{"id":9365,"date":"2019-06-15T14:37:38","date_gmt":"2019-06-15T11:37:38","guid":{"rendered":"http:\/\/mathland.gr\/?p=9365"},"modified":"2019-06-15T14:37:38","modified_gmt":"2019-06-15T11:37:38","slug":"%ce%b1%cf%83%ce%ba%ce%ae%cf%83%ce%b5%ce%b9%cf%82-%ce%bc%ce%b9%ce%b3%ce%b1%ce%b4%ce%b9%ce%ba%cf%8e%ce%bd","status":"publish","type":"post","link":"https:\/\/dpanagiotis.sites.sch.gr\/?p=9365","title":{"rendered":"\u0391\u03c3\u03ba\u03ae\u03c3\u03b5\u03b9\u03c2 \u039c\u03b9\u03b3\u03b1\u03b4\u03b9\u03ba\u03ce\u03bd"},"content":{"rendered":"\n

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[nextpage title=”\u0394\u03c5\u03bd\u03ac\u03bc\u03b5\u03b9\u03c2 \u03c4\u03bf\u03c5 i”]<\/p>\n

\u0386\u03c3\u03ba\u03b7\u03c3\u03b71:<\/strong><\/p>\n

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    \n
  1. \"i^6+i^{16}+i^{26}+i^{36}+i^{46}+i^{56}\"<\/li>\n
  2. \"\frac{1}{i^{11}}+\frac{1}{i^{41}}+\frac{1}{i^{75}}+\frac{1}{i^{1023}}\"<\/li>\n<\/ol>\n

    \u039b\u03cd\u03c3\u03b7:<\/em><\/p>\n

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      \n
    1. \n\n\n\n\n\n\n\n\n\n
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        <\/span>   <\/span>\"\[i^6+i^{16}+i^{26}+i^{36}+i^{46}+i^{56}=\]\"<\/p>

        <\/span>   <\/span>\"\[-1+1-1+1-1+1=0\]\"<\/p><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n

    2. \n\n\n\n\n\n\n\n
      i11<\/sup>=i3<\/sup>= -i<\/td>\n\u03b1\u03c6\u03bf\u03cd \u03b7 \u03b4\u03b9\u03b1\u03af\u03c1\u03b5\u03c3\u03b7 \u03c4\u03bf\u03c5 11 \u03bc\u03b5 \u03c4\u03bf 4 \u03b1\u03c6\u03ae\u03bd\u03b5\u03b9 \u03c5\u03c0\u03cc\u03bb\u03bf\u03b9\u03c0\u03bf 3<\/td>\n<\/tr>\n
      i41<\/sup>=i1<\/sup>=i<\/td>\n\u03b1\u03c6\u03bf\u03cd \u03b7 \u03b4\u03b9\u03b1\u03af\u03c1\u03b5\u03c3\u03b7 \u03c4\u03bf\u03c5 41 \u03bc\u03b5 \u03c4\u03bf 4 \u03b1\u03c6\u03ae\u03bd\u03b5\u03b9 \u03c5\u03c0\u03cc\u03bb\u03bf\u03b9\u03c0\u03bf 1<\/td>\n<\/tr>\n
      i75<\/sup>=i3<\/sup> -i<\/td>\n\u03b1\u03c6\u03bf\u03cd \u03b7 \u03b4\u03b9\u03b1\u03af\u03c1\u03b5\u03c3\u03b7 \u03c4\u03bf\u03c5 75 \u03bc\u03b5 \u03c4\u03bf 4 \u03b1\u03c6\u03ae\u03bd\u03b5\u03b9 \u03c5\u03c0\u03cc\u03bb\u03bf\u03b9\u03c0\u03bf 3<\/td>\n<\/tr>\n
      i1023<\/sup>=i3<\/sup>= -i<\/td>\n\u03b1\u03c6\u03bf\u03cd \u03b7 \u03b4\u03b9\u03b1\u03af\u03c1\u03b5\u03c3\u03b7 \u03c4\u03bf\u03c5 1023 \u03bc\u03b5 \u03c4\u03bf 4 \u03b1\u03c6\u03ae\u03bd\u03b5\u03b9 \u03c5\u03c0\u03cc\u03bb\u03bf\u03b9\u03c0\u03bf 3<\/td>\n<\/tr>\n

