Migadikoi Georga Full
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ΦΘΙΝΟΠΩΡΟ 2008 Μ Α Θ Η Μ Α Τ Ι Κ Α ΜΙΑ ΑΠΟΠΕΙΡΑ ΔΙΔΑΣΚΑΛΙΑΣ ΓΙΑΝΝΗΣ Δ ΓΕΩΡΓΑΣ ΜΑΘΗΜΑΤΙΚΟΣ ΜΙΓΑΔΙΚΟΙ ΑΡΙΘΜΟΙ ΘΕΩΡΙΑ‐ΜΕΘΟΔΟΙ‐ΑΣΚΗΣΕΙΣ
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ΜΙΓΑΔΙΚΟΙ
Transcript of Migadikoi Georga Full
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2008
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1
1. : ={0, 1, 2, 3, ,,} : = {.-, -3, -2, -1, 0, 1, 2, 3, , , } : 0 . :
+ ={0, 1, 2, 3, , , .}= . .(1) Q: Q: {
/ * }
:
.
:
. , : 21 ,
42 ,
63 ,
84
105 ,
, .: . . : . : 2, -3,
12 ,
13- . Q, Q,
(1):
Q. : 2 , 7 8 , , .
R: . (1): ( ), .
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(2): . , . II. ( - ) ( ) . : x, : x0 xx-0 xx-0 : * . 57 55, 5
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3
( 22 , )
300, 450, 600, 900,
1800, 2700, 3600 6
, 4 ,
3 ,
2 , ,
23 , 2,
. , , ,
, 2
. ( 10 e)
!! a ! ! .
2. ( ) -(!!!) !
x x20. x2=-1 (), x . 2 , x2+x+=0, 0 (), =2-4
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, X1=-21 +
23 i,
x2= -21 -
23 i, +i
. * 1- 3- , . , . ( ) . , , , . , . .
3. C :
i i2=-1. i
( i), , I .
={i, }. . +i, ,
( i). () ,
, , (0 1 ), ( ), , .
C . , C={+i, , }.
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C +0i i C 0+i. 0 0+0i, 1 1+0i. x,,, , , ,W ,1,2 . :
C : . 4. C =+i (,) , : 1=+i 2=+i :
o 1=2 +i = +i = = ( ) o 1=0 +i = 0+0i =0 =0. o 10 +i 0+0i 0 0. o !!!!
5. C 1.-: 1+2 = (+i) + (+i) = (+)+(+)i, Z1-Z2 = 1+(-Z2) = (+i) + (+i) = (-)+(-)i
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2.: 12=(+i)(+i)=+i+i+i2=(-)+(+)i.( , R, ii =i2 -1).
!!!. C . 3. : 10, W=x+yi 1W=1 (+i)(x+yi) =1+0i (ax-y)+(y+x)i =1+0i x-y =1 (1) x+y =0 (2). (1) (2) D=2+2 0 ( ?), , :
x = 22 +
y = 22 -+
. 1
W, 1Z
1
1Z1
= 22 +
+
22 -+
i.
4.: 20 2
1
ZZ
= 12Z
1.
, : 2
1
ZZ
= ii
++
=
i-
2222 ++
++
.
!!! .( 9) 5.: C : = -1 N 2 1=. 0, 0 =1 -= Z
1 . 6. i : : i0=1. i4=1 . : i=i =0,1,2,3 I1=i i5=I I2=-1 i6=-1 4.(?) I3=-i i7=-i
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6. Z = +i
1. = Re() = Im() .
2. (,),
, . (). =+0 (,0), xx , i=0+i yy . . ! , R.
3. OM=(,) . .
4. OM
Z . Z = OM = 22 + =d(,).
0 .
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=0 Z = 0.
. Z = ,>0 , . : Z =. 1 2 : !!! 1=2 => 1Z = 2Z , (?). : i) 2121 zzzz = ,
ii) 2
1
2
1
zz
zz
= , 20
iii) 21 z-z 21 zz + 1Z + 2Z . i) iii) , 3 . , 1(1) 2(2) : 21 z-z =d(1,2). : ) , 0z-z =, >0, (0) , () .
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) , 1z-z = 2z-z , (1) (2) () . ) , 1z-z + 2z-z =2,>0, 1(1) 2(2), () . 1,2 .
5. : Re(Z) Z Im(Z) Z ( ?)
6. (,) 1(,-)
-i, Z . : 1) +Z = 2 = 2Re().
2) - Z = 2i = 2Im()i. 3) Z =2+2= Z 2 .
, . W , W= +Z ,W= Z W= - Z . : ) ZR Im(Z)=0 Z=Z ) ZI Re(Z)=0 Z=-Z
: i) 2121 zzzz +=+ , ii) 2121 zzzz = ,
iii) (2
1
2
1
zz
zz
=) .
i) ii) ,3 .
!!! =+i :
i.
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10
.
.
7. (,)
2(-,-) -i , .
8. (,) 3(-,)
+i -Z .
9. =+i, Z =-i, -=--i, -Z =-+i
: Z-Z-ZZ === .
7. 1. : ) ) ) || ) . 2. : ) ) ) || ) . 3. 2,3 . 3 =1, 310 = (3)103 = . 4. (. ) () =x+yi .
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. () ; | | | ||| 10|| 15 ||
.
3/2. .. 5. , : = R. 1.) . 6.
||
.
, || ,
.
|| , || , || , : , | | | | - ! ( , ). - ;
| | | | | | !! . 7. . , ( ), A | | | |. - | | | | | | | |
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| | . E 9 | |; 0 : | | .
