Askiseis stous migadikous (diabathmismenes)

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Transcript of Askiseis stous migadikous (diabathmismenes)

  • 1

    () i2 = 1i2 = 1i2 = 1

    1.1 + i, , R :) i(1 2i) + 2(i 1) ) (i2 2i+ 3)(i2 + i 1)) (i2 + i)(1+ 2i) ) (i2 i)(1 i)(2+ i2)(2 i2)) i(2 i)+2i(1 i)3i(i22) ) (i1)(3i2+2i+5)) (3i2 7i+ 5)(4i2 2i+ 7) + (1 i2)(2+ 3i 5i2)) ix(i x 2) i(2x 3) + i(x2 3), x R

    1.2 + i, , R :

    )5

    1 2i) i

    1i

    )1+ 2i2 i

    )3 7i1+ i

    +7 3i1 i

    )3+ i2i

    )1+ i

    2

    2 i)

    3i2 + i2i 1

    ) i + i

    )1 i

    3

    3+ i)

    (2+ 3i)(1 i)1 2i

    )1 2i3+ i

    1+ 2i1+ 3i

    )(2 5i)(1+ i) (2+ 3i)

    3+ 4i)

    1+ 2i2 i

    +2i

    1.3 + i, , R :) (i 3)2 + (2+ i)2 ) (3+ 4i)2 + (4 3i)2 + i2) (1

    2i)2 ) (3+ 4i)2(1 i) + (3 4i)2(1+ i)

    ) (1 i)(1 2i)(1+ i)(1+ 2i) + i2 ) (1 2i)3) (1+ i)3 (1 i)3 ) (2+ 3i)3 + (2 3i)3) (2 i)5(2+ i)6

    1.4 pi :) (i 1)i1 2(1 i)2 ) (1 2i)2 + (1+ 2i)2) (6 4i)(1 i)1 + (17+ 19i)(1 5i)1

    1.5 + i, , R :

    )(2+ i)2

    3+ i)

    (1 i1+ i

    )2)

    (1 5i3 2i

    )2+

    (4 2i1 3i

    )2)

    (1 i)3 + (1+ i)3

    (1 i)3 (1+ i)3)

    (i 32

    )3

    (i+ 32

    )3)

    (3+ 3ii 1

    )2)

    22 +22i4 22

    22i4

    1.6 x R, pi :

    (x 1 i)(x 1+ i)(x+ 1+ i)(x+ 1 i) = x4 + 4

    1.7 z = 3+4i w = 23i, + i, , R, :

    z+ w, zw,26w

    25wz

    1.8 z = 2 3i, + i,

    , R, w = z+ 2iz 2i

    1.9 :

    z =(1 i)2

    1+ i w =

    z2 + 1z

    pi w

    1.10 z =1

    1 i+

    11+ i

    +i2.

    pi 4z2 8z+ 5 = 0

    1.11 z = 9+ xi, x > 0. x z2 z3

    1.12 a, b, c C :a

    7+ 4i=

    b1+ 2i

    =c

    1+ 3i

    pi a

    a 6b+ 2c= 1+ 2i

    1.13 pi :

    )1+ i1

    1 i+

    1 i1

    1+ i+

    1+ i1 i1

    +1 i1+ i1

    = 4

    )1i

    (1i 1

    )

    (1i i

    ) (1i+ 1

    )= 1+ 3i

    )(1+ i)n

    (1 i)n2= 2in1, n N

    ) (1+ i)4n = (1 i)4n, n N

    1.14 z C, : Re(z) > 0 Re ( 1z

    )> 0

    1.15 , > 0 n N (+i)n = (i)n) pi n 3) n = 3, pi

    =

    33

    1.16 :

    z =1

    1+ + i, pi , 2pi+ pi, Z

    pi (Re(z))2+(Im(z))2 =1

    2(1+ )

    1.17 z =12+

    32

    i w = 12+

    32

    i. pi- pi:

    1

  • A =1z3+

    1w3

    +3z2w

    +3zw2

    1.18 z1 = 1 + i z2 = 1 i pi

    E(z) =z2 + z+ 1z3 + z2 + 1

    pi E(z1) E(z2)

    1.19 (1, 1), pi - :

    ) z =1+ + i

    1

    1+ i1

    1 + i

    1+

    1 i1+

    ) w =1 i

    1+

    1 + i1+

    1+ i

    1

    1+ + i1

    1.20 z = 12+ i

    32

    w =12+ i

    32.

    pi :) 1+ z+ z2 = 0 z3 = 1) 1+ z+ z2 + z3 + z4 + z5 + z6 = 1) w2n = zn, n N) w300 = 1 w333 = 1) z2016 +

    1z2016

    = z2013 +1

    z2013= 2

    1.21 z =1+ i

    3

    2

    pi:

    E = (+ z+ z2)(+ z2 + z), , , R

    ) pi z3 = 1) pi pi E

    1.22 a, b, c pi c = (a+ bi)3 107i

    ()

    1.23 z = (2x + y + 5) + 2i w = (4x 2y + 4) + (3x 4y)i. z = w, pi pi x, y

    1.24 :

    z = 6i (3 4i)x 3yi (3i 2)x+ (4 2yi)

    x, y R, pi z = 0

    1.25 pi x, y pi :) x 2+ (y 3)i = (1+ 3i)(y+ i)) (x+ y) + (x2 y2)i = 5+ 5i) x(1+ xi) + 2(1+ xi) = 2(3y 1)i y(1+ yi)) x(x+ 2y+ i) + 1+ yi = 6x+ 3i) (x2 + y2i)i+ x2 = 9(1+ i) + y

    ) (x+ yi)2 =12+ 5i

    i

    )x

    2+ i+

    y1 2i

    =6 2i4 3i

    ) (x+ i)2 +4i= (i x)2 + y+ 1

    )x yi1+ i

    +1i=

    x2+

    11 i

    ) (x+ yi)2 =55+ 10i1 2i

    )x

    1+ 2i+

    y3+ 2i

    =5+ 6i1+ 8i

    1.26 f (z) = z3 + 2z2 + z+ 4 i. , R, pi f (1+ i) = 0

    () z R Im(z) = 0z R Im(z) = 0z R Im(z) = 0 z I Re(z) = 0z I Re(z) = 0z I Re(z) = 01.27 pi pi

    pi:

    ) (1 i)24 ) (1+ i3)3 )

    (1

    1 i

    11+ i

    )2) (3+ i)6

    1.28 z =1+ i1 i

    , pi R. pi :

    ) w = z+1z

    pi

    ) u = z1z

    1.29 :

    z =(1+ 3i)(2 3i) (4+ i)

    2x+ i, x R

    ) pi - z) x, pi :

    ) z R ) z I1.30 pi pi pipi,

    x R pi :) z R ) z I

    2

  • ) z = 6+2x2i3(x+6i) ) z = (2 x+3xi)(i x)) z = (2x+ i)(1+ i)2 + (x2 + 4)i+ x

    ) z =x+ 2x+ i

    ) z =1 i

    3

    x+ (x+ 1)i) z =

    x+ ix xi

    ) z =x+ ix

    3+ i

    1.31 z = x+ yi, x, y R, pi :

    (z 1)(iz 1) R x = y x y = 11.32 , R :

    z =+ i i

    w =+ i i

    pi :) z+ w I ) zw R ) z = w || = ||

    () z 0 Re(z) 0z 0 Re(z) 0z 0 Re(z) 0 Im(z) = 0Im(z) = 0Im(z) = 01.33 z C, z2 0, pi z R1.34 z , z , i, -

    pi z3 iz+ i

    < 0

    1.35 z, pi - pi :