        <\/span>   <\/span>\"\[\frac{1}{i^{11}}+\frac{1}{i^{41}}+\frac{1}{i^{75}}+\frac{1}{i^{1023}}=\]\"<\/p>

        <\/span>   <\/span>\"\[\frac{1}{-i}+\frac{1}{i}+\frac{1}{-i}+\frac{1}{-i}=-\frac{2}{i}\]\"<\/p><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n

      \u0386\u03c3\u03ba\u03b7\u03c3\u03b72:<\/strong><\/p>\n

      \u039d\u03b1 \u03b1\u03c0\u03bf\u03b4\u03b5\u03af\u03be\u03b5\u03c4\u03b5 \u03cc\u03c4\u03b9 \"(1+i)^{20}=(1-i)^{20}\"<\/p>\n

      \u039b\u03cd\u03c3\u03b7:<\/em><\/p>\n

      \u03a4\u03bf \u03bd\u03b1 \u03c5\u03c0\u03bf\u03bb\u03bf\u03b3\u03af\u03c3\u03bf\u03c5\u03bc\u03b5 \u03c4\u03bf (1+i)20<\/sup> \u03b1\u03c0\u03b1\u03b9\u03c4\u03b5\u03af \u03c7\u03c1\u03ae\u03c3\u03b7 \u03c4\u03b1\u03c5\u03c4\u03cc\u03c4\u03b7\u03c4\u03b1\u03c2 \u03c0\u03bf\u03c5 \u03b4\u03b5\u03bd \u03b3\u03bd\u03c9\u03c1\u03af\u03b6\u03bf\u03c5\u03bc\u03b5, \u03ba\u03ac\u03bd\u03bf\u03c5\u03bc\u03b5 \u03b5\u03bc\u03b5\u03af\u03c2 \u03bb\u03bf\u03b9\u03c0\u03cc\u03bd \u03c4\u03b7\u03bd \u03c4\u03b1\u03c5\u03c4\u03cc\u03c4\u03b7\u03c4\u03b1 \u03c0\u03bf\u03c5 \u03be\u03ad\u03c1\u03bf\u03c5\u03bc\u03b5 ( \u03ac\u03b8\u03c1\u03bf\u03b9\u03c3\u03bc\u03b1 \u03c3\u03c4\u03bf \u03c4\u03b5\u03c4\u03c1\u03ac\u03b3\u03c9\u03bd\u03bf) \u03b5\u03bb\u03c0\u03af\u03b6\u03bf\u03bd\u03c4\u03b1\u03c2 \u03bd\u03b1 \u03b2\u03b3\u03b5\u03b9 \u03ba\u03ac\u03c4\u03b9 \u03c7\u03c1\u03ae\u03c3\u03b9\u03bc\u03bf (\u03c0\u03bf\u03c5 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03af\u03b3\u03bf\u03c5\u03c1\u03bf \u03cc\u03c4\u03b9 \u03b8\u03b1 \u03b2\u03b3\u03b5\u03b9)

        <\/span>   <\/span>\"\[(1+i)^2=1^2+2i+i^2=\]\"<\/p>

        <\/span>   <\/span>\"\[=1+2i-1=2i\]\"<\/p><\/p>\n

      \u03ac\u03c1\u03b1

        <\/span>   <\/span>\"\[(1+i)^{20}=(1+i)^{2\cdot 10}=\]\"<\/p>

        <\/span>   <\/span>\"\[=((1+i)^2)^{10}=(2i)^{10}=\]\"<\/p><\/p>\n

        <\/span>   <\/span>\"\[=2^{10}\cdot i^{10}=2^{10}\cdot i^2=-2^{10}\]\"<\/p><\/p>\n

      \u038c\u03bc\u03bf\u03b9\u03b1

        <\/span>   <\/span>\"\[(1-i)^2=1^2-2i+i^2=\]\"<\/p>

        <\/span>   <\/span>\"\[=1-2i-1=-2i\]\"<\/p><\/p>\n

      \u03ac\u03c1\u03b1

        <\/span>   <\/span>\"\[(1-i)^{20}=(1-i)^{2\cdot 10}=\]\"<\/p>

        <\/span>   <\/span>\"\[=((1-i)^2)^{10}=(-2i)^{10}=\]\"<\/p><\/p>\n

        <\/span>   <\/span>\"\[=2^{10}\cdot i^{10}=2^{10}\cdot i^2=-2^{10}\]\"<\/p><\/code><\/p>\n

