{"id":1036,"date":"2024-01-30T12:14:21","date_gmt":"2024-01-30T06:14:21","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=1036"},"modified":"2024-11-26T16:52:15","modified_gmt":"2024-11-26T10:52:15","slug":"square-roots-of-complex-numbers","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/square-roots-of-complex-numbers\/","title":{"rendered":"\u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u09ac\u09b0\u09cd\u0997\u09ae\u09c2\u09b2 \u098f\u09ac\u0982 \u098f\u0995\u0995\u09c7\u09b0 \u0998\u09a8\u09ae\u09c2\u09b2 | Square Roots of Complex Numbers and Cubic Roots of Unity"},"content":{"rendered":"

\u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u09ac\u09b0\u09cd\u0997\u09ae\u09c2\u09b2 (Square Roots of Complex Numbers)<\/b><\/span><\/h2>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, a+ib<\/span><\/span> \u098f\u0995\u099f\u09bf \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be<\/a><\/span> (<\/span>Complex Numbers) \u098f\u09ac\u0982 \u09a4\u09be\u09b0 \u09ac\u09b0\u09cd\u0997\u09ae\u09c2\u09b2 x+iy<\/span><\/span>, \u09af\u09c7\u0996\u09be\u09a8\u09c7 x<\/span><\/span> \u0993 y<\/span><\/span> \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09b8\u0982\u0996\u09cd\u09af\u09be<\/a><\/span>\u0964 \u0985\u09b0\u09cd\u09a5\u09be\u09ce,<\/span><\/p>\na+ib=x+iy\\\\\n\na+ib=(x+iy)^2 [\u0989\u09ad\u09af\u09bc\\; \u09aa\u0995\u09cd\u09b7\u0995\u09c7\\; \u09ac\u09b0\u09cd\u0997\\; \u0995\u09b0\u09c7]\\\\\n\na+ib=(x^2-y^2)+2ixy\\\\\n\n\\therefore a=x^2-y^2......(i) \\;\u098f\u09ac\u0982\\; 2xy=b......(ii)\\\\\n\n\u098f\u0996\u09a8,\\; (x^2+y^2)^2=(x^2-y^2)^2+4x^2y^2\\\\\n\n=a^2+b^2 [(i) \u0993 (ii)\\; \u09a8\u0982\\; \u098f\u09b0\\; \u09b8\u09be\u09b9\u09be\u09af\u09cd\u09af\u09c7]\\\\\n\n\\therefore x^2+y62=a^2+b^2......(iii)\u00a0<\/span>\n

\u09af\u09c7\u09b9\u09c7\u09a4\u09c1, x<\/span><\/span> \u0993 y<\/span><\/span> \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09b8\u0982\u0996\u09cd\u09af\u09be, \u09b8\u09c1\u09a4\u09b0\u09be\u0982 x^2+y^2<\/span><\/span> \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u09b8\u0982\u0996\u09cd\u09af\u09be \u0985\u09b0\u09cd\u09a5\u09be\u09ce a^2+b^2<\/span><\/span>\u00a0\u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995\u0964<\/span><\/p>\n

(i)<\/span><\/span> \u0993 (ii)<\/span><\/span>\u00a0\u09a8\u0982 \u09a5\u09c7\u0995\u09c7 \u09aa\u09be\u0987,<\/span><\/p>\nx^2=\\frac{1}{2}(a+\\sqrt{a^2+b^2})\\; \u098f\u09ac\u0982\\; y^2=\\frac{1}{2}(-a+\\sqrt{a^2+b^2})\\\\\n\n\\therefore x=\\pm \\{\\frac{1}{2}(a+\\sqrt{a^2+b^2})\\}^\\frac{1}{2}\\; \u098f\u09ac\u0982\\; y=\\pm \\{\\frac{1}{2}(-a+\\sqrt{a^2+b^2})\\}^\\frac{1}{2}<\/span>\n

(ii)<\/span><\/span> \u09a8\u0982 \u09a5\u09c7\u0995\u09c7 \u0986\u09ae\u09b0\u09be \u09a8\u09bf\u09b6\u09cd\u099a\u09bf\u09a4 \u09af\u09c7, b<\/span><\/span> \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u09b9\u09b2\u09c7, x<\/span><\/span> \u098f\u09ac\u0982 y<\/span><\/span> \u098f\u0995\u0987 \u099a\u09bf\u09b9\u09cd\u09a8\u09af\u09c1\u0995\u09cd\u09a4 \u09b9\u09ac\u09c7\u0964<\/span><\/p>\n

\u098f\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u09a8\u09bf\u09b0\u09cd\u09a3\u09c7\u09af\u09bc \u09ac\u09b0\u09cd\u0997\u09ae\u09c2\u09b2 =\\pm\\left[\\left\\{\\frac{1}{2}\\left(a+\\sqrt{a^{2}+b^{2}}\\right)\\right\\}^{1 \/ 2}+i\\left\\{\\frac{1}{2}\\left(-a+\\sqrt{a^{2}+b^{2}}\\right)\\right\\}^{1 \/ 2}\\right]<\/span><\/span><\/p>\n

