{"id":10937,"date":"2019-07-16T19:35:09","date_gmt":"2019-07-16T16:35:09","guid":{"rendered":"http:\/\/mathland.gr\/?p=10937"},"modified":"2019-07-16T19:35:09","modified_gmt":"2019-07-16T16:35:09","slug":"the-power-of-i","status":"publish","type":"post","link":"https:\/\/dpanagiotis.sites.sch.gr\/?p=10937","title":{"rendered":"The Power of i"},"content":{"rendered":"

\u03a3\u03c4\u03bf \u03ac\u03c1\u03b8\u03c1\u03bf \u03b1\u03c5\u03c4\u03cc \u03b8\u03b1 \u03b1\u03c3\u03c7\u03bf\u03bb\u03b7\u03b8\u03bf\u03cd\u03bc\u03b5 \u03bc\u03b5 \u03c4\u03b9\u03c2 \u03b4\u03c5\u03bd\u03ac\u03bc\u03b5\u03b9\u03c2 \u03c4\u03bf\u03c5 \"i\" (\u03b4\u03b9\u03b1\u03b2\u03ac\u03b6\u03b5\u03c4\u03b1\u03b9 “\u03b3\u03b9\u03ce\u03c4”). \u039f \"i\" \u03b5\u03af\u03bd\u03b1\u03b9 \u03ad\u03bd\u03b1\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2 \u03c0\u03bf\u03c5 \u03b4\u03b5\u03bd \u03c4\u03bf\u03bd \u03b3\u03bd\u03c9\u03c1\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03b1\u03bb\u03bb\u03ac \u03b1\u03c5\u03c4\u03cc \u03c3\u03c4\u03b1 \u03bc\u03b1\u03b8\u03b7\u03bc\u03b1\u03c4\u03b9\u03ba\u03ac \u03b4\u03b5\u03bd \u03bc\u03b1\u03c2 \u03b1\u03c0\u03b1\u03c3\u03c7\u03bf\u03bb\u03b5\u03af \u03ba\u03b1\u03b9 \u03c4\u03cc\u03c3\u03bf. \u0386\u03bb\u03bb\u03c9\u03c3\u03c4\u03b5 \u03bc\u03ad\u03c7\u03c1\u03b9 \u03c4\u03ce\u03c1\u03b1 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b5\u03b9 \u03ba\u03b1\u03b9 \u03ac\u03bb\u03bb\u03bf\u03c5\u03c2 \u03c0\u03bf\u03bb\u03bb\u03bf\u03cd\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03cd\u03c2 \u03c4\u03bf\u03c5\u03c2 \u03bf\u03c0\u03bf\u03af\u03bf\u03c5\u03c2 \u03b4\u03b5\u03bd \u03be\u03ad\u03c1\u03bf\u03c5\u03bc\u03b5 \u03b1\u03ba\u03c1\u03b9\u03b2\u03ce\u03c2 \u03c0\u03bf\u03b9\u03bf\u03af \u03b5\u03af\u03bd\u03b1\u03b9 \u03cc\u03c0\u03c9\u03c2 \u03b3\u03b9\u03b1 \u03c0\u03b1\u03c1\u03ac\u03b4\u03b5\u03b9\u03b3\u03bc\u03b1 \u03c4\u03bf\u03bd \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc \"\sqrt{2}\" \u03b3\u03b9\u03ac \u03c4\u03bf\u03bd \u03bf\u03c0\u03bf\u03af\u03bf \u03be\u03ad\u03c1\u03bf\u03c5\u03bc\u03b5 \u03bc\u03cc\u03bd\u03bf \u03bc\u03b9\u03b1 \u03b9\u03b4\u03b9\u03cc\u03c4\u03b7\u03c4\u03ac \u03c4\u03bf\u03c5 (\u03cc\u03c4\u03b9 \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae \"\sqrt{2}^2=2\") \u03b1\u03bb\u03bb\u03ac \u03b1\u03c5\u03c4\u03ae \u03bc\u03b1\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b1\u03c1\u03ba\u03b5\u03c4\u03ae \u03b3\u03b9\u03b1 \u03bd\u03b1 \u03bc\u03c0\u03bf\u03c1\u03bf\u03cd\u03bc\u03b5 \u03bd\u03b1 \u03b4\u03bf\u03c5\u03bb\u03ad\u03c8\u03bf\u03c5\u03bc\u03b5 \u03bc\u03b5 \u03b1\u03c5\u03c4\u03cc\u03bd \u03c3\u03b1\u03bd \u03bd\u03b1 \u03be\u03ad\u03c1\u03b1\u03bc\u03b5 \u03b1\u03ba\u03c1\u03b9\u03b2\u03ce\u03c2 \u03c0\u03bf\u03b9\u03bf\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9. \u03a4\u03bf \u03af\u03b4\u03b9\u03bf \u03c3\u03c5\u03bc\u03b2\u03b1\u03af\u03bd\u03b5\u03b9 \u03ba\u03b1\u03b9 \u03bc\u03b5 \u03c4\u03bf\u03bd \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc \"e\" \u03b1\u03bb\u03bb\u03ac \u03ba\u03b1\u03b9 \u03bc\u03b5 \u03c4\u03bf\u03bd \"i\" \u03c0\u03bf\u03c5 \u03b8\u03b1 \u03b4\u03bf\u03cd\u03bc\u03b5 \u03c3\u03ae\u03bc\u03b5\u03c1\u03b1. \u03a4\u03bf \u03bc\u03cc\u03bd\u03bf \u03b3\u03bd\u03c9\u03c3\u03c4\u03cc \u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u03af\u03bf \u03c0\u03bf\u03c5 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b3\u03b9\u03b1 \u03b1\u03c5\u03c4\u03cc\u03bd \u03b5\u03af\u03bd\u03b1\u03b9 \u03cc\u03c4\u03b9 \u03ad\u03c7\u03b5\u03b9 \u03c4\u03b7\u03bd \u03b9\u03b4\u03b9\u03cc\u03c4\u03b7\u03c4\u03b1, \u03b1\u03bd \u03c5\u03c8\u03c9\u03b8\u03b5\u03af \u03c3\u03c4\u03bf \u03c4\u03b5\u03c4\u03c1\u03ac\u03b3\u03c9\u03bd\u03bf \u03b5\u03af\u03bd\u03b1\u03b9 \u03af\u03c3\u03bf\u03c2 \u03bc\u03b5 -1, \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9: <\/p>\n

