 22 2
4 4 270 4
3 15 4 330 4(cos330 sin 330 ) 4( ) 2 3 22 2
i i
k z cis i i i
k z cis i
k z cis i i i
+ = +
= = = + = =
= = =
= = = + = =
: 8
0,1 2 3 4 5 6 70 , 2 3 2 , 4 , 2 3 2 , 2 3 2 , 4 , 2 3 2z z i z i z i z i z i z i= = + = = + = = =
z ', . rcis= 0 r 0r - = :
1
1
, ( )( ) ( )
360360 ( 1) 360 , 0,1, 2,...,1
0 1 1360 , 0,1, 2,...,
1360 , 0,1, 2,...,
1
n
n
nn n
n n
k
z zz rcis z rcisr cis n rcis
r rr r r rkn k n k k n
nr r r r r
k k nn
z cis k k nn
+
== =
=
= = = = + + = = = +
= = =
= = +
= = +
20 0nr z += =
2n . +
-
, "
9
3: . 3 2( ) 3 4p x x ix i= + 1x . + i= .
.
:3
1x :
.
1x - i= x i -
x i: 2
3 2
3 2
2
2
4 4
3 4
4 4
4 4
4 4
4 4
x ix
x ix i x i
x ix
ix i
ix x
x i
x i
+
+ +
+
+
+
+
= =
2: 4 4x ix+ .
:2
2,3
4 4 0
4 16 4 ( 4) 4 22 2
x ix
i ix i
+ =
= = =
.2 2i - 1 i:
2: . 4 11 2 22
116 170 , 3408
zz z cis cisz
= =
. 2z 1z .
14 , 2z . 22
zxz
= .
:4
1: 1 1 2 2 2,z rcis z r cis = =
: .
-
, "
10
2 21 2 1 2 1 2
1 11 22 2
2 2
(2 ) 16 1701( 2 ) 3408
z z r r cis cisz r cis cisz r
= =
= =
:
42 2 5 52
1 2 1 2 2222 2
1 22 212 21 1
12
1 2 2 1 2 1 2 1
1 2 1 2 1 1
16 16 1024 416 4641 28 8 88
2 170 2 170 2 170 2 1
2 340 2 340 4 340 340
rr r r r rr rr r r rrr rrr
= = = == = = == = =
= = = = = = + = 1
2 12 1
11
70
3 0 360
2 170 2 170360 120 , 0,1,2
3
k
k k k
= +
= = = ==
:2z 1z 3 1 3
1 2
1 2
1 2
1 2
1 2
1 2
. 0 0 170 190
2 0 2 , 4 190
. 1 120 70
2 120 , 4 70
. 2 240 310
2 240 , 4 310
I kz cis z cis
II kz cis z cis
III kz cis z cis
= = = =
= = =
= = =
= =
= = =
= =
14 . II , 2z. 22
zxz
: =
( ) ( )1 4 4 444 22
2 120 2 1 1120 ( 140 ) 120 140 26016 ( 140 ) 16 8 8
z cis cis cis cisz cis
= = = + =
: z n
-
, "
11
( )4 4 4
40
41
42
43
1 1 260 360 1260 65 90 , 0,1,2,38 8 4 8
10 658
11 155812 245813 3358
kcis cis cis k k
k x cis
k x cis
k x cis
k x cis
+= = + =
= =
= =
= =
= =
5.
1 . 2 3, , . 3 2005 2005 20051 2 3 + +
1 . 20, ,z z 7 20 20 7i . 1 2 20arg( )z z z
. 3600 - 0
:5
1 . 2 3, , 3 3 1j - . j = 1,2,3 =
2005 = 3k + 1 :
( ) ( ) ( )2005 2005 2005 3 1 3 1 3 1 3 3 31 2 3 1 2 3 1 1 2 2 3 3 1 2 3k k kk k k + + ++ + = + + = + + = + +
: 0 -
2005 2005 20051 2 3 1 2 3 0 + + = + + =
1 . 20z z z .
) ) 11 n z+ 7 7z i= : 21
1 20
1 2 20
( 1) (7 7 ) 7 7
arg( ) arg( 7 7 ) 135
z z i iz z z i
= = +
= + =