{"id":1034,"date":"2024-01-30T12:15:55","date_gmt":"2024-01-30T06:15:55","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=1034"},"modified":"2024-11-26T17:11:40","modified_gmt":"2024-11-26T11:11:40","slug":"identification-of-i-and-complex-number","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/identification-of-i-and-complex-number\/","title":{"rendered":"i \u098f\u09b0 \u09aa\u09b0\u09bf\u099a\u09af\u09bc \u098f\u09ac\u0982 \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be (Identification of i and Complex Number)"},"content":{"rendered":"\r\n

\u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be i \u098f\u09b0 \u09ae\u09be\u09a8 \u0995\u09a4 (Identification of i and Complex Number)<\/span><\/h2>\r\n\r\n\r\n\r\n

\u09af\u0996\u09a8, x^2-1=0\\Rightarrow x^2=1 \\Rightarrow x=\\pm \\sqrt{1} \\therefore x=\\pm1<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u098f\u09b0\u09c2\u09aa \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09b8\u09ae\u09be\u09a7\u09be\u09a8 \u0995\u09b0\u09a4\u09c7 \u0995\u09cb\u09a8\u09cb \u09b8\u09ae\u09b8\u09cd\u09af\u09be \u09b9\u09af\u09bc \u09a8\u09be\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

\u0995\u09bf\u09a8\u09cd\u09a4\u09c1, \u09af\u0996\u09a8 x^2+1=0\\Rightarrow x^2=-1\\Rightarrow x=\\sqrt{\\pm-1}......(i)<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u0995\u09cb\u09a8\u09cb \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u09ac\u09b0\u09cd\u0997 \u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995 \u09b9\u09af\u09bc \u09a8\u09be\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

\u2234<\/span> \u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995 \u09b8\u0982\u0996\u09cd\u09af\u09be\u0995\u09c7 \u09ac\u09b0\u09cd\u0997\u09ae\u09c2\u09b2 \u0995\u09b0\u09b2\u09c7 \u0995\u09cb\u09a8 \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09b8\u0982\u0996\u09cd\u09af\u09be \u09aa\u09be\u0993\u09af\u09bc\u09be \u09af\u09be\u09af\u09bc \u09a8\u09be\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

\u09a4\u09be\u0987, \u09a7\u09b0\u09be \u09b9\u09af\u09bc, \\sqrt{-1}=i,\u00a0i^2=-1......(ii)<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u09af\u09c7\u0996\u09be\u09a8\u09c7 i<\/span><\/span> \u098f\u0995\u099f\u09bf \u0995\u09be\u09b2\u09cd\u09aa\u09a8\u09bf\u0995 \u09b8\u0982\u0996\u09cd\u09af\u09be (Imaginary number) \u09a4\u09be\u09b9\u09b2\u09c7 (i)<\/span><\/span> \u09a8\u0982 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7 \\sqrt{-1}<\/span><\/span> \u098f\u09b0 \u09ae\u09be\u09a8 \u09ac\u09b8\u09be\u09b2\u09c7, x=\\pm i<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u098f\u0996\u09a8 (ii)<\/span><\/span> \u09a8\u0982 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7, i^2=-1<\/span><\/span> \u0995\u09bf\u09a8\u09cd\u09a4\u09c1 \u098f\u0996\u09be\u09a8 \u09a5\u09c7\u0995\u09c7, i=\\pm 1<\/span><\/span> \u09b2\u09c7\u0996\u09be \u09af\u09be\u09ac\u09c7 \u09a8\u09be \u0995\u09be\u09b0\u09a3 i=-1<\/span><\/span> (\u09b8\u0982\u099c\u09cd\u099e\u09be\u09af\u09bc\u09bf\u09a4) \u09a4\u09be\u0987 \u098f\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \\pm<\/span>\u00a0<\/span> \u09a8\u09c7\u0993\u09af\u09bc\u09be \u09af\u09be\u09ac\u09c7 \u09a8\u09be, \u09b6\u09c1\u09a7\u09c1\u09ae\u09be\u09a4\u09cd\u09b0 +<\/span><\/span> \u099a\u09bf\u09b9\u09cd\u09a8 \u09a8\u09bf\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

