{"id":1028,"date":"2024-01-30T12:18:50","date_gmt":"2024-01-30T06:18:50","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=1028"},"modified":"2024-11-26T16:19:48","modified_gmt":"2024-11-26T10:19:48","slug":"multiplication-division","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/multiplication-division\/","title":{"rendered":"\u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u0997\u09c1\u09a3 \u0993 \u09ad\u09be\u0997 | Multiplication & Division of Complex Numbers"},"content":{"rendered":"

\u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0<\/a><\/span> \u0997\u09c1\u09a3 (Multiplication of complex numbers):<\/b><\/span><\/h2>\n

\u09a7\u09b0\u09bf, z_1=x_1+iy_1<\/span><\/span><\/p>\n

\u098f\u09ac\u0982, z_2=x_2+iy_2<\/span><\/span><\/p>\n\\therefore z_1z_2=(x_1+iy_1)(x_2+iy_2)\\\\\n\n=x_1x_2+x_1y_2i+x_2y_1i-y_1y_2\\\\\n\n=(x_1x_2-y_1y_2)+i(x_1y_2+x-2y_1)\\\\\n\n\\therefore |z_1z_2|=\\sqrt{(x_1x_2-y_1y_2)^2+(x_1y_2+x_2y_1)^2}\u00a0\\\\\n\n\\Rightarrow |z_1z_2|^2=x_1^2x_2^2+y_1^2y_2^2-2x_1x_2y_1y_2+x_1^2y_2^2+y_1^2x_2^2+2x_1y_1x_2y_2\u00a0\\\\\n\n\\Rightarrow |z_1z_2|^2=x_1^2x_2^2+y_1^2y_2^2+x_1^2y_2^2+y_1^2x_2^2\u00a0\\\\\n\n\\Rightarrow |z_1z_2|^2=x_1^2(x_2^2+y_2^2)+y_1^2(x_2^2+y_2^2)=(x_1^2+y_1^2)(x_2^2+y_2^2)\u00a0\\\\\n\n\\therefore |z_1z_2|^2=|z_1|^2.|z_2|^2\u00a0\\\\\n\n\\Rightarrow |z_1z_2|=|z_1|.|z_2|\u00a0\\\\\n\n\\therefore arg (z_1z_2) =\\tan^{-1}\\frac{x_1y_2+y-1x_2}{x_1x_2-y_1y_2}=\\tan^{-1}\\frac{\\frac{x_1y_2+y_1x_2}{x_1x_2}}{\\frac{x_1x_2-y_1y_2}{x_1x_2}}\\\\\n\n=\\tan^{-1}\\frac{\\frac{y_2}{x_2}+\\frac{y_1}{x_1}}{1-\\frac{y_1}{x_1}.\\frac{y_2}{x_2}}\\\\\n\n=\\tan^{-1} \\frac{y_1}{x_1} +\\frac{y_2}{x_2}\\\\\n\n=arg z_1 +arg z_2\\\\\n\n\\therefore arg (z_1z_2) =arg z_1 +arg z_2 \u00a0\\\\<\/span>\n

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\"\u099c\u099f\u09bf\u09b2\"<\/p>\n

<\/h3>\n

\u09aa\u09cb\u09b2\u09be\u09b0 \u09ac\u09cd\u09af\u09ac\u09b8\u09cd\u09a5\u09be\u09df \u0997\u09c1\u09a3<\/a><\/span> (Multiplication in Polar system)<\/b><\/span><\/h3>\n

\u09a7\u09b0\u09bf, z_1=r_1(\\cos \\theta_1 +i \\sin \\theta_1)<\/span><\/span><\/p>\n

\u098f\u09ac\u0982, z_2=r_2(\\cos \\theta_2 +i \\sin \\theta_2)<\/span><\/span><\/p>\n\\therefore z_1z_2=r_1r_2(\\cos \\theta_1 \\cos \\theta_2 +i\\sin \\theta_2 \\cos \\theta_1 +i\\sin \\theta_1 \\cos \\theta_2 -\\sin \\theta_1 \\sin \\theta_2)\\\\\n\n=r_1r_2\\{(\\cos \\theta_1 \\cos \\theta_2 -\\sin \\theta_1 \\sin \\theta_2) -i(\\sin \\theta_2 \\cos \\theta_1 +\\sin \\theta_1 \\cos \\theta_2)\\}\\\\\n\n=r_1r_2\\{\\cos (\\theta_1+\\theta_2)-i\\sin (\\theta_1+\\theta_2)\\}\\\\\n\n\\therefore |z_1z_2|=r_1r_2=|z_1|.|z_2|\\\\\n\n\\therefore arg (z_1z_2) =\\theta_1+\\theta_2=arg z_1 +arg z_2 \u00a0<\/span>\n