    ) z2 + z 0 ) z2 4z+ 3 < 0 ) z+ 1z< 0

    () in = iin = iin = i

    1.36 + i, , R pi :

    )1+ ii5

    +i2 ii7

    +i4 ii9

    +i 1i3

    )(1i3+ i

    ) (1i7 1

    )+

    (1i5 i

    ) (1i9 1

    )1.37 pi pi pi:

    ) i31 + i46 + i265 ) i2012 + i2013 + i2014 + i2015

    ) 5i2 + 7i78 4i31 + 7i 3i44 + 15

    )1i+

    2i2+

    3i3+

    4i4+

    5i5+

    6i6+

    7i7

    )14i3

    1

    5i13+

    12i23

    +1

    3i37+

    16i49

    ) i10 +1i15

    + i20 +1i25

    + i30 +1i35

    1.38 + i, , R pi :) (1+ i)20 ) (1+ i)21 ) (2+ 2i)20 + (2 2i)20

    1.39 , , , 4 pipi, pi i+++ = 1

    1.40 pi pi pi:) i57 + i58 + i59 + + i213 ) i i2 i3 i2015) i+ i2 + i3 + + i4, N

    1.41 + i, , R :

    z =(1 9i5 4i

    )28+

    (2+ 3i3+ 2i

    )171.42 :

    z =(13+ 7i7 13i

    )36+

    (21 4i4+ 21i

    )53 pi [z Re(z)]2015 = i

    1.43 z =12

    12i, pi z127

    1.44 n Z, pi (3+ i)6n = (3 i)6n

    1.45 n Z, pi :) i3n+31 = in5 ) i3n13 = i3n ) i3n2 = i32n

    1.46 n N, pi :) 2in + (1)n = n 1 ) 2in 3(1)n = n 3) 8in + 5(1)n = n+ 5 ) 2i2n+4 + 5(1)n = 4n 1

    1.47 pi , . pi- : (+ i)2014 + ( i)2014 = 0

    1.48 f (n) = inz, z C n N. pi f (3) + f (8) + f (13) + f (18) = 0

    1.49 2in = 3n 20, n N, pi :) n = 6 ) (3+ 5i)n + (5 3i)n = 0

    1.50 in+2 i = 0, n N, pi- (1+ i)n+1 pi

    1.51 f (n) = in(1 i), n N. pi :) f (4n) + f (4n+ 1) + f (4n+ 2) + f (4n+ 3) = 0) f (1) + f (2) + f (3) + + f (101) = 1+ i

    1.52 f (z) = i(z2 + 1) + z, z C. pi:) f (i) f (i)) pi ( f (i))n + ( f (i))n, n N

    3

  • 1.53 z =3+ i w = 1

    3i.

    ) pi :

    )zw= i )

    z wz+ w

    =zw, R

    ) z22 + w22 = 0) n N, pi zn + wn = 0

    1.54 n N, pi :) in + in+1 + in+2 + in+3 =

    1in+

    1in+1

    +1

    in+2+

    1in+3

    ) i2n+i2n+1+i2n+2+i2n+3 =1i2n

    1

    i2n+1+

    1i2n+2

    1

    i2n+3

    1.55 n N, pi pi-:) i5n+7 ) i3n+1 + i2n + i5n+3 ) (1+ in)(1+ i2n)

    ) in+ i3n+2+ i2n+1+ in+3 ) in+1+ i2n+3+ i3n+ in+2

    1.56 z =2(1 i)2

    . pi:) z8 = 1 ) 1+ z+ z2 + z3 + + z79 = 0

    1.57 z C z2 iz 1 = 0. pi pi:

    A = z63 z42 + z33 + z120

    1.58 pi 3+ 1

    22

    +

    3 122

    i72 = 1

    1.59 n N, pi :