      \u03ba\u03b9 \u03ad\u03c4\u03c3\u03b9 \u03b4\u03b5\u03af\u03be\u03b1\u03bc\u03b5 \u03cc\u03c4\u03b9 \"(1+i)^{20}=(1-i)^{20}\"<\/p>\n

      \u00a0<\/p>\n

      [su_note note_color=”#2ec4b6″]\u0397 \u03bc\u03b5\u03b8\u03bf\u03b4\u03bf\u03bb\u03bf\u03b3\u03af\u03b1 \u03b5\u03b4\u03ce \u03b5\u03af\u03bd\u03b1\u03b9: \u03b3\u03b9\u03b1 \u03bd\u03b1 \u03c5\u03c0\u03bf\u03bb\u03bf\u03b3\u03af\u03c3\u03bf\u03c5\u03bc\u03b5 \u03bc\u03b5\u03b3\u03ac\u03bb\u03b5\u03c2 \u03b4\u03c5\u03bd\u03ac\u03bc\u03b5\u03b9\u03c2 \u03b5\u03bd\u03cc\u03c2 \u03bc\u03b9\u03b3\u03b1\u03b4\u03b9\u03ba\u03bf\u03cd z, \u03c5\u03c0\u03bf\u03bb\u03bf\u03b3\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03c0\u03c1\u03ce\u03c4\u03b1 \u03bc\u03b9\u03ba\u03c1\u03ad\u03c2 \u03b4\u03c5\u03bd\u03ac\u03bc\u03b5\u03b9\u03c2 \"z^2\", \"z^3\", … \u03bc\u03ad\u03c7\u03c1\u03b9 \u03bd\u03b1 \u03c0\u03b5\u03c4\u03cd\u03c7\u03bf\u03c5\u03bc\u03b5 \u03ba\u03ac\u03c0\u03bf\u03b9\u03b1 \u03b4\u03cd\u03bd\u03b1\u03bc\u03b7 \u03c4\u03bf\u03c5 z \u03c0\u03bf\u03c5 \u03bd\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc\u03c2 \u03ae \u03c6\u03b1\u03bd\u03c4\u03b1\u03c3\u03c4\u03b9\u03ba\u03cc\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2. \u03a3\u03c4\u03b7 \u03c3\u03c5\u03bd\u03ad\u03c7\u03b5\u03b9\u03b1 \u03b5\u03ba\u03bc\u03b5\u03c4\u03b1\u03bb\u03bb\u03b5\u03c5\u03cc\u03bc\u03b1\u03c3\u03c4\u03b5 \u03b1\u03c5\u03c4\u03cc \u03c4\u03bf \u03b1\u03c0\u03bf\u03c4\u03ad\u03bb\u03b5\u03c3\u03bc\u03b1 \u03ce\u03c3\u03c4\u03b5 \u03bd\u03b1 \u03b2\u03c1\u03bf\u03cd\u03bc\u03b5 \u03c4\u03bf\u03bd \"z^\nu\".[\/su_note]<\/p>\n

      [nextpage title=”\u0388\u03bd\u03bd\u03bf\u03b9\u03b1 \u03c4\u03bf\u03c5 \u03bc\u03b9\u03b3\u03b1\u03b4\u03b9\u03ba\u03bf\u03cd – \u03a0\u03c1\u03ac\u03be\u03b5\u03b9\u03c2”]<\/p>\n

      \u0386\u03c3\u03ba\u03b7\u03c3\u03b71:<\/strong><\/p>\n

      \u0394\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03bf \u03bc\u03b9\u03b3\u03b1\u03b4\u03b9\u03ba\u03cc\u03c2 \"z=6i-(3-4i)x-3yi-(3i-2)x+(4-2yi)\" \u03bc\u03b5 \"x,y \in \mathbb{R}\". \u039d\u03b1 \u03b3\u03c1\u03ac\u03c8\u03b5\u03c4\u03b5 \u03c4\u03bf\u03bd z \u03c3\u03c4\u03b7 \u03bc\u03bf\u03c1\u03c6\u03ae a+bi \u03ba\u03b1\u03b9 \u03c3\u03c4\u03b7 \u03c3\u03c5\u03bd\u03ad\u03c7\u03b5\u03b9\u03b1 \u03bd\u03b1 \u03b2\u03c1\u03b5\u03af\u03c4\u03b5:<\/p>\n