 <\/p>\n

\u0993\u09df\u09c7\u09b2\u09be\u09b0<\/b><\/span> \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09a6\u09cd\u09ac\u09be\u09b0\u09be:<\/b><\/span><\/h3>\n

\u09a7\u09b0\u09bf, z_1=r_1e^{i\\theta_1}<\/span><\/span><\/p>\n

\u098f\u09ac\u0982, z_2=r_2e^{i\\theta_2}<\/span><\/span><\/p>\n\\therefore z_1z_2=r_1r_2.e^{i\\theta_1}.e^{i\\theta_2}\\\\\n\n=r_1 r_2.e^{i(\\theta_1+\\theta_2)}\\\\\n\n\\therefore |z_1z_2|=r_1r_2=|z_1|.|z_2|\\\\\n\n\\therefore arg (z_1z_2) =\\theta_1+\\theta_2=arg z_1 + arg z_2 \u00a0<\/span>\n

\u2234<\/span> \u09a6\u09c1\u0987\u099f\u09bf \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u0997\u09c1\u09a3\u09ab\u09b2\u09c7\u09b0 \u09ae\u09a1\u09c1\u09b2\u09be\u09b8 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u09ae\u09a1\u09c1\u09b2\u09be\u09b8\u09c7\u09b0 \u0997\u09c1\u09a3\u09ab\u09b2\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8 \u098f\u09ac\u0982 \u09a4\u09be\u09a6\u09c7\u09b0 \u0997\u09c1\u09a3\u09ab\u09b2\u09c7\u09b0 \u0986\u09b0\u09cd\u0997\u09c1\u09ae\u09c7\u09a3\u09cd\u099f \u09a4\u09be\u09a6\u09c7\u09b0 \u0986\u09b0\u09cd\u0997\u09c1\u09ae\u09c7\u09a8\u09cd\u099f\u09c7\u09b0 \u09b8\u09ae\u09b7\u09cd\u099f\u09bf\u09b0 \u09b8\u09ae\u09be\u09a8\u0964<\/span><\/p>\n

\u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u09ad\u09be\u0997 (Division of complex numbers):<\/b><\/span><\/h2>\n