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

\"\"
\u039f\u03b9 \u0394\u03c5\u03bd\u03ac\u03bc\u03b5\u03b9\u03c2 \u03c4\u03bf\u03c5 i<\/figcaption><\/figure>\n

\u0391\u03c5\u03c4\u03ae \u03b7 \u03b9\u03b4\u03b9\u03cc\u03c4\u03b7\u03c4\u03b1 \u03c4\u03bf\u03c5 \"i\", \u03c0\u03bf\u03c5 \u03c3\u03c4\u03b7\u03bd \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc\u03c4\u03b7\u03c4\u03b1 \u03b4\u03b5\u03bd \u03b5\u03af\u03bd\u03b1\u03b9 \u03b9\u03b4\u03b9\u03cc\u03c4\u03b7\u03c4\u03b1 \u03b1\u03bb\u03bb\u03ac \u03bf \u03bf\u03c1\u03b9\u03c3\u03bc\u03cc\u03c2 \u03c4\u03bf\u03c5, \u03bc\u03b1\u03c2 \u03b4\u03af\u03bd\u03b5\u03b9 \u03ba\u03ac\u03b8\u03b5 \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03af\u03b1 \u03b3\u03b9\u03b1 \u03c4\u03bf \u03c0\u03c9\u03c2 \u03c3\u03c5\u03bc\u03c0\u03b5\u03c1\u03b9\u03c6\u03ad\u03c1\u03b5\u03c4\u03b1\u03b9 \u03b1\u03c5\u03c4\u03cc\u03c2 \u03bf \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2 \u03ba\u03b1\u03b9 \u03b1\u03c5\u03c4\u03cc \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03bf\u03c5 \u03bc\u03b1\u03c2 \u03b5\u03bd\u03b4\u03b9\u03b1\u03c6\u03ad\u03c1\u03b5\u03b9 \u03ba\u03b9\u03cc\u03bb\u03b1\u03c2. \u039a\u03b1\u03c4’ \u03b1\u03c1\u03c7\u03ac\u03c2 \u03c4\u03bf \u03c0\u03c1\u03ce\u03c4\u03bf \u03c3\u03c5\u03bc\u03c0\u03ad\u03c1\u03b1\u03c3\u03bc\u03b1 \u03c0\u03bf\u03c5 \u03bc\u03c0\u03bf\u03c1\u03bf\u03cd\u03bc\u03b5 \u03bd\u03b1 \u03b2\u03b3\u03ac\u03bb\u03bf\u03c5\u03bc\u03b5 \u03b5\u03af\u03bd\u03b1\u03b9 \u03cc\u03c4\u03b9 \u03b4\u03b5\u03bd \u03bc\u03c0\u03bf\u03c1\u03b5\u03af \u03bd\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2 \u03b1\u03c6\u03bf\u03cd \u03bc\u03b1\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 \u03ae\u03b4\u03b7 \u03b3\u03bd\u03c9\u03c3\u03c4\u03cc \u03cc\u03c4\u03b9 \u03c3\u03c4\u03bf \u03c3\u03cd\u03bd\u03bf\u03bb\u03bf \u03c4\u03c9\u03bd \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ce\u03bd \u03b1\u03c1\u03b9\u03b8\u03bc\u03ce\u03bd \u03b9\u03c7\u03c3\u03cd\u03b5\u03b9 \"x^2\geq0\", \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc x. \u0386\u03c1\u03b1 \u03bf \"i\" \u03b2\u03c1\u03af\u03c3\u03ba\u03b5\u03c4\u03b1\u03b9 \u03ad\u03be\u03c9 \u03b1\u03c0\u03cc \u03c4\u03bf \u03c3\u03cd\u03bd\u03bf\u03bb\u03bf \u03c4\u03c9\u03bd \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ce\u03bd \u03b1\u03c1\u03b9\u03b8\u03bc\u03ce\u03bd \u03c3\u03b5 \u03ba\u03ac\u03c0\u03bf\u03b9\u03bf \u03ac\u03bb\u03bb\u03bf \u03c3\u03cd\u03bd\u03bf\u03bb\u03bf (\u03bc\u03cc\u03bd\u03bf\u03c2 \u03c4\u03bf\u03c5 \u03ae \u03cc\u03c7\u03b9 \u03b4\u03b5\u03bd \u03ad\u03c7\u03b5\u03b9 \u03c3\u03b7\u03bc\u03b1\u03c3\u03af\u03b1) \u03c0\u03bf\u03c5 \u03bc\u03b1\u03c2 \u03c6\u03ac\u03bd\u03b7\u03ba\u03b5 \u03ba\u03b1\u03bb\u03cc \u03bd\u03b1 \u03c4\u03bf \u03bf\u03bd\u03bf\u03bc\u03ac\u03c3\u03bf\u03c5\u03bc\u03b5 \u03c3\u03cd\u03bd\u03bf\u03bb\u03bf \u03c4\u03c9\u03bd “\u03a6\u03b1\u03bd\u03c4\u03b1\u03c3\u03c4\u03b9\u03ba\u03ce\u03bd \u03b1\u03c1\u03b9\u03b8\u03bc\u03ce\u03bd<\/strong>” (\u03c3\u03b5 \u03b1\u03bd\u03c4\u03b9\u03b4\u03b9\u03b1\u03c3\u03c4\u03bf\u03bb\u03ae \u03bc\u03b5 \u03c4\u03bf \u03c3\u03cd\u03bd\u03bf\u03bb\u03bf \u03c4\u03c9\u03bd \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ce\u03bd) \u03ba\u03b1\u03b9 \u03c4\u03bf\u03bd \"i\" \u03c4\u03bf\u03bd \u03bf\u03bd\u03bf\u03bc\u03ac\u03c3\u03b1\u03bc\u03b5 “\u03c6\u03b1\u03bd\u03c4\u03b1\u03c3\u03c4\u03b9\u03ba\u03ae \u03bc\u03bf\u03bd\u03ac\u03b4\u03b1<\/strong>“.
\n\u039c\u03b5 \u03c4\u03b7 \u03b2\u03bf\u03ae\u03b8\u03b5\u03b9\u03b1 \u03c4\u03ce\u03c1\u03b1 \u03c4\u03c9\u03bd \u03b3\u03bd\u03c9\u03c3\u03c4\u03ce\u03bd \u03bc\u03b1\u03c2 \u03c0\u03c1\u03ac\u03be\u03b5\u03c9\u03bd (\u03c0\u03c1\u03cc\u03c3\u03b8\u03b5\u03c3\u03b7 – \u03c0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03b1\u03c3\u03b9\u03b1\u03c3\u03bc\u03cc) \u03bc\u03c0\u03bf\u03c1\u03bf\u03cd\u03bc\u03b5 \u03bd\u03b1 \u03b4\u03bf\u03cd\u03bc\u03b5 \u03ba\u03b1\u03b9 \u03c4\u03b7 \u03c3\u03c5\u03bd\u03ad\u03c7\u03b5\u03b9\u03b1:
\n\"i+i=2i\" ,\"2i+i=3i\" \u03ba.\u03bf.\u03ba. \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03af \u03ba\u03b9 \u03b1\u03c5\u03c4\u03bf\u03af \u03c6\u03b1\u03bd\u03c4\u03b1\u03c3\u03c4\u03b9\u03ba\u03bf\u03af, \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae \u03b1\u03bd\u03ae\u03ba\u03bf\u03c5\u03bd \u03ba\u03b9 \u03b1\u03c5\u03c4\u03bf\u03af \u03c3\u03c4\u03bf \u03c3\u03cd\u03bd\u03bf\u03bb\u03bf \u03c4\u03c9\u03bd \u03c6\u03b1\u03bd\u03c4\u03b1\u03c3\u03c4\u03b9\u03ba\u03ce\u03bd \u03b1\u03c1\u03b9\u03b8\u03bc\u03ce\u03bd. \u03a4\u03b5\u03bb\u03b9\u03ba\u03ac \u03bf \"i\" \u03b4\u03b5\u03bd \u03b5\u03af\u03bd\u03b1\u03b9 \u03bc\u03cc\u03bd\u03bf\u03c2 \u03c4\u03bf\u03c5 \u03b1\u03bb\u03bb\u03ac \u03ad\u03c7\u03b5\u03b9 \u03c0\u03b1\u03c1\u03ad\u03b1 \u03ba\u03ac\u03b8\u03b5 \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc \u03c4\u03b7\u03c2 \u03bc\u03bf\u03c1\u03c6\u03ae\u03c2 \"ki\" , \u03bc\u03b5 k \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc. \u039a\u03b1\u03b9 \u03bc\u03ac\u03bb\u03b9\u03c3\u03c4\u03b1 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \"(ki)^2=k^2i^2=-k^2\".
\n\u0391\u03c2 \u03b4\u03bf\u03cd\u03bc\u03b5 \u03c4\u03ce\u03c1\u03b1 \u03c4\u03b9 \u03b3\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03bc\u03b5 \u03c4\u03b9\u03c2 \u03b4\u03c5\u03bd\u03ac\u03bc\u03b5\u03b9\u03c2 \u03c4\u03bf\u03c5 \"i\", \u03bc\u03ad\u03c7\u03c1\u03b9 \u03c3\u03c4\u03b9\u03b3\u03bc\u03ae\u03c2 \u03be\u03ad\u03c1\u03bf\u03c5\u03bc\u03b5 \u03cc\u03c4\u03b9 <\/p>\n