x<\/span> \u098f\u09ac\u0982 y<\/span> \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09b8\u0982\u0996\u09cd\u09af\u09be \u09b9\u09b2\u09c7, x+iy<\/span><\/span> \u0986\u0995\u09be\u09b0\u09c7\u09b0 \u09b8\u0995\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u0995\u09c7 \u09ac\u09b2\u09be \u09b9\u09af\u09bc \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be<\/a><\/span> (<\/span>Complex Number) \u09a6\u09cd\u09ac\u09bf\u09ae\u09be\u09a4\u09cd\u09b0\u09bf\u0995 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0982\u0995 \u09ac\u09cd\u09af\u09ac\u09b8\u09cd\u09a5\u09be\u09af\u09bc (x,y)<\/span>(x,y) \u098f\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u09af\u09c7\u0996\u09be\u09a8\u09c7 x+iy<\/span> \u098f\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8\u0993 \u09b8\u09c7\u0996\u09be\u09a8\u09c7\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

\r\n
\"\u099c\u099f\u09bf\u09b2<\/figure>\r\n<\/div>\r\n\r\n\r\n\r\n

 <\/p>\r\n

x+iy<\/span><\/span> \u0986\u0995\u09be\u09b0\u09c7\u09b0 \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u0995\u09cd\u09b0\u09ae\u09af\u09c1\u0997\u09b2 (x,y)<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

3+4i<\/span><\/span> \u0986\u0995\u09be\u09b0\u09c7\u09b0 \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u0995\u09cd\u09b0\u09ae\u09af\u09c1\u0997\u09b2 (3,4)<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

5=5+0.i<\/span><\/span> \u0986\u0995\u09be\u09b0\u09c7\u09b0 \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u0995\u09cd\u09b0\u09ae\u09af\u09c1\u0997\u09b2 (5,0)<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

5i=0+5.i<\/span><\/span> \u0986\u0995\u09be\u09b0\u09c7\u09b0 \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u0995\u09cd\u09b0\u09ae\u09af\u09c1\u0997\u09b2 (0,5)<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u09a4\u09be\u09b9\u09b2\u09c7 \u09a6\u09c7\u0996\u09be \u09af\u09be\u099a\u09cd\u099b\u09c7, 5<\/span><\/span> \u098f\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 x<\/span><\/span> \u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u0989\u09aa\u09b0 \u098f\u09ac\u0982 5i<\/span><\/span> \u098f\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 y<\/span><\/span> \u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u0989\u09aa\u09b0\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

\u0985\u09b0\u09cd\u09a5\u09be\u09ce \u099a\u09bf\u09a4\u09cd\u09b0 \u09b9\u09a4\u09c7 \u09a6\u09c7\u0996\u09be \u09af\u09be\u09af\u09bc \u09af\u09c7, 5<\/span><\/span> \u0995\u09c7 i<\/span><\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0997\u09c1\u09a3 \u0995\u09b0\u09be\u09b0 \u09ab\u09b2\u09c7 \u09b8\u09c7\u099f\u09bf \u0998\u09a1\u09bc\u09bf\u09b0 \u0995\u09be\u0981\u099f\u09be\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4\u09c7 \\frac{\\pi}{2}<\/span><\/span> \u0995\u09cb\u09a3\u09c7 \u0998\u09c1\u09b0\u09c7 \u09af\u09be\u09af\u09bc\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