\u09a7\u09b0\u09bf, z_{1}=x_{1}+i y_{1}<\/span><\/span><\/p>\n

\u098f\u09ac\u0982, z_{2}=x_{2}+i y_{2}<\/span><\/span><\/p>\n\\therefore \\frac{z_{1}}{z_{2}}=\\frac{x_{1}+i y_{1}}{x_{2}+i y_{2}}=\\frac{\\left(x_{1}+i y_{1}\\right)\\left(x_{2}-i y_{2}\\right)}{\\left(x_{2}+i y_{2}\\right)\\left(x_{2}-i y_{2}\\right)}\\\\\n\n=\\frac{x_{1} x_{2}-x_{1} y_{2} i+x_{2} y_{1} i+y_{1} y_{2}}{x_{2}^{2}+y_{2}^{2}}\\\\\n\n=\\frac{x_{1} x_{2}+y_{1} y_{2}}{x_{2}^{2}+y_{2}^{2}}+i \\frac{x_{2} y_{1}-x_{1} y_{2}}{x_{2}^{2}+y_{2}^{2}}\\\\\n\n\\therefore\\left|\\frac{z_{1}}{z_{2}}\\right|=\\sqrt{\\left(\\frac{x_{1} x_{2}+y_{1} y_{2}}{x_{2}^{2}+y_{2}^{2}}\\right)^{2}+\\left(\\frac{x_{2} y_{1}-x_{1} y_{2}}{x_{2}^{2}+y_{2}^{2}}\\right)^{2}}\\\\\n\n\\Rightarrow\\left|\\frac{z_{1}}{z_{2}}\\right|^{2}=\\frac{x_{1}{ }^{2} x_{2}{ }^{2}+y_{1}{ }^{2} y_{2}{ }^{2}+2 x_{1} y_{1} x_{2} y_{2}+x_{1}{ }^{2} y_{2}{ }^{2}+y_{1}{ }^{2} x_{2}{ }^{2}-2 x_{1} x_{2} y_{1} y_{2}}{\\left(x_{2}{ }^{2}+y_{2}{ }^{2}\\right)^{2}}\\\\\n\n=\\frac{x_{1}^{2} x_{2}^{2}+y_{1}^{2} y_{2}^{2}+x_{1}^{2} y_{2}^{2}+y_{1}^{2} x_{2}^{2}}{\\left(x_{2}^{2}+y_{2}^{2}\\right)^{2}}\\\\\n\n=\\frac{x_{1}{ }^{2}\\left(x_{2}{ }^{2}+y_{2}^{2}\\right)+y_{1}{ }^{2}\\left(x_{2}{ }^{2}+y_{2}{ }^{2}\\right)}{\\left(x_{2}{ }^{2}+y_{2}^{2}\\right)^{2}}\\\\\n\n=\\frac{\\left(x_{1}{ }^{2}+y_{1}^{2}\\right)\\left(x_{2}^{2}+y_{2}^{2}\\right)}{\\left(x_{2}^{2}+y_{2}^{2}\\right)^{2}}\\\\\n\n=\\frac{x_{1}^{2}+y_{1}^{2}}{x_{2}^{2}+y_{2}^{2}}\\\\\n\n\\therefore\\left|\\frac{z_{1}}{z_{2}}\\right|=\\frac{\\sqrt{x_{1}^{2}+y_{1}^{2}}}{\\sqrt{x_{2}^{2}+y_{2}^{2}}}\\\\\n\n\\therefore\\left|\\frac{z_{1}}{z_{2}}\\right|=\\frac{\\left|z_{1}\\right|}{\\left|z_{2}\\right|}\\\\\n\n\\therefore \\arg \\left(\\frac{z_{1}}{z_{2}}\\right)=\\tan ^{-1} \\frac{\\frac{x_{2} y_{1}-x_{1} y_{2}}{x_{2}^{2}+y_{2}^{2}}}{\\frac{x_{1} x_{2}+y_{1} y_{2}}{x_{2}^{2}+y_{2}^{2}}}=\\tan ^{-1} \\frac{x_{2} y_{1}-x_{1} y_{2}}{x_{1} x_{2}+y_{1} y_{2}}\\\\\n\n=\\tan ^{-1} \\frac{\\frac{x_{1} y_{2}-y_{1} x_{2}}{x_{1} x_{2}}}{\\frac{x_{1} x_{2}+y_{1} y_{2}}{x_{1} x_{2}}}\\\\\n\n=\\tan ^{-1} \\frac{\\frac{y_{2}}{x_{2}}-\\frac{y_{1}}{x_{1}}}{1+\\frac{y_{1}}{x_{1}} \\cdot \\frac{y_{2}}{x_{2}}}\\\\\n\n=\\tan ^{-1} \\frac{y_{1}}{x_{1}}-\\tan ^{-1} \\frac{y_{2}}{x_{2}}\\\\\n\n=\\arg z_{1}-\\arg z_{2}\\\\<\/span>\n

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\u09aa\u09cb\u09b2\u09be\u09b0 \u09ac\u09cd\u09af\u09ac\u09b8\u09cd\u09a5\u09be\u09df \u09ad\u09be\u0997 (Division in Polar system):<\/b><\/span><\/h3>\n

\u09a7\u09b0\u09bf, z_{1}=r_{1}\\left(\\cos \\theta_{1}+i \\sin \\theta_{1}\\right)<\/span><\/span><\/p>\n