\u0986\u09ac\u09be\u09b0, b<\/span><\/span> \u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995 \u09b9\u09b2\u09c7, x<\/span><\/span> \u098f\u09ac\u0982 y<\/span><\/span> \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u099a\u09bf\u09b9\u09cd\u09a8\u09af\u09c1\u0995\u09cd\u09a4 \u09b9\u09ac\u09c7,<\/span><\/p>\n

\u098f\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u09a8\u09bf\u09b0\u09cd\u09a3\u09c7\u09af\u09bc \u09ac\u09b0\u09cd\u0997\u09ae\u09c2\u09b2 =\\pm\\left[\\left\\{\\frac{1}{2}\\left(a+\\sqrt{a^{2}+b^{2}}\\right)\\right\\}^{1 \/ 2}-i\\left\\{\\frac{1}{2}\\left(-a+\\sqrt{a^{2}+b^{2}}\\right)\\right\\}^{1 \/ 2}\\right]<\/span><\/span><\/p>\n

\u098f\u0995\u0995\u09c7\u09b0 \u0998\u09a8\u09ae\u09c2\u09b2 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df <\/b>(Determining the cube root of the unit):<\/b><\/span><\/h2>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \\sqrt[3]{1}=x<\/span><\/span><\/p>\n

\u09a4\u09be\u09b9\u09b2\u09c7, x^3=1\\Rightarrow x^3-1=0\\Rightarrow (x-1)(x^2+x+1)=0\\\\\n\n\\therefore x-1=0 \\therefore x=1\\; \u0985\u09a5\u09ac\u09be,\\; x^2+x+1=0\\\\\n\n\u09ac\u09be,\\; x=\\frac{-1 \\pm \\sqrt{1-4.1}}{2}=\\frac{-1 \\pm \\sqrt{-3}}{2}<\/span><\/span><\/p>\n

\u2234<\/span> \u098f\u0995\u0995\u09c7\u09b0 \u09a4\u09bf\u09a8\u099f\u09bf \u0998\u09a8\u09ae\u09c2\u09b2 1, \\frac{1}{2}(-1+\\sqrt{-3}), \\frac{1}{2}(-1-\\sqrt{-3})<\/span><\/span><\/p>\n

\u098f\u09a6\u09c7\u09b0 \u09ae\u09a7\u09cd\u09af\u09c7 1<\/span><\/span> \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u098f\u09ac\u0982 \u0985\u09aa\u09b0 \u09a6\u09c1\u0987\u099f\u09bf \u099c\u099f\u09bf\u09b2\u0964<\/span><\/p>\n

\"\u09ac\u09b0\u09cd\u0997\u09ae\u09c2\u09b2\"<\/p>\n

\u09a7\u09b0\u09cd\u09ae (Properties):<\/b><\/span><\/h2>\n

\u098f\u0995\u0995\u09c7\u09b0 \u099c\u099f\u09bf\u09b2 \u0998\u09a8\u09ae\u09c2\u09b2 \u09a6\u09c1\u0987\u099f\u09bf\u09b0 \u098f\u0995\u099f\u09bf \u0985\u09aa\u09b0\u099f\u09bf\u09b0 \u09ac\u09b0\u09cd\u0997:<\/b><\/span><\/h3>\n\\begin{aligned}\n\n\\left\\{\\frac{1}{2}(-1+\\sqrt{-3})\\right\\}^{2} &=\\frac{1}{4}\\left\\{1+2(-1) \\sqrt{-3}+(\\sqrt{-3})^{2}\\right\\} \\\\\n\n&=\\frac{1}{4}\\{1-2 \\sqrt{-3}-3\\} \\\\\n\n&=\\frac{1}{4}\\{-2-2 \\sqrt{-3}\\} \\\\\n\n&=\\frac{1}{2}(-1-\\sqrt{-3})\n\n\\end{aligned}\\\\\n\n\u0986\u09ac\u09be\u09b0,\\\\ \\begin{aligned} \\left\\{\\frac{1}{2}(-1-\\sqrt{-3})\\right\\}^{2}\\\\=\\frac{1}{4}\\left\\{1-2(-1) \\sqrt{-3}+(\\sqrt{-3})^{2}\\right\\}\n\n&=\\frac{1}{4}\\{1+2 \\sqrt{-3}-3\\} \\\\\n\n&=\\frac{1}{4}\\{-2+2 \\sqrt{-3}\\} \\\\\n\n&=\\frac{1}{2}(-1+\\sqrt{-3})\n\n\\end{aligned}<\/span>\n

\u0985\u09a4\u098f\u09ac \u098f\u0995\u0995\u09c7\u09b0 \u098f\u0995\u099f\u09bf \u099c\u099f\u09bf\u09b2 \u09ae\u09c2\u09b2 \\omega<\/span> <\/span>\u09b9\u09b2\u09c7 \u0985\u09aa\u09b0\u099f\u09bf \u09b9\u09ac\u09c7 \\omega^2<\/span><\/span>\u0964 \u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u098f\u0995\u0995\u09c7 \u0998\u09a8\u09ae\u09c2\u09b2 \u09a4\u09bf\u09a8\u099f\u09bf 1, \\omega, \\omega^2<\/span><\/span>\u00a0\u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09b8\u09c2\u099a\u09bf\u09a4 \u0995\u09b0\u09be \u09af\u09be\u09df\u0964<\/span><\/p>\n