    )(1+ i1 i

    )2n+

    (1 i1+ i

    )2n= 2(1)n

    )(4 2i1+ 2i

    )n+

    (6+ 2i1 3i

    )n=

    {0, n pi2n+1in, n

    )( x+ i1 xi

    )2n+

    ( i x1+ xi

    )2n= 2(1)n, x, y R

    )(xi yx+ yi

    )4n

    (x+ yixi y

    )4n+1= 1+ i, x, y R

    1.60 n N, :) (2 3i)n + (3+ 2i)n = 0 )

    (2 i1+ 2i

    )3n+1= 1

    )(2 3i3+ 2i

    )2n+

    (4+ 5i5 4i

    )2n= 2n

    27n+13

    1.61 f (z) = z3n, n N) n = 10, pi pi- f (1+ i

    3) f (

    3 i)

    ) n pi f (3+ 4i) + f (4 3i) = 0

    1.62 pi 99=0

    (1)(100 )i+1

    () z = w, zn = wn z = w, zn = wn z = w, zn = wn

    1.63 z,w z2 + w2 = 0. pi- z2014 + w2014 = 0

    1.64 z,w pi 2z = (1+ i)w. pi z4 + w4 = 0.

    1.65 z C, z2 + z+ 10 = 0. pi (2z+ 1)10 < 0

    ()

    1.66 a, b, c a, b , 0, pi- :

    b+ ca R c+ a

    b R a

    b< R

    pi a+ b+ c = 0

    1.67 z,w C z + w = 1 z3 + w3 = 2. pi z100 + w100 = 1

    1.68 z C z4 + z2 + 1 = 0 , , N, pi z6 + z6+2 + z6+4 = 0

    1.69 a, b, c :

    a+ b+ c = a2 + b2 + c2 = 0

    pi :) ab+ bc+ ca = 0 ) a4 + b4 + c4 = 0) a4b4 + b4c4 + c4a4 = 0

    1.70 z C z2 + z + 1 = 0. , , C, pi :) (+ )(z+ z2)(z2 + z) = 3 + 3) (z2 + z+ )3 = (z2 + z+ )3

    () pi

    1.71 f : C C, pi 2 f (z) + f (iz) = z2, z C. pi f (z) = z2, z C

    1.72 f : C C, pi - f (z) + f (iz) + f (iz) = z, z C. pi f (z) = z, z C

    1.73 f : C C, pi f (z) + f (z) = z, z C, 3 = 1. pi f (z) =

    z + 1

    4

  • 2

    ()

    2.74 z =1+ i

    3

    2. pi

    z2 = z

    2.75 z1 = 2 i z2 = 3+ 2i. w = z1 + z1 z2 u =

    z1z2

    2.76 x + yi, x, y R :

    ) w = (1+ i)3 (1 i)3 ) z =(2+ i)2 + (2 i)2

    1+ i

    2.77 z C, pi :(1+ zz

    )

    (z+

    1z

    )= 1 z

    2.78 x, y R, z,w , :) z = x2 + (y+ 3)i w = xy 2(x+ i)) z = 3+ (3x+ y)i z = 2y x 5i) z = (x2 + i)i w = x(5i x) + 4i

    2.79 z, x+ yi, x, y R, :

    ) z =(1+ 2i)2(3 i)

    i(1+ i)5) z =

    1 i1+ i

    (1+ i)20

    (1 i)10

    2.80 z =(3+ i)2

    1+ i.

    pi -

    w =1+ zz

    2.81 z C I, pi Re( zz+ z

    )=

    12

    2.82 z,w C, pi (zw zw)2 0

    2.83 f (z) =2z+ iz 2i

    , z , 2i. f (9 5i) = w, :

    ) pi w =32

    32i

    ) pi 23 w

    20142.84 z,w Re(z) Im(w) , 0.

    (z + w) I (z2 + w2) R, pi z+ w = 0

    2.85 z,w C z+ w R zw R, pi z = w

    2.86 z , 0 w =zzzz.

    pi : ) w I ) 2 Im(w) 22.87 a, b, c C, pi :

    ) a Im(bc) + b Im