\u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09b8\u09bf\u09a6\u09cd\u09a7\u09be\u09a8\u09cd\u09a4 \u09a8\u09c7\u0993\u09af\u09bc\u09be \u09af\u09be\u09af\u09bc \u09af\u09c7, i<\/span><\/span> \u098f\u09ae\u09a8\u0987 \u098f\u0995\u099f\u09bf \u0985\u09aa\u09be\u09b0\u09c7\u099f\u09b0 \u09af\u09be \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09af\u09c7\u0995\u09cb\u09a8\u09cb \u09b8\u0982\u0996\u09cd\u09af\u09be\u0995\u09c7 \u0997\u09c1\u09a3 \u0995\u09b0\u09b2\u09c7 \u09b8\u0982\u0996\u09cd\u09af\u09be\u099f\u09bf \u0998\u09a1\u09bc\u09bf\u09b0 \u0995\u09be\u0981\u099f\u09be\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4\u09c7 \\frac{\\pi}{2}<\/span><\/span> \u0995\u09cb\u09a3\u09c7 \u0998\u09c1\u09b0\u09c7 \u09af\u09be\u09af\u09bc\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

\u2234<\/span> \u0995\u09cb\u09a8\u09cb \u09b8\u0982\u0996\u09cd\u09af\u09be\u0995\u09c7 i^2<\/span><\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0997\u09c1\u09a3 \u0995\u09b0\u09b2\u09c7 \u09b8\u09c7\u099f\u09bf \u0998\u09a1\u09bc\u09bf\u09b0 \u0995\u09be\u0981\u099f\u09be\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4\u09c7 2.\\frac{\\pi}{2}<\/span><\/span>\u00a0\u0995\u09cb\u09a3\u09c7 \u0998\u09c1\u09b0\u09c7 \u09af\u09be\u09af\u09bc\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

\u0995\u09cb\u09a8\u09cb \u09b8\u0982\u0996\u09cd\u09af\u09be\u0995\u09c7 i^3<\/span><\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0997\u09c1\u09a3 \u0995\u09b0\u09b2\u09c7 \u09b8\u09c7\u099f\u09bf \u0998\u09a1\u09bc\u09bf\u09b0 \u0995\u09be\u0981\u099f\u09be\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4\u09c7 3.\\frac{\\pi}{2}<\/span><\/span>\u00a0\u0995\u09cb\u09a3\u09c7 \u0998\u09c1\u09b0\u09c7 \u09af\u09be\u09af\u09bc\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

\u0995\u09cb\u09a8\u09cb \u09b8\u0982\u0996\u09cd\u09af\u09be\u0995\u09c7 i^4<\/span><\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0997\u09c1\u09a3 \u0995\u09b0\u09b2\u09c7 \u09b8\u09c7\u099f\u09bf \u0998\u09a1\u09bc\u09bf\u09b0 \u0995\u09be\u0981\u099f\u09be\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4\u09c7 4.\\frac{\\pi}{2}<\/span><\/span>\u00a0\u0995\u09cb\u09a3\u09c7 \u0998\u09c1\u09b0\u09c7 \u09af\u09be\u09af\u09bc\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

\u0995\u09cb\u09a8\u09cb \u09b8\u0982\u0996\u09cd\u09af\u09be\u0995\u09c7 i^n<\/span><\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0997\u09c1\u09a3 \u0995\u09b0\u09b2\u09c7 \u09b8\u09c7\u099f\u09bf \u0998\u09a1\u09bc\u09bf\u09b0 \u0995\u09be\u0981\u099f\u09be\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4\u09c7 n.\\frac{\\pi}{2}<\/span><\/span>\u00a0\u0995\u09cb\u09a3\u09c7 \u0998\u09c1\u09b0\u09c7 \u09af\u09be\u09af\u09bc\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