        \n
      1. \u03a4\u03bf x \u03ce\u03c3\u03c4\u03b5 \u03bf z \u03bd\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c6\u03b1\u03bd\u03c4\u03b1\u03c3\u03c4\u03b9\u03ba\u03cc\u03c2<\/li>\n
      2. \u03a4\u03b7 \u03c3\u03c7\u03ad\u03c3\u03b7 \u03c0\u03bf\u03c5 \u03c3\u03c5\u03bd\u03b4\u03ad\u03b5\u03b9 \u03c4\u03b1 x \u03ba\u03b1\u03b9 y \u03ce\u03c3\u03c4\u03b5 \u03bf z \u03bd\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc\u03c2<\/li>\n
      3. \u03a4\u03b1 x \u03ba\u03b1\u03b9 y \u03ce\u03c3\u03c4\u03b5 \u03bd\u03b1 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 z=0<\/li>\n<\/ol>\n

        \u039b\u03cd\u03c3\u03b7:<\/em><\/p>\n

        \u039a\u03b1\u03c4’ \u03b1\u03c1\u03c7\u03ac\u03c2 \u03b1\u03c0\u03b1\u03bb\u03bb\u03b1\u03c3\u03c3\u03cc\u03bc\u03b1\u03c3\u03c4\u03b5 \u03b1\u03c0\u03cc \u03c4\u03b9\u03c2 \u03c0\u03b1\u03c1\u03b5\u03bd\u03b8\u03ad\u03c3\u03b5\u03b9\u03c2 \u03ba\u03b1\u03b9 \u03c3\u03c5\u03bc\u03bc\u03b1\u03b6\u03b5\u03cd\u03bf\u03c5\u03bc\u03b5 (\u03c4\u03bf\u03c5\u03c2 \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03bf\u03cd\u03c2 \u03bc\u03b5 \u03c4\u03bf\u03c5\u03c2 \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03bf\u03cd\u03c2 \u03ba\u03b1\u03b9 \u03c4\u03bf\u03c5\u03c2 \u03c6\u03b1\u03bd\u03c4\u03b1\u03c3\u03c4\u03b9\u03ba\u03bf\u03cd\u03c2 \u03bc\u03b5 \u03c4\u03bf\u03c5\u03c2 \u03c6\u03b1\u03bd\u03c4\u03b1\u03c3\u03c4\u03b9\u03ba\u03bf\u03cd\u03c2).

          <\/span>   <\/span>\"\[z=6i-(3-4i)x-3yi-(3i-2)x+(4-2yi)\]\"<\/p>

          <\/span>   <\/span>\"\[z=6i-3x+4xi-3yi-3xi+2x+4-2yi\]\"<\/p> \u03bf\u03c0\u03cc\u03c4\u03b5

          <\/span>   <\/span>\"\[(4-x)+(x-5y+6)i\]\"<\/p><\/p>\n

          \n
        1. \u0393\u03b9\u03b1 \u03bd\u03b1 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \"z\in \mathbb{I}\" \u03c0\u03c1\u03ad\u03c0\u03b5\u03b9 \u03c4\u03bf \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc \u03c4\u03bf\u03c5 \u03bc\u03ad\u03c1\u03bf\u03c2 \u03bd\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03af\u03c3\u03bf \u03bc\u03b5 \u03bc\u03b7\u03b4\u03ad\u03bd, \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae 4 – x=0, \u03ac\u03c1\u03b1 x=4<\/li>\n
        2. \u0393\u03b9\u03b1 \u03bd\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bf z \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc\u03c2 \u03c0\u03c1\u03ad\u03c0\u03b5\u03b9 \u03c4\u03bf \u03c6\u03b1\u03bd\u03c4\u03b1\u03c3\u03c4\u03b9\u03ba\u03cc \u03c4\u03bf\u03c5 \u03ba\u03bf\u03bc\u03bc\u03ac\u03c4\u03b9 \u03bd\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 0. \u03a0\u03c1\u03ad\u03c0\u03b5\u03b9 \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae \u03bd\u03b1 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 x – 5y+6=0 \u03ba\u03b9 \u03b1\u03c5\u03c4\u03ae \u03b5\u03af\u03bd\u03b1\u03b9 \u03b7 \u03c3\u03c7\u03ad\u03c3\u03b7 \u03c0\u03bf\u03c5 \u03c3\u03c5\u03bd\u03b4\u03ad\u03b5\u03b9 \u03c4\u03b1 x \u03ba\u03b1\u03b9 y.<\/li>\n
        3. \u0393\u03b9\u03b1 \u03bd\u03b1 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03c4\u03ce\u03c1\u03b1 z=0 \u03b8\u03b1 \u03c0\u03c1\u03ad\u03c0\u03b5\u03b9 \u03ba\u03b1\u03b9 \u03c4\u03bf \u03c6\u03b1\u03bd\u03c4\u03b1\u03c3\u03c4\u03b9\u03ba\u03cc \u03b1\u03bb\u03bb\u03ac \u03ba\u03b1\u03b9 \u03c4\u03bf \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc \u03bc\u03ad\u03c1\u03bf\u03c2 \u03c4\u03bf\u03c5 z \u03bd\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03af\u03c3\u03bf \u03bc\u03b5 0. \u0394\u03b7\u03bb\u03b1\u03b4\u03ae,