!!!!. , , :
max| | . - , , . . , , , . :
| | | | | | | | | | | |
| | , | | 0 | | , | | . ,
| | , : | | . . : ) - ) - ,W , , min max |-W|. , - . |-2+3|=4 |-(2-3)|=4 (2,-3) 4, |-2+3| 4 . .
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8. i i ,
i=i4+, =0,1,2,3 =4,=4+1,=4+2,=4+3. . =i3+1+i5+3+i2, ,>0. =(i3)i+(i5)i3+(i2) = (-i)i+i(-i)+(-1) . ) =4 =(-i)4i+i4(-i)+(-1)4=i+(-i)+1=1 ) =4+1 ) =4+2 ,) =4+3 , i4=(-i)4=1. . (1+i)2=2i, (1-i)2=-2i, (1+i)3=-2(1-i) , (1+i)104=((1+i)2)52=(2i)52=252i52=252i0=252 . . 9. , , . ! . S=1+i+i2++i, . . 8., , !!! 10. . : |1+2|2+|1-2|2= 2|1|2+2|2|2 1,2. : 1. 1,2,3 1+2+3=0 |1|=|2|=|3|=>0, (1),(2),(3)
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2. z1, z2, z3, z4C z1+z2+z3+z4=1 z1=z2=z3=z4=1. (z1), (z2), (z3), (z4) .
1. 4+5, 1+, -1+, -,5+0, 0-4, -1-, 1-, 2+2, -2+2,4-5, -4+5, -2-2. 2. 1. 3. 3+4, 2-7, 5+ . 4. 1. , , . 5. = (2+1)+(||-2) . : ) , ) , ) , ) , ) . 6. : ) 1+, -2+2 ) 1+2, 3+ 7. =+ W= - , ) 2+W2. ? ) . 8. : ) Re(z) |z| ) Im(z) |z|, ) Re(z) =
, Im(z)
=
. 9. ||2=2 . 10. ||2=-2 .
1. , W W : Re
=
||,
Im
|| .
2. ,W , : Z + W = 2Re(Z ). 3. 3+2++=0, ,,, , 0, 0. 4. 3.
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5. ;
.
6. , (+2)2=, = + . . 7. ||2=|2-1|, Re(Z2)=1/2. 8. W = + () -+2=0 = + =2W+3- , . Wo , =2Wo+3-i. () (Wo) . 9. ) 22>-1, ) 22-+1>0. 10. W |W| =2, =
.
1. z13= 4+2 12 i, z25= 30 +2 31i, z34=14+2 15 i. : |z1z2 + z2z3 + z3z1|= 2|z1+z2+z3| 2. zC : z+16=4z+1 z=4. 3. : (1-2i)+(i+2)=0 4. zC z2+z+1=0. , z3+2+z6+1+1=0. 5. zC, z0 z+
z1 =1
() z6
() : z6+2 + 26z +1
= -1.
6. (z+2) zv =0, zC, *. z1 , : Re (z1)= -1.
7. N*, (i+)+(-i)=0,
, 2+2 >0.
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8. z=x+(y-2)I, x, y, (x,y)(21 ,0).
4i1-z2z
w+
= wI M(x, y) xx
9. :( )( )
1-2-
i2i-1i1
=+
.
10. 1,2 :|1-1-2i|=1 |2+2+2i|=2, |1-2|.
11. 0. 3ziz,-z, . 32 , . 12. =(2-4+3)+(2+2-3)i. :
. . . V. = 2.
13. ) |+i|=2, : 2-2 +1 2+ 2 . ) |-2+i|=4, : 1 |+2+4i| 9. 14. ) : ||2=2 . ) : |+|||+|-||| =2||,
. 15. Z1,Z2 Z12+Z22 = 0,
: (i) 21 zz = (ii) 2121 z-zzz =+ (iii)
, ,, z1,z2,z1+z2 .
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16. : =
2
i-1i
+ +2
i1-i
+ N.
17. : () )Re( 21
22
21
221 zz2zzzz ++=+
() )212
22
12
21 z2Re(z-zzz-z += .
18. z, w .
19. zC :z-1-i 5, :z-10-13i. . 20. z1, z2 C z2-z+9=0, z1,z2. () |z1|, |z2|. ()
2
1
zz +
1
2
zz = -2, .
() =0, z : |z-z1| + |z-z2|=10. 21. :2+2=4. )
) w=Z+
2+2=8. 22. 1,2,3,4 |1|=|2|=|3|=|4|=10. : |1-2|+|2-3|+|3-4|+|4-1|
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-
1. .
2. .
3. (0,0)
4. .
5. Z =1 C . 6. , . 7. =x+ix 1. 8. , . 9. = +i0 0 0 . 10. Re(2Z+1)=Re(2Z)+1. 11. R Im(Z)=0. 12. I Re(Z)=0. 13. 2121 ZZZZ +=+ 1,2 C. 14. Z , >0 . 15. |1|=|2| 1,2 16. . 17. 90. 18. 0=+, , . ) |-0|= . ) |-0|=||
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C KAI R.
R C R C 1. 1. 2. x = x2= 2. 3. x - x ,>0 3. 4. x > x 4. 5. x = x= 5. 6. - x x x , xR 6. 7. ,22 xx = xR 7. . zC, zzz 2 = 8.
8. ( )
9. ( )
9. ( )
10. 2
-
..
200
8
.
.
z2+1=0 z=