  <\/span>   <\/span>\"\[i^0=1\]\"<\/p>\n

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

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

\u03ba\u03b1\u03c4\u03ac \u03c3\u03c5\u03bd\u03ad\u03c0\u03b5\u03b9\u03b1 \u03b8\u03b1 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5: <\/p>\n

  <\/span>   <\/span>\"\[i^3=i^2i=-i\]\"<\/p>\n

  <\/span>   <\/span>\"\[i^4=(i^2)^2=(-1)^2=1\]\"<\/p>\n

  <\/span>   <\/span>\"\[i^5=i^2i^3=-(-i)=i\]\"<\/p>\n

\u03ba.\u03bf.\u03ba.
\n\u03c0\u03b1\u03c1\u03b1\u03c4\u03b7\u03c1\u03bf\u03cd\u03bc\u03b5 \u03bb\u03bf\u03b9\u03c0\u03cc\u03bd \u03cc\u03c4\u03b9 \u03c4\u03b1 \u03b4\u03c5\u03bd\u03b1\u03c4\u03b1 \u03b1\u03c0\u03bf\u03c4\u03b5\u03bb\u03ad\u03c3\u03bc\u03b1\u03c4\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bc\u03cc\u03bd\u03bf \u03c4\u03ad\u03c3\u03c3\u03b5\u03c1\u03b1 <\/p>\n