            <\/span>   <\/span>\"\[z=0\Leftrightarrow x=4 \: \kappa\alpha\iota \: x-5y+6=0 \Leftrightarrow\]\"<\/p>

            <\/span>   <\/span>\"\[x=4 \: \kappa\alpha\iota \: 4-5y+6=0\]\"<\/p> \u03ac\u03c1\u03b1 \"x=4\" \u03ba\u03b1\u03b9 \"y=2\"<\/li>\n<\/ol>\n

          \u0386\u03c3\u03ba\u03b7\u03c3\u03b72:<\/strong><\/p>\n

          \u0391\u03bd \"z=\frac{5-9i}{7+4i}\" \u03ba\u03b1\u03b9 \"w=\frac{5+9i}{7-4i}\" \u03bd\u03b1 \u03b4\u03b5\u03af\u03be\u03b5\u03c4\u03b5 \u03cc\u03c4\u03b9 \u03bf z+w \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc\u03c2 \u03ba\u03b1\u03b9 \u03cc\u03c4\u03b9 \u03bf z – w \u03b5\u03af\u03bd\u03b1\u03b9 \u03c6\u03b1\u03bd\u03c4\u03b1\u03c3\u03c4\u03b9\u03ba\u03cc\u03c2.<\/p>\n

          \u039b\u03cd\u03c3\u03b7:<\/em><\/p>\n

          \u03a0\u03c1\u03ce\u03c4\u03b1 \u03b8\u03b1 \u03c6\u03ad\u03c1\u03bf\u03c5\u03bc\u03b5 \u03c4\u03bf\u03c5\u03c2 \u03bc\u03b9\u03b3\u03b1\u03b4\u03b9\u03ba\u03bf\u03cd\u03c2 \u03c3\u03c4\u03b7 \u03bc\u03bf\u03c1\u03c6\u03ae a+bi \u03ba\u03b9 \u03b5\u03c0\u03b5\u03b9\u03b4\u03ae \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03b7\u03bb\u03af\u03ba\u03bf \u03c0\u03bf\u03bb\/\u03b6\u03bf\u03c5\u03bc\u03b5 \u03b1\u03c1\u03b9\u03b8\u03bc\u03b7\u03c4\u03ae \u03ba\u03b1\u03b9 \u03c0\u03b1\u03c1\u03bf\u03bd\u03bf\u03bc\u03b1\u03c3\u03c4\u03ae \u03bc\u03b5 \u03c4\u03bf\u03bd \u03c3\u03c5\u03b6\u03c5\u03b3\u03ae \u03c4\u03bf\u03c5 \u03c0\u03b1\u03c1\u03bf\u03bd\u03bf\u03bc\u03b1\u03c3\u03c4\u03ae. \u03a0\u03b1\u03c1\u03ac\u03bb\u03bb\u03b7\u03bb\u03b1 \u03b3\u03b9\u03b1 \u03bd\u03b1 \u03b3\u03bb\u03c5\u03c4\u03ce\u03c3\u03bf\u03c5\u03bc\u03b5 \u03c0\u03c1\u03ac\u03be\u03b5\u03b9\u03c2 \u03ba\u03b1\u03bb\u03cc \u03b5\u03af\u03bd\u03b1\u03b9 \u03bd\u03b1 \u03b8\u03c5\u03bc\u03b7\u03b8\u03bf\u03cd\u03bc\u03b5 \u03cc\u03c4\u03b9 \u03b1\u03bd z=a+bi, \u03c4\u03cc\u03c4\u03b5 \"z \cdot \overline{z}=|z|^2=a^2+b^2\". \u0388\u03c7\u03bf\u03c5\u03bc\u03b5 \u03bb\u03bf\u03b9\u03c0\u03cc\u03bd,