\u09a4\u09be\u09b9\u09b2\u09c7 \u09b2\u0995\u09cd\u09b7\u09cd\u09af\u09a3\u09c0\u09af\u09bc \u09af\u09c7, i^2<\/span><\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0995\u09cb\u09a8\u09cb \u09b8\u0982\u0996\u09cd\u09af\u09be\u0995\u09c7 \u0997\u09c1\u09a3 \u0995\u09b0\u09b2\u09c7 \u09b8\u09c7\u099f\u09bf \u0998\u09a1\u09bc\u09bf\u09b0 \u0995\u09be\u0981\u099f\u09be\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4\u09c7 2.\\frac{\\pi}{2}<\/span><\/span> \u09ac\u09be 180^{\\circ}<\/span><\/span> \u0995\u09cb\u09a3\u09c7 \u0998\u09c1\u09b0\u09c7 \u09af\u09be\u09af\u09bc\u0964 \u0986\u09b0 \u0986\u09ae\u09b0\u09be \u099c\u09be\u09a8\u09bf \u09af\u09c7, \u09af\u09c7 \u0995\u09cb\u09a8\u09cb \u099c\u09bf\u09a8\u09bf\u09b8\u0995\u09c7 \u09a4\u09be\u09b0 \u09b8\u09cd\u09ac\u09be\u09ad\u09be\u09ac\u09bf\u0995 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u09a5\u09c7\u0995\u09c7 180^{\\circ}<\/span><\/span> \u0995\u09cb\u09a3\u09c7 \u0998\u09c1\u09b0\u09be\u09b2\u09c7 \u09b8\u09c7\u099f\u09bf \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09b9\u09af\u09bc\u09c7 \u09af\u09be\u09af\u09bc\u0964 \u09b8\u0982\u0996\u09cd\u09af\u09be \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u09b8\u0982\u0996\u09cd\u09af\u09be\u0995\u09c7 180^{\\circ}<\/span><\/span>\u00a0\u0995\u09cb\u09a3\u09c7 \u0998\u09c1\u09b0\u09be\u09b2\u09c7 \u09b8\u09c7\u099f\u09bf \u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995 \u09b9\u09af\u09bc\u09c7 \u09af\u09be\u09af\u09bc\u0964<\/span><\/p>\r\n\r\n\r\n\\therefore 5i^2=-5\\\\ \\therefore i^2=-1\u00a0<\/span>\n

 <\/p>\n\r\n\r\n\r\n

\r\n
\"\u099c\u099f\u09bf\u09b2<\/figure>\r\n<\/div>\r\n\r\n\r\n\r\n

 <\/p>\r\n

\u098f\u09ac\u09be\u09b0 \u0989\u09aa\u09b0\u09c7\u09b0 \u099a\u09bf\u09a4\u09cd\u09b0\u09c7 \u09b2\u0995\u09cd\u09b7\u09cd\u09af \u0995\u09b0\u09cb-<\/span><\/p>\r\n\r\n\r\n\r\n

1<\/span><\/span> \u0995\u09c7 i<\/span><\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0997\u09c1\u09a3 \u0995\u09b0\u09b2\u09c7 i<\/span><\/span>, A<\/span><\/span> \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u09a5\u09c7\u0995\u09c7 B<\/span><\/span> \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8\u09c7 \u09af\u09be\u09df\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

i<\/span><\/span> \u0995\u09c7 i<\/span><\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0997\u09c1\u09a3 \u0995\u09b0\u09b2\u09c7 i^2<\/span><\/span>, B<\/span><\/span> \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u09a5\u09c7\u0995\u09c7 C<\/span><\/span> \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8\u09c7 \u09af\u09be\u09df\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

i^2<\/span><\/span> \u0995\u09c7 i<\/span><\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0997\u09c1\u09a3 \u0995\u09b0\u09b2\u09c7 i^3<\/span><\/span>, C<\/span><\/span> \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u09a5\u09c7\u0995\u09c7 D<\/span><\/span> \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8\u09c7 \u09af\u09be\u09df\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

i^3<\/span><\/span> \u0995\u09c7 i<\/span><\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0997\u09c1\u09a3 \u0995\u09b0\u09b2\u09c7 i^4<\/span><\/span>, D<\/span>\u2026\u2026<\/span> \u09aa\u09c1\u09a8\u09b0\u09be\u09df A<\/span><\/span> \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8\u09c7 \u09af\u09be\u09df\u0964<\/span><\/p>\r\n\r\n\r\n\\therefore i^2=-1,i^3=-i,i^4=1,i^5=i,i^6=-1,i^7=-i,i^8=1\u00a0<\/span>\n\r\n\r\n\r\n