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

, \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae<\/p>\n\n\n\n\n\n\n\n
\n

  <\/span>   <\/span>\"\[i^0=1\]\"<\/p>\n<\/td>\n

\n

  <\/span>   <\/span>\"\[i^1=i\]\"<\/p>\n<\/td>\n

\n

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

\n

  <\/span>   <\/span>\"\[i^3=-i\]\"<\/p>\n<\/td>\n<\/tr>\n

\n

  <\/span>   <\/span>\"\[i^4=1\]\"<\/p>\n<\/td>\n

\n

  <\/span>   <\/span>\"\[i^5=i\]\"<\/p>\n<\/td>\n

\n

  <\/span>   <\/span>\"\[i^6=-1\]\"<\/p>\n<\/td>\n

\n

  <\/span>   <\/span>\"\[i^7=-i\]\"<\/p>\n<\/td>\n<\/tr>\n

\n

  <\/span>   <\/span>\"\[i^8=1\]\"<\/p>\n<\/td>\n

\n

  <\/span>   <\/span>\"\[i^9=i\]\"<\/p>\n<\/td>\n

\n

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

\n

  <\/span>   <\/span>\"\[i^{11}=-i\]\"<\/p>\n<\/td>\n<\/tr>\n

\n

  <\/span>   <\/span>\"\[i^{12}=1\]\"<\/p>\n<\/td>\n

\n

  <\/span>   <\/span>\"\[i^{13}=i\]\"<\/p>\n<\/td>\n

\n

  <\/span>   <\/span>\"\[i^{14}=-1\]\"<\/p>\n<\/td>\n

\n

  <\/span>   <\/span>\"\[i^{15}=-i\]\"<\/p>\n<\/td>\n<\/tr>\n

\n

  <\/span>   <\/span>\"\[i^{16}=1\]\"<\/p>\n<\/td>\n

\n

  <\/span>   <\/span>\"\[i^{17}=i\]\"<\/p>\n<\/td>\n

\n

  <\/span>   <\/span>\"\[i^{18}=-1\]\"<\/p>\n<\/td>\n

\n

  <\/span>   <\/span>\"\[i^{19}=-i\]\"<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u0392\u03bb\u03ad\u03c0\u03bf\u03bd\u03c4\u03b1\u03c2 \u03c4\u03b1 \u03c0\u03b1\u03c1\u03b1\u03c0\u03ac\u03bd\u03c9 \u03bc\u03c0\u03bf\u03c1\u03bf\u03cd\u03bc\u03b5 \u03bd\u03b1 \u03b3\u03b5\u03bd\u03b9\u03ba\u03b5\u03cd\u03c3\u03bf\u03c5\u03bc\u03b5 \u03ba\u03b1\u03b9 \u03c0\u03bf\u03cd\u03bc\u03b5 \u03cc\u03c4\u03b9 \u03b1\u03c5\u03c4\u03cc \u03c0\u03bf\u03c5 \u03ba\u03b1\u03b8\u03bf\u03c1\u03af\u03b6\u03b5\u03b9 \u03c4\u03b5\u03bb\u03b9\u03ba\u03ac \u03c4\u03bf \u03b1\u03c0\u03bf\u03c4\u03ad\u03bb\u03b5\u03c3\u03bc\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03bf \u03c5\u03c0\u03cc\u03bb\u03bf\u03b9\u03c0\u03bf \u03c4\u03b7\u03c2 \u03b4\u03b9\u03b1\u03af\u03c1\u03b5\u03c3\u03b7\u03c2 \u03c4\u03bf\u03c5 \u03b5\u03ba\u03b8\u03ad\u03c4\u03b7 \u03bc\u03b5 \u03c4\u03bf 4. \u03a0\u03c1\u03ac\u03b3\u03bc\u03b1\u03c4\u03b9 \u03b1\u03bd \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b5\u03ba\u03b8\u03ad\u03c4\u03b7 \u03c3\u03c4\u03bf \"i\" \u03c4\u03bf\u03bd \u03c6\u03c5\u03c3\u03b9\u03ba\u03cc \"n\" (\"i^n\"), \u03b1\u03c5\u03c4\u03cc \u03c0\u03bf\u03c5 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03bd\u03b1 \u03ba\u03ac\u03bd\u03bf\u03c5\u03bc\u03b5 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bd\u03b1 \u03b4\u03b9\u03b1\u03b9\u03c1\u03ad\u03c3\u03bf\u03c5\u03bc\u03b5 \u03c4\u03bf\u03bd \"n\" \u03bc\u03b5 \u03c4\u03bf 4, \u03ba\u03b9 \u03ad\u03c3\u03c4\u03c9 \u03cc\u03c4\u03b9 \u03b2\u03c1\u03af\u03c3\u03ba\u03bf\u03c5\u03bc\u03b5 \u03c0\u03b7\u03bb\u03af\u03ba\u03bf \"p\" \u03ba\u03b1\u03b9 \u03c5\u03c0\u03cc\u03bb\u03bf\u03b9\u03c0\u03bf \"u\", \u03c4\u03cc\u03c4\u03b5 \u03b8\u03b1 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 (\u03b1\u03c0\u03cc \u03c4\u03b7\u03bd \u03c4\u03b1\u03c5\u03c4\u03cc\u03c4\u03b7\u03c4\u03b1 \u03c4\u03b7\u03c2 \u03b4\u03b9\u03b1\u03af\u03c1\u03b5\u03c3\u03b7\u03c2 \u0394=\u03b4\u03c0+\u03c5) \"n=4p+u\", \u03bf\u03c0\u03cc\u03c4\u03b5 \u03c0\u03c1\u03bf\u03ba\u03cd\u03c0\u03c4\u03b5\u03b9,<\/p>\n

  <\/span>   <\/span>\"\[i^n=i^{4p+u}\]\"<\/p>\n

  <\/span>   <\/span>\"\[i^n=i^{4p}i^u\]\"<\/p>\n

  <\/span>   <\/span>\"\[i^n=(i^4)^Pi^u\]\"<\/p>\n

  <\/span>   <\/span>\"\[i^n=1^pi^u\]\"<\/p>\n

  <\/span>   <\/span>\"\[i^n=i^u\]\"<\/p>\n

\u03ba\u03b9 \u03b5\u03c0\u03b5\u03b9\u03b4\u03ae \u03c4\u03bf \u03c5\u03c0\u03cc\u03bb\u03bf\u03b9\u03c0\u03bf \u03c4\u03b7\u03c2 \u03b4\u03b9\u03b1\u03af\u03c1\u03b5\u03c3\u03b7\u03c2 \u03ba\u03ac\u03b8\u03b5 \u03c6\u03c5\u03c3\u03b9\u03ba\u03bf\u03cd \u03bc\u03b5 \u03c4\u03bf 4 \u03bc\u03c0\u03bf\u03c1\u03b5\u03af \u03bd\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bc\u03cc\u03bd\u03bf 0,1,2 \u03ae 3 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b5\u03bb\u03b9\u03ba\u03ac<\/p>\n\n\n\n\n
u: \u03c4\u03bf \u03c5\u03c0\u03cc\u03bb\u03bf\u03b9\u03c0\u03bf\u03c4\u03b7\u03c2 \u03b4\u03b9\u03b1\u03af\u03c1\u03b5\u03c3\u03b7\u03c2 n:4<\/td>\nu=0<\/td>\nu=1<\/td>\nu=2<\/td>\nu=3<\/td>\n<\/tr>\n
\"i^n\"<\/td>\n\"1\"<\/td>\n\"i\"<\/td>\n\"-1\"<\/td>\n\"-i\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u03ba\u03b1\u03b9 \u03c4\u03b5\u03bb\u03b5\u03b9\u03ce\u03bd\u03bf\u03c5\u03bc\u03b5 \u03bc\u03b5 \u03ad\u03bd\u03b1 \u03c0\u03b1\u03c1\u03ac\u03b4\u03b5\u03b9\u03b3\u03bc\u03b1 \u03c3\u03c4\u03bf \u03bf\u03c0\u03bf\u03af\u03bf \u03b8\u03b1 \u03c5\u03c0\u03bf\u03bb\u03bf\u03b3\u03af\u03c3\u03bf\u03c5\u03bc\u03b5 \u03c4\u03bf \"i^{1821}\".
\n\u0394\u03b9\u03b1\u03b9\u03c1\u03bf\u03cd\u03bc\u03b5 \u03c4\u03bf 1821 \u03bc\u03b5 \u03c4\u03bf 4 \u03ba\u03b1\u03b9 \u03b2\u03c1\u03af\u03c3\u03ba\u03bf\u03c5\u03bc\u03b5 \u03c0\u03b7\u03bb\u03af\u03ba\u03bf 455(\u03ac\u03c7\u03c1\u03b7\u03c3\u03c4\u03bf) \u03ba\u03b1\u03b9 \u03c5\u03c0\u03cc\u03bb\u03bf\u03b9\u03c0\u03bf 1(\u03b1\u03c5\u03c4\u03cc \u03bc\u03b1\u03c2 \u03c7\u03c1\u03b5\u03b9\u03ac\u03b6\u03b5\u03c4\u03b1\u03b9) \u03ac\u03c1\u03b1 \u03b8\u03b1 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5: \"i^{1821}=i^1=i\".<\/p>\n","protected":false},"excerpt":{"rendered":"