            <\/span>   <\/span>\"\[z=\frac{5-9i}{7+4i}\]\"<\/p>

            <\/span>   <\/span>\"\[z=\frac{(5-9i)(7-4i)}{(7+4i)(7-4i)}\]\"<\/p>

            <\/span>   <\/span>\"\[z=\frac{35-20i-63i+36i^2}{7^2+4^2}\]\"<\/p>

            <\/span>   <\/span>\"\[z=\frac{(35-36)+(-20-63)i}{65}\]\"<\/p>

            <\/span>   <\/span>\"\[z=-\frac{1}{65}-\frac{83}{65}i\]\"<\/p> \u038c\u03bc\u03bf\u03b9\u03b1

            <\/span>   <\/span>\"\[w=\frac{(5+9i)(7+4i)}{(7-4i)(7+4i)}\]\"<\/p>

            <\/span>   <\/span>\"\[w=\frac{35+20i+63i+36i^2}{65}\]\"<\/p>

            <\/span>   <\/span>\"\[w=\frac{(35-36)+(20+63)i}{65}\]\"<\/p>

            <\/span>   <\/span>\"\[w=-\frac{1}{65}+\frac{83}{65}i\]\"<\/p><\/p>\n

          \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03c0\u03c1\u03ac\u03b3\u03bc\u03b1\u03c4\u03b9 \u03b9\u03c3\u03c7\u03cd\u03bf\u03c5\u03bd \"z+w=(-\frac{1}{65}-\frac{83}{65}i)+(-\frac{1}{65}+\frac{83}{65}i)\" \"=-\frac{2}{65}\" \u03c0\u03bf\u03c5 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc\u03c2 \u03ba\u03b1\u03b9 \"z-w=(-\frac{1}{65}-\frac{83}{65}i)-(-\frac{1}{65}+\frac{83}{65}i)\" \"=-2\frac{83}{65}i\" \u03c0\u03bf\u03c5 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c6\u03b1\u03bd\u03c4\u03b1\u03c3\u03c4\u03b9\u03ba\u03cc\u03c2.<\/p>\n

          [nextpage title=”\u03a3\u03c5\u03b6\u03c5\u03b3\u03ae\u03c2 \u03ba\u03b1\u03b9 \u039c\u03ad\u03c4\u03c1\u03bf \u039c\u03b9\u03b3\u03b1\u03b4\u03b9\u03ba\u03bf\u03cd”]<\/p>\n

          \u0386\u03c3\u03ba\u03b7\u03c3\u03b71:<\/strong><\/p>\n

          (\u03b1) \u039d\u03b1 \u03b2\u03c1\u03b5\u03af\u03c4\u03b5 \u03c4\u03bf\u03c5\u03c2 \u03bc\u03b9\u03b3\u03b1\u03b4\u03b9\u03ba\u03bf\u03cd\u03c2 \u03c0\u03bf\u03c5 \u03b5\u03c0\u03b1\u03bb\u03b7\u03b8\u03b5\u03cd\u03bf\u03c5\u03bd \u03c4\u03b7\u03bd \u03b9\u03c3\u03cc\u03c4\u03b7\u03c4\u03b1 \"z \cdot \overline{z} +(z-\overline{z})=3+2i\"<\/p>\n

          (\u03b2) \u039d\u03b1 \u03b2\u03c1\u03b5\u03b8\u03b5\u03af \u03bf \u03bc\u03b9\u03b3\u03b1\u03b4\u03b9\u03ba\u03cc\u03c2 \u03b3\u03b9\u03b1 \u03c4\u03bf\u03bd \u03bf\u03c0\u03bf\u03af\u03bf \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \"\overline{z}=z^2\"<\/p>\n

