ΜΕΘΟΔΟΛΟΓΙΑ ΑΣΚΗΣΕΩΝ ΜΙΓΑΔΙΚΩΝ ΑΡΙΘΜΩΝ

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1 . bi a z + . , , , 0 ) Im( b z z z z z , ..z z w + z z z w , zz22 z 2 . bi a z + ., , ,0 ) Re( a z , z z , , z zz22 3 , 4 . =0,1,2,3 -+i i i i i iu u u p u p u +14 4 : v 3i 1i 1z ,_

+, N v. 1 | ) z Im( ) z Re( | + . i2i 22i 2 1 11 1i 2 i 1) i 1 )( i 1 () i 1 )( i 1 (i 1i 12 + + + + , ( ) [ ] [ ] ( ) [ ]' + + + -1 Im(z) , 0 Re(z) , 3 2k v , i0 Im(z) , 1 Re(z) , 2 2k v , 11 Im(z) , 0 Re(z) , 1 2k v , i0 Im(z) , 1 Re(z) , 2kv , 1i i i i ) i ( zv vv3v3 v 3 1 | ) z Im( ) z Re( | +.4 , 4 ,4+1, 4+2,4+3. .5 . , , , , , . . i>0 , -1=ii>0 . i 1 ( i)( i) 0 > ( +i>0 > =0) ( +i + + > + + > + + + > + (1) : ' > + ' > + ' > + 2 x 0 y0 8 x 4 y x0 ) 2 x ( y 20 8 x 4 y x0 y 4 xy 20 8 x 4 y x2 2 2 2 2 2 '> + '> + 0 y0 8 x 4 x0 y0 8 x 4 y x2 2 2 0 8 x 4 x2> + , 0 16 < 0 8 x 4 x2> + R x . R x , ) 0 , x ( ) y , x ( ''< < + 2 x2 y 22 x4 y2 x0 8 2 4 y 22 x0 8 x 4 y x2 2 2 2 26 z z 2 1, , : z1=+i z2=+i ,, , R ,: = = ) z Re( ) z Re(2 1 ) z Im( ) z Im(2 1.7 z ,, , .8 z, , z +i , , .9 z z zz 2 z z2 .10 ( )Nbi avv*, + R = ( ), ibi ak + =+ ,k u 011 R z z=x+yi x,y ,12 . , . . .13 f(z) z . , . z z 2 1t 14 w=f(z) . M(z) I w w _ _ w w w w , , w x+yi :0 , 0 x I w y wH .15w=f(z) . M(z) C1 C2 M(w), w=x+yi, z=a+bi w=f(z) . b C2 M(w) ( , ). M(z) C1 w=a+bi , z=x+yi. : . z=+1+(-2)i , : z M(+1,-2) , : '' + 3 x y1 x 2 y1 x .. z : y=x-3 : z=+1+(-2)i y=2x-1 M(+1,-2) y=2x-1 , -2=2(+1)-1. : 2 21 x x + 16 z w w=f(z) . z w.)A z , w : z=x+yi w=f(z) a+bi w=a+bi ,b b=0 w =0 , b=0 =0 . z ) y , (x , ) y , B(x , ) y , x ( A3 3 2 2 1 1 . AB ( )1 2 1 2y y , x x AB ,, : ( ) ( ) ( ) ( ) 0 y y x x y y x x 0y y x xy y x x0 , AB det1 2 1 2 1 3 1 21 3 1 21 2 1 2

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z=x+yi (x,y) .

17 (Re(z)) (Im(z)) :1/ 2 2i 2Re(z) 2iIm(z) .2/ _ _z - z 2iIm(z) , z z ) z Re( 2 +

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w zzw zzIm iw - zzRe(1) (1) 2 , :. ,w zz2w zz2w zz2w zzw zzw zzw zzw zz2w zzw zzw zzw zzw zz2w zzIm i 2w - zzRe 2w zzw zzIm iw - zzRe

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18 ( ) . : z i ,, R + , z (0,0) (,) z , OM (,) uuuur.19 .1. , z w , 2. Re(z) Re(w)Im(z) Im(w) ' : R y , x , ( ) yi x 3 z , i 4 y x z2221+ + + . 2 1z , z ''''' + +4 x 1 xx 3 y0 4 x 3 xx 3 y) x 3 ( x 4x 3 yy x 4x 3 yy x 43 y x2 222 422 222222 ''t 1 x4 y1 x1 3 y2 ' + 4 x4 3 y2 (x,y)=(1,-4) (x,y)=(-1,-4) 20 21 i =+i z=+i=-i2+i=i(-i). : : 2008 2008(x yi) (y xi) + . [ ]2008 2008 502 4 2008 20082008 2008 2008 2008 2 2008 2008) yi x ( ) yi x ( i ) yi x ( i) yi x ( ) i ( ) yi x ( i ) ix y i ( ) xi y 1 ( ) xi y (+ + + + + 0 ) yi x ( ) yi x ( ) xi y ( ) yi x (2008 2008 2008 2008 + + +. : _ _z - z 2iIm(z) , z z ) z Re( 2 + 21 i : ,xR * 2xi 1x ixi 1i x v 4 v 4

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++

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+. ( )( )v 4 v 4v 4v 4 v 4xi 1i x xi 1i x 1xi 1i x xi 1x i

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+ 1]1

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+ v 4 v 4 v 4 v 4 v 4xi 1i x Re 2xi 1i x xi 1i x xi 1x ixi 1i x

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+ ( ) 1 1 i ixi 1) xi 1 ( ixi 1i x ixi 1i x vv4 v 4v 4v 42v 4 1]1

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+2 1 2 ) 1 Re( 2xi 1i x Re 2xi 1x ixi 1i x v 4 v 4 v 4 ,_

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+22 3 5 z | z i | + |z+2|=6 : w=z+2 z=w-2 . 3 5 z | z i | + :6 2 z 6 w .. .......... i 5 2 w 5 wi 5 2 w 5 w i 5 2 w 3 2 w | i 5 z | 3 z2 2 + + + + + 23 , , . :20 z z ,, R , 0 + + , :2224z _+ , (1) 24 ) > (1) : 2 z2 , 1t ) =0 (1) : 2z2 , 1 )