\u03a3\u03c4\u03bf \u03ac\u03c1\u03b8\u03c1\u03bf \u03b1\u03c5\u03c4\u03cc \u03b8\u03b1 \u03b1\u03c3\u03c7\u03bf\u03bb\u03b7\u03b8\u03bf\u03cd\u03bc\u03b5 \u03bc\u03b5 \u03c4\u03b9\u03c2 \u03b4\u03c5\u03bd\u03ac\u03bc\u03b5\u03b9\u03c2 \u03c4\u03bf\u03c5 i. \u039f i \u03b5\u03af\u03bd\u03b1\u03b9 \u03ad\u03bd\u03b1\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2 \u03c0\u03bf\u03c5 \u03b4\u03b5\u03bd \u03c4\u03bf\u03bd \u03b3\u03bd\u03c9\u03c1\u03af\u03b6\u03bf\u03c5\u03bc\u03b5, \u03c4\u03bf \u03bc\u03cc\u03bd\u03bf \u03b3\u03bd\u03c9\u03c3\u03c4\u03cc \u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u03af\u03bf \u03c0\u03bf\u03c5 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b3\u03b9\u03b1 \u03b1\u03c5\u03c4\u03cc\u03bd \u03b5\u03af\u03bd\u03b1\u03b9 \u03cc\u03c4\u03b9 \u03ad\u03c7\u03b5\u03b9 \u03c4\u03b7\u03bd \u03b9\u03b4\u03b9\u03cc\u03c4\u03b7\u03c4\u03b1, \u03b1\u03bd \u03c5\u03c8\u03c9\u03b8\u03b5\u03af \u03c3\u03c4\u03bf \u03c4\u03b5\u03c4\u03c1\u03ac\u03b3\u03c9\u03bd\u03bf \u03b5\u03af\u03bd\u03b1\u03b9 \u03af\u03c3\u03bf\u03c2 \u03bc\u03b5 -1<\/p>\n","protected":false},"author":2,"featured_media":10942,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[13],"tags":[61],"class_list":["post-10937","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-clyk","tag-imaginary"],"_links":{"self":[{"href":"https:\/\/dpanagiotis.sites.sch.gr\/index.php?rest_route=\/wp\/v2\/posts\/10937","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/dpanagiotis.sites.sch.gr\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/dpanagiotis.sites.sch.gr\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/dpanagiotis.sites.sch.gr\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/dpanagiotis.sites.sch.gr\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=10937"}],"version-history":[{"count":0,"href":"https:\/\/dpanagiotis.sites.sch.gr\/index.php?rest_route=\/wp\/v2\/posts\/10937\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/dpanagiotis.sites.sch.gr\/index.php?rest_route=\/wp\/v2\/media\/10942"}],"wp:attachment":[{"href":"https:\/\/dpanagiotis.sites.sch.gr\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=10937"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/dpanagiotis.sites.sch.gr\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=10937"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/dpanagiotis.sites.sch.gr\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=10937"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}