          \u039b\u03cd\u03c3\u03b7:<\/em><\/p>\n

          \u03a3\u03c4\u03b9\u03c2 \u03c0\u03b5\u03c1\u03b9\u03c3\u03c3\u03cc\u03c4\u03b5\u03c1\u03b5\u03c2 \u03b1\u03c3\u03ba\u03ae\u03c3\u03b5\u03b9\u03c2 \u03b8\u03ad\u03c4\u03bf\u03c5\u03bc\u03b5 z=x+yi \u03ba\u03b1\u03b9 \u03c0\u03c1\u03bf\u03c3\u03c0\u03b1\u03b8\u03bf\u03cd\u03bc\u03b5 \u03b1\u03c0\u03cc \u03c4\u03b7 \u03c3\u03c7\u03ad\u03c3\u03b7 \u03c0\u03bf\u03c5 \u03bc\u03b1\u03c2 \u03ad\u03c7\u03bf\u03c5\u03bd \u03b4\u03ce\u03c3\u03b5\u03b9 \u03bd\u03b1 \u03c5\u03c0\u03bf\u03bb\u03bf\u03b3\u03af\u03c3\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b1 x \u03ba\u03b1\u03b9 y \u03bf\u03c0\u03cc\u03c4\u03b5 \u03b8\u03b1 \u03bc\u03ac\u03b8\u03bf\u03c5\u03bc\u03b5 \u03ba\u03b1\u03b9 \u03c0\u03bf\u03b9\u03bf\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bf z. \u03a4\u03bf \u03af\u03b4\u03b9\u03bf \u03b8\u03b1 \u03ba\u03ac\u03bd\u03bf\u03c5\u03bc\u03b5 \u03ba\u03b9 \u03b5\u03b4\u03ce.<\/p>\n

          (\u03b1) \u0388\u03c3\u03c4\u03c9 \u03bb\u03bf\u03b9\u03c0\u03cc\u03bd \u03cc\u03c4\u03b9 \u03bf z=x+yi, \u03c4\u03cc\u03c4\u03b5 \u03b8\u03b1 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5

            <\/span>   <\/span>\"\[z \cdot \overline{z}=|z|^2=x^2+y^2\]\"<\/p> \u03ba\u03b1\u03b9

            <\/span>   <\/span>\"\[z-\overline{z}=2yi\]\"<\/p> \u03ad\u03c4\u03c3\u03b9 \u03b7 \u03c3\u03c7\u03ad\u03c3\u03b7

            <\/span>   <\/span>\"\[z \cdot \overline{z} +(z-\overline{z})=3+2i\]\"<\/p> \u03b3\u03af\u03bd\u03b5\u03c4\u03b1\u03b9

            <\/span>   <\/span>\"\[x^2+y^2+2yi=3+2i\]\"<\/p> \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd \u03bf\u03c0\u03bf\u03af\u03b1 \u03c0\u03b1\u03af\u03c1\u03bd\u03bf\u03c5\u03bc\u03b5 \u03b4\u03cd\u03bf \u03b9\u03c3\u03cc\u03c4\u03b7\u03c4\u03b5\u03c2 (\u03b3\u03b9\u03b1\u03c4\u03af \u03b3\u03b9\u03b1 \u03bd\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03af\u03c3\u03bf\u03b9 \u03b4\u03cd\u03bf \u03bc\u03b9\u03b3\u03b1\u03b4\u03b9\u03ba\u03bf\u03af \u03c0\u03c1\u03ad\u03c0\u03b5\u03b9 \u03bd\u03b1 \u03ad\u03c7\u03bf\u03c5\u03bd \u03af\u03c3\u03b1 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ac \u03c4\u03bf\u03c5\u03c2 \u03bc\u03ad\u03c1\u03b7 \u03b1\u03bb\u03bb\u03ac \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03c6\u03b1\u03bd\u03c4\u03b1\u03c3\u03c4\u03b9\u03ba\u03ac \u03c4\u03bf\u03c5\u03c2) \u03c4\u03b9\u03c2:<\/p>\n

          (\u03c3\u03c7\u03ad\u03c3\u03b7 1) \"x^2+y^2=3\"<\/p>\n

          (\u03c3\u03c7\u03ad\u03c3\u03b7 2) \"2y=2\"<\/p>\n

          \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd \u03bf\u03c0\u03bf\u03af\u03b1 \u03c0\u03c1\u03bf\u03ba\u03cd\u03c0\u03c4\u03b5\u03b9 y=1. \u03a4\u03b7\u03bd \u03c4\u03b9\u03bc\u03ae \u03b1\u03c5\u03c4\u03ae \u03c4\u03bf\u03c5 y \u03b1\u03bd\u03c4\u03b9\u03ba\u03b1\u03b8\u03b9\u03c3\u03c4\u03bf\u03cd\u03bc\u03b5 \u03c3\u03c4\u03b7\u03bd \u03c3\u03c7\u03ad\u03c3\u03b7 1 \u03c0\u03bf\u03c5 \u03bc\u03b1\u03c2 \u03b4\u03af\u03bd\u03b5\u03b9