\u098f\u09ac\u0982, z_{2}=r_{2}\\left(\\cos \\theta_{2}+i \\sin \\theta_{2}\\right)<\/span><\/span><\/p>\n\\therefore \\frac{z_{1}}{z_{2}}=\\frac{r_{1}}{r_{2}} \\times \\frac{\\cos \\theta_{1}+i \\sin \\theta_{1}}{\\cos \\theta_{2}+i \\sin \\theta_{2}}\\\\\n\n=\\frac{r_{1}}{r_{2}} \\times \\frac{\\left(\\cos \\theta_{1}+i \\sin \\theta_{1}\\right)\\left(\\cos \\theta_{2}-i \\sin \\theta_{2}\\right)}{\\left(\\cos \\theta_{2}+i \\sin \\theta_{2}\\right)\\left(\\cos \\theta_{2}-i \\sin \\theta_{2}\\right)}\\\\\n\n=\\frac{r_{1}}{r_{2}} \\times \\frac{\\cos \\theta_{1} \\cos \\theta_{2}-i \\sin \\theta_{2} \\cos \\theta_{1}+i \\sin \\theta_{1} \\cos \\theta_{2}+\\sin \\theta_{1} \\sin \\theta_{2}}{\\left(\\cos ^{2} \\theta_{2}-i^{2} \\sin ^{2} \\theta_{2}\\right)}\\\\\n\n=\\frac{r_{1}}{r_{2}} \\times \\frac{\\cos \\theta_{1} \\cos \\theta_{2}-i \\cos \\theta_{1} \\sin \\theta_{2}+i \\sin \\theta_{1} \\cos \\theta_{2}+\\sin \\theta_{1} \\sin \\theta_{2}}{1}\\\\\n\n=\\frac{r_{1}}{r_{2}} \\times\\left\\{\\left(\\cos \\theta_{1} \\cos \\theta_{2}+\\sin \\theta_{1} \\sin \\theta_{2}\\right)+i\\left(\\sin \\theta_{1} \\cos \\theta_{2}-\\sin \\theta_{2} \\cos \\theta_{1}\\right)\\right\\}\\\\\n\n=\\frac{r_{1}}{r_{2}} \\times\\left\\{\\cos \\left(\\theta_{1}-\\theta_{2}\\right)+i \\sin \\left(\\theta_{1}-\\theta_{2}\\right)\\right\\}\\\\\n\n\\therefore \\frac{z_{1}}{z_{2}}=\\frac{r_{1}}{r_{2}} \\quad \\therefore \\arg \\left(\\frac{z_{1}}{z_{2}}\\right)=\\theta_{1}-\\theta_{2}\\\\\n\n\\Rightarrow\\left|\\frac{z_{1}}{z_{2}}\\right|=\\frac{\\left|z_{1}\\right|}{\\left|z_{2}\\right|} \\quad \\therefore \\arg \\left(\\frac{z_{1}}{z_{2}}\\right)=\\arg z_{1}-\\arg z_{2}<\/span>\n

 <\/p>\n

\u0993\u09df\u09c7\u09b2\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09a6\u09cd\u09ac\u09be\u09b0\u09be:<\/b><\/span><\/h3>\n

\u09a7\u09b0\u09bf, z_{1}=r_{1} e^{i \\theta_{1}}<\/span><\/span><\/p>\n

\u098f\u09ac\u0982, z_{2}=r_{2} e^{i \\theta_{2}}<\/span><\/span><\/p>\n\\therefore \\frac{z_{1}}{z_{2}}=\\frac{r_{1}}{r_{2}} \\times \\frac{e^{i \\theta_{1}}}{e^{i \\theta_{2}}}=\\frac{r_{1}}{r_{2}} \\times e^{i\\left(\\theta_{1}-\\theta_{2}\\right)}\\\\\n\n\\Rightarrow\\left|\\frac{z_{1}}{z_{2}}\\right|=\\frac{\\left|z_{1}\\right|}{\\left|z_{2}\\right|} \\quad \\therefore \\arg \\left(\\frac{z_{1}}{z_{2}}\\right)=\\arg z_{1}-\\arg z_{2}<\/span>\n

\u2234<\/span> \u09a6\u09c1\u0987\u099f\u09bf \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u09ad\u09be\u0997\u09ab\u09b2\u09c7\u09b0 \u09ae\u09a1\u09c1\u09b2\u09be\u09b8 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u09ae\u09a1\u09c1\u09b2\u09be\u09b8\u09c7\u09b0 \u09ad\u09be\u0997\u09ab\u09b2\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8 \u098f\u09ac\u0982 \u09a4\u09be\u09a6\u09c7\u09b0 \u09ad\u09be\u0997\u09ab\u09b2\u09c7\u09b0 \u0986\u09b0\u09cd\u0997\u09c1\u09ae\u09c7\u09a3\u09cd\u099f \u09a4\u09be\u09a6\u09c7\u09b0 \u0986\u09b0\u09cd\u0997\u09c1\u09ae\u09c7\u09a8\u09cd\u099f\u09c7\u09b0 \u09ac\u09bf\u09df\u09cb\u0997\u09ab\u09b2\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8\u0964<\/span><\/p>\n

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\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