\\therefore n<\/span><\/span> \u09aa\u09c2\u09b0\u09cd\u09a3\u09b8\u0982\u0996\u09cd\u09af\u09be \u09b9\u09b2\u09c7, i^{4n}=1<\/span><\/span><\/p>\r\n\r\n\r\n\\therefore i^{4n+1}=i^{4n}.i=i,i^{4n+2}=i^{4n}.i^2=-1 \\therefore i^{4n+p}=i^p<\/span>\n\r\n\r\n\r\n

\u0986\u09ac\u09be\u09b0, i^{4n}+i^{4n+1}+i^{4n+2}+i^{4n+3}=1+i-1-i=0<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

Note<\/b>: 4n,\u00a04n+1,\u00a04n+2,\u00a04n+3<\/span><\/span>\u00a0\u099a\u09be\u09b0\u099f\u09bf \u0995\u09cd\u09b0\u09ae\u09bf\u0995 \u09aa\u09c2\u09b0\u09cd\u09a3\u09b8\u0982\u0996\u09cd\u09af\u09be<\/span><\/p>\r\n\r\n\r\n\r\n

a,b,c,d<\/span><\/span> \u099a\u09be\u09b0\u099f\u09bf \u0995\u09cd\u09b0\u09ae\u09bf\u0995 \u09aa\u09c2\u09b0\u09cd\u09a3\u09b8\u0982\u0996\u09cd\u09af\u09be \u09b9\u09b2\u09c7 ia+ib+ic+id=0<\/span><\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u098f\u09ac\u09be\u09b0 \u099a\u09b2\u09cb, i^{01717722655}=i^{4\\times 439430663+3}=i^3=-i<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u09af\u09a6\u09bf \u0995\u09b7\u09cd\u099f \u09b9\u09af\u09bc \u09a4\u09ac\u09c7 \u09b6\u09cb\u09a8\u09cb-<\/span><\/p>\r\n\r\n\r\n\r\n

i<\/span><\/span>-\u098f\u09b0 Power \u098f\u09b0 \u09b6\u09c7\u09b7 \u09a6\u09c1\u099f\u09bf \u09a1\u09bf\u099c\u09bf\u099f\u0995\u09c7 4<\/span><\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09ad\u09be\u0997 \u0995\u09b0\u09b2\u09c7 \u0985\u09ac\u09b6\u09bf\u09b7\u09cd\u099f \u09af\u09be \u09b9\u09ac\u09c7 \u09b8\u09c7\u099f\u09be\u0987 i<\/span><\/span> \u098f\u09b0 Power \u09b9\u09ac\u09c7 \u098f\u09ac\u0982 \u09b8\u09c7\u099f\u09bf\u0987 \u09b9\u09ac\u09c7 \u09b8\u09a0\u09bf\u0995 Answer: \u09af\u09c7\u09ae\u09a8, 55<\/span><\/span> \u0995\u09c7 4<\/span><\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09ad\u09be\u0997 \u0995\u09b0\u09b2\u09c7 \u09ad\u09be\u0997\u09b6\u09c7\u09b7 =3<\/span><\/span><\/p>\r\n\r\n\r\n\\therefore i^{01717722655}=i^3=-i (Answer)<\/span>\n\r\n\r\n\r\n

\u0986\u09ac\u09be\u09b0, i<\/span><\/span> \u098f\u09b0 \u09a7\u09be\u09b0\u09be:<\/span><\/p>\r\n\r\n\r\n\r\n