            <\/span>   <\/span>\"\[x^2+1^2=3 \Leftrightarrow x^2=2\]\"<\/p> \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae

            <\/span>   <\/span>\"\[x=-\sqrt{2}\]\"<\/p> \u03ae

            <\/span>   <\/span>\"\[x=\sqrt{2}\]\"<\/p>. \u03a4\u03b5\u03bb\u03b9\u03ba\u03ac \u03bf\u03b9 \u03bb\u03cd\u03c3\u03b5\u03b9\u03c2 \u03c0\u03bf\u03c5 \u03b2\u03c1\u03ae\u03ba\u03b1\u03bc\u03b5 \u03b1\u03c0\u03cc \u03c4\u03b7 \u03bb\u03cd\u03c3\u03b7 \u03c4\u03bf\u03c5 \u03c0\u03b1\u03c1\u03b1\u03c0\u03ac\u03bd\u03c9 \u03c3\u03c5\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9

            <\/span>   <\/span>\"\[(x,y)=(-\sqrt{2},1)\]\"<\/p> \u03ae

            <\/span>   <\/span>\"\[(x,y)=(\sqrt{2},1)\]\"<\/p> \u03b5\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03bf\u03b9 \u03bc\u03b9\u03b3\u03b1\u03b4\u03b9\u03ba\u03bf\u03af \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03af \u03c0\u03bf\u03c5 \u03c8\u03ac\u03c7\u03bd\u03b1\u03bc\u03b5 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bf\u03b9

            <\/span>   <\/span>\"\[z_1=-\sqrt{2}+i\]\"<\/p> \u03ae

            <\/span>   <\/span>\"\[z_2=\sqrt{2}+i\]\"<\/p><\/p>\n

          (\u03b2) \u039a\u03b9 \u03b5\u03b4\u03ce \u03b8\u03ad\u03c4\u03bf\u03c5\u03bc\u03b5 z=x+yi \u03ba\u03b9 \u03b1\u03bd\u03b1\u03b6\u03b7\u03c4\u03bf\u03cd\u03bc\u03b5 \u03c4\u03b1 x \u03ba\u03b1\u03b9 y \u03ce\u03c3\u03c4\u03b5 \u03bd\u03b1 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \u03b7

            <\/span>   <\/span>\"\[\overline{z}=z^2\Leftrightarrow\]\"<\/p>

            <\/span>   <\/span>\"\[x-yi=(x+yi)^2 \Leftrightarrow\]\"<\/p>

            <\/span>   <\/span>\"\[x-yi=x^2+2xyi+y^2i^2\Leftrightarrow\]\"<\/p>

            <\/span>   <\/span>\"\[x-yi=(x^2-y^2)+2xyi\]\"<\/p> \u03b1\u03c5\u03c4\u03cc \u03cc\u03bc\u03c9\u03c2 \u03bc\u03c0\u03bf\u03c1\u03b5\u03af \u03bd\u03b1 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \u03bc\u03cc\u03bd\u03bf \u03b1\u03bd \u03b9\u03c3\u03c7\u03cd\u03bf\u03c5\u03bd \u03c4\u03b1\u03c5\u03c4\u03cc\u03c7\u03c1\u03bf\u03bd\u03b1 \u03bf\u03b9 \u03c3\u03c7\u03ad\u03c3\u03b5\u03b9\u03c2:<\/p>\n

          (\u03c3\u03c7\u03ad\u03c3\u03b7 1) \"x^2-y^2=x \Leftrightarrow x^2-x=y^2\Leftrightarrow x(x-1)=y^2\"<\/p>\n

          (\u03c3\u03c7\u03ad\u03c3\u03b7 2) \"2xy=-y \Leftrightarrow 2xy+y=0 \Leftrightarrow y(2x+1)=0\"<\/p>\n

          \u03b7 \u03c3\u03c7\u03ad\u03c3\u03b7 2 \u03bc\u03b1\u03c2 \u03b4\u03af\u03bd\u03b5\u03b9 \u03b4\u03cd\u03bf \u03bb\u03cd\u03c3\u03b5\u03b9\u03c2 \u03c4\u03b9\u03c2<\/p>\n