\u098f\u0995\u0995\u09c7\u09b0 \u099c\u099f\u09bf\u09b2 \u0998\u09a8\u09ae\u09c2\u09b2 \u09a6\u09c1\u0987\u099f\u09bf\u09b0 \u0997\u09c1\u09a3\u09ab\u09b2 <\/b>1<\/span> \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u098f\u0995\u099f\u09bf \u0985\u09aa\u09b0\u099f\u09bf\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4:<\/b><\/span><\/h3>\n\\omega=\\frac{1}{2}(-1+\\sqrt{-3}) \u098f\u09ac\u0982 \\omega^{2}=\\frac{1}{2}(-1-\\sqrt{-3})\\; \u09b9\u09b2\u09c7,\\\\\n\n\\begin{aligned}\n\n\\omega . \\omega^{2} &=\\frac{1}{2}(-1+\\sqrt{-3}) \\cdot \\frac{1}{2}(-1-\\sqrt{-3}) \\\\\n\n&=\\frac{1}{4}\\left\\{(-1)^{2}-(\\sqrt{-3})^{2}\\right\\}=\\frac{1}{4}(1+3)=1\n\n\\end{aligned}\\\\\n\n\\therefore \\omega^{3}=1; \u0986\u09ac\u09be\u09b0, \\omega=\\frac{1}{\\omega^{2}} \u0985\u09a5\u09ac\u09be, \\omega^{2}=\\frac{1}{\\omega}<\/span>\n

 <\/p>\n

\u098f\u0995\u0995\u09c7\u09b0 \u0998\u09a8\u09ae\u09c2\u09b2 \u09a4\u09bf\u09a8\u099f\u09bf\u09b0 \u09b8\u09ae\u09b7\u09cd\u099f\u09bf \u09b6\u09c2\u09a8\u09cd\u09af:<\/b><\/span><\/h3>\n\\omega=\\frac{1}{2}(-1+\\sqrt{-3})\\; \u098f\u09ac\u0982\\; \\omega^{2}=\\frac{1}{2}(-1-\\sqrt{-3})\\; \u09b9\u09b2\u09c7,\\; \\\\\n\n\\begin{aligned}\n\n1+\\omega+\\omega^{2} &=1+\\frac{1}{2}(-1+\\sqrt{-3})+\\frac{1}{2}(-1-\\sqrt{-3}) \\\\\n\n&=1+\\frac{1}{2}(-1+\\sqrt{-3}-1-\\sqrt{-3}) \\\\\n\n&=1+\\frac{1}{2}(-2) \\\\\n\n&=1-1=0\n\n\\end{aligned}<\/span>\n\\therefore 1+\\omega+\\omega^2=0<\/span>\n

Note:<\/b> \u09af\u09c7\u0995\u09cb\u09a8\u09cb \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09b8\u0982\u0996\u09cd\u09af\u09be \u2018a^3<\/span><\/span>\u2019 \u098f\u09b0 \u09a4\u09bf\u09a8\u099f\u09bf \u0998\u09a8\u09ae\u09c2\u09b2 \u09b9\u099a\u09cd\u099b\u09c7 a, a\\omega<\/span><\/span> \u098f\u09ac\u0982 a\\omega^2<\/span><\/span><\/p>\n

\\omega<\/span> \u098f\u09b0 \u09b6\u0995\u09cd\u09a4\u09bf (Power of \\omega<\/span><\/b>)<\/b>:<\/span><\/span><\/h3>\n\\omega^{3}=1, \\omega^{4}=\\omega^{3} \\cdot \\omega=\\omega, \\omega^{5}=\\omega^{3} \\cdot \\omega^{2}=\\omega^{2}, \\omega^{6}=\\left(\\omega^{3}\\right)^{2}=1\\\\\n\n\u0985\u09a4\u098f\u09ac,\\; n \\in \\mathbb{N} \\;\u098f\u09b0\\; \u099c\u09a8\u09cd\u09af,\\; \\omega^{3 n}=\\left(\\omega^{3}\\right)^{n}=1^{n}=1\\\\\n\n\\omega^{3 n+1}=\\omega^{3 n} \\cdot \\omega=1 . \\omega=\\omega, \\omega^{3 n+2}=\\omega^{3 n} \\cdot \\omega^{2}=1 . \\omega^{2}=\\omega^{2}<\/span>\n

\u0985\u09b0\u09cd\u09a5\u09be\u09ce \\omega^n=1,\\omega, \\omega^2<\/span><\/span> \u09b9\u09ac\u09c7 \u09af\u09a6\u09bf n<\/span><\/span> \u0995\u09c7 3<\/span><\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09ad\u09be\u0997 \u0995\u09b0\u09b2\u09c7 \u09ad\u09be\u0997\u09b6\u09c7\u09b7 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 0,1,2<\/span><\/span>\u00a0\u09b9\u09df\u0964<\/span><\/p>\n

\n

 <\/p>\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>\n