\u09b2\u0995\u09cd\u09b7\u09cd\u09af \u0995\u09b0: <\/span><\/p>\r\n\r\n\r\ni+i^2=i-1\\\\ i+i^2+i^3+i^4+i^5=i^1\\\\ i+i^2+i^3+i^4+i^5+i^6=i^5+i^6=i+i^2\\\\ i+i^2+i^3+i^4+i^5+i^6+i^7=i^5+i^6+i^7=i+i^2+i^3<\/span>\n\r\n\r\n\r\n

\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u09b2\u0995\u09cd\u09b7\u09cd\u09af \u0995\u09b0\u09c7 \u09ac\u09b2\u09cb\u09a4\u09cb ……….. \u0986\u09ae\u09bf\u0987 \u09ac\u09b2\u09bf,<\/span><\/p>\r\n\r\n\r\n\r\n

\u09a7\u09be\u09b0\u09be\u099f\u09bf\u09b0 \u09b8\u09b0\u09cd\u09ac\u09cb\u099a\u09cd\u099a Power \u0995\u09c7 4<\/span><\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09ad\u09be\u0997 \u0995\u09b0\u09b2\u09c7 \u09ad\u09be\u0997\u09b6\u09c7\u09b7 \u09b8\u0982\u0996\u09cd\u09af\u09be\u099f\u09bf \u09af\u09a4 \u09aa\u09be\u0993\u09af\u09bc\u09be \u09af\u09be\u099a\u09cd\u099b\u09c7 \u09aa\u09cd\u09b0\u09a5\u09ae \u09b9\u09a4\u09c7 i<\/span><\/span> \u098f\u09b0 Power \u09af\u09c7\u0996\u09be\u09a8\u09c7 \u09ad\u09be\u0997\u09b6\u09c7\u09b7\u09c7\u09b0 …. \u09a4\u09a4\u09a6\u09c2\u09b0 \u09aa\u09b0\u09cd\u09af\u09a8\u09cd\u09a4 \u09b0\u09be\u0996\u09b2\u09c7\u0987 \u09a7\u09be\u09b0\u09be\u099f\u09bf\u09b0 Answer \u09aa\u09be\u0993\u09af\u09bc\u09be \u09af\u09be\u099a\u09cd\u099b\u09c7\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

\u098f\u09ac\u09be\u09b0 \u098f\u0995\u099f\u09be \u09b8\u09ae\u09b8\u09cd\u09af\u09be \u09b8\u09ae\u09be\u09a7\u09be\u09a8<\/span><\/a> \u0995\u09b0\u09bf-<\/span><\/p>\r\n\r\n\r\n1+i+i^2+i^3+...+i^{202}=?\u00a0<\/span>\n\r\n\r\n\r\n

\u098f\u0996\u09be\u09a8\u09c7 \u09b8\u09b0\u09cd\u09ac\u09cb\u099a\u09cd\u099a Power 202<\/span><\/span> \u0995\u09c7 4<\/span><\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09ad\u09be\u0997 \u0995\u09b0\u09b2\u09c7 \u09ad\u09be\u0997\u09b6\u09c7\u09b7 =2<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u2234<\/span> Answer \u09b9\u09ac\u09c7 i+i^2<\/span><\/span><\/p>\r\n

\r\n

 <\/p>\r\n

\u098f\u0987\u099a\u098f\u09b8\u09b8\u09bf \u0993 \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u09aa\u09b0\u09c0\u0995\u09cd\u09b7\u09be\u09b0\u09cd\u09a5\u09c0\u09a6\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u0986\u09ae\u09be\u09a6\u09c7\u09b0 \u0995\u09cb\u09b0\u09cd\u09b8\u09b8\u09ae\u09c2\u09b9\u0983<\/strong><\/em><\/p>\r\n