{"id":37069,"date":"2022-11-26T20:19:32","date_gmt":"2022-11-26T14:49:32","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=37069"},"modified":"2022-11-29T18:27:32","modified_gmt":"2022-11-29T12:57:32","slug":"unit-ii-complex-numbers-formulae","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=37069","title":{"rendered":"UNIT \u2013 II COMPLEX NUMBERS (Formulae)"},"content":{"rendered":"\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {ALGEBRA\\ OF\\ COMPLEX\\ NUMBERS}}\\]<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/hooks.min.js?ver=dd5603f07f9220ed27f1\" id=\"wp-hooks-js\"><\/script>\n<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/i18n.min.js?ver=c26c3dc7bed366793375\" id=\"wp-i18n-js\"><\/script>\n<script id=\"wp-i18n-js-after\">\nwp.i18n.setLocaleData( { 'text direction\\u0004ltr': [ 'ltr' ] } );\n\/\/# sourceURL=wp-i18n-js-after\n<\/script>\n<script  async src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\" id=\"mathjax-js\"><\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Definition\\ of\\ complex\\ number}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[If\\ Z = a + ib ,\\  then\\ a\\ is\\ called\\ the\\ real\\ part\\ of\\ Z\\  and\\ b\\ is\\ called\\ the\\ imaginary\\ part\\ of\\ Z.\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Re ( Z ) =  a\\   and\\   Im ( Z ) =  b\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Algebra\\ of\\ Complex\\ numbers}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {brown} {(i)\\ Addition\\ of\\ two\\ Complex\\ numbers}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ Z_1=  a + ib,\\  Z_2 =  c + id\\  be\\ any\\ two\\ complex\\ numbers.\\ Then\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Z_1  +  Z_2  =   a + ib\\  +\\   c + id\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ a + c + i(b + d )\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {brown} {(ii)\\ Difference\\ of\\ two\\ Complex\\ numbers}:\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ Z_1=  a + ib,\\  Z_2 =  c + id\\  be\\ any\\ two\\ complex\\ numbers.\\ Then\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Z_1  &#8211;  Z_2  =   a + ib\\  -\\   (c + id)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  =   a + ib  &#8211; c &#8211; id\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ a &#8211; c + i(b &#8211; d )\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {brown} {(ii)\\ Multiplication\\ of\\ two\\ Complex\\ numbers}:\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ Z_1=  a + ib,\\  Z_2 =  c + id\\  be\\ any\\ two\\ complex\\ numbers.\\ Then\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Z_1  Z_2  =   (a + ib) (c + id)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =   ac + iad + ibc + i^2bd\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =   ac + i(ad + bc) &#8211; bd\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =   (ac &#8211; bd) + i(ad + bc)\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:300px;height:250px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"5990643685\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {brown} {(ii)\\ Division\\ of\\ two\\ Complex\\ numbers}:\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ Z_1=  a + ib,\\  Z_2 =  c + id\\  be\\ any\\ two\\ complex\\ numbers.\\ Then\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{z_1}{z_2}\\ =\\frac{a+ib}{c+id}\\ \u00d7\\ \\frac{c-id}{c-id}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{ac- iad + ibc &#8211; i^2bd}{c^2 &#8211; i^2d^2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{ac + i (bc &#8211; ad) + bd}{c^2 &#8211; i^2d^2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{ac +  bd + i(bc &#8211; ad)}{c^2 + d^2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{ac +  bd} {c^2 + d^2}\\ +\\  i\\ \\frac{bc &#8211; ad}{c^2 + d^2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Remember\\ (a +ib)(a &#8211; ib)\\ =\\ (a)^2 + (b)^2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Modulus\\ and\\ Amplitude\\ (or)\\ Argument\\ of\\ a\\ Complex\\ number}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[If\\ Z = a + ib\\ then\\ Modulus is |z| = \\sqrt{a^2 + b^2}\\ and\\ Amplitude\\ is\\ \u03b8  = tan^{-1} (\\frac{b}{a})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Distance\\ between\\ two\\ Complex\\ numbers}:\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[If\\ Z_1\\ =  a + ib,\\  Z_2\\ =  c + id\\  be\\ any\\ two\\ complex\\ numbers,\\ then\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Z_1 Z_2  =\\ \\sqrt{ (a-  c)^2  +  (b- d)^2 )}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Condition\\ for\\ collinear\\ points}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[If\\ the\\ three\\ complex\\ numbers\\ say,\\ A(x_1 +iy_1),\\ B(x_2 +iy_2),\\ and\\ C(x_3 +iy_3),\\ are\\ collinear\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[if\\ \\frac{1}{2}[\\ x_1(y_2 &#8211; y_3)\\ +\\ x_2(y_3 &#8211; y_1)\\ +\\ x_3(y_1 &#8211; y_2)] = 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Condition\\ for\\ Square}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[If\\ A,\\ B,\\ C\\ and\\ D\\ are\\ any\\ four\\ complex\\ numbers\\ representing\\ the\\ vertices\\ of\\ a\\ square\\]\\[ then\\ the\\ required\\ conditions\\ are\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[( i  )\\ AB = BC =  CD = DA\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[( ii  )\\ AC = BD\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Condition\\ for\\ rhombus}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[If\\ A,\\ B,\\ C\\ and\\ D\\ are\\ any\\ four\\ complex\\ numbers\\ representing\\ the\\ vertices\\ of\\ a\\ rhombus\\]\\[ then\\ the\\ required\\ conditions\\ are\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[( i  )\\ AB = BC =  CD = DA\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[( ii  )\\ AC \\neq BD\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Condition\\ for\\ rectangle}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[If\\ A,\\ B,\\ C\\ and\\ D\\ are\\ any\\ four\\ complex\\ numbers\\ representing\\ the\\ vertices\\ of\\ a\\ rectangle\\]\\[ then\\ the\\ required\\ conditions\\ are\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[( i  )\\ AB = CD\\ and\\  BC= DA\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[( ii  )\\ AC = BD\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Condition\\ for\\ parallelogram}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[If\\ A,\\ B,\\ C\\ and\\ D\\ are\\ any\\ four\\ complex\\ numbers\\ representing\\ the\\ vertices\\ of\\ a\\ parallelogram\\]\\[ then\\ the\\ required\\ conditions\\ are\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[( i  )\\ AB = CD\\ and\\  BC= DA\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[( ii  )\\ AC \\neq BD\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block; text-align:center;\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"6565358754\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {DE-MOIVRE\u2019S\\ THEOREM}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {De-Moivre\u2019s\\ Theorem( Statement\\ only)}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[If\\ n\\ is\\ an\\ integer\\ positive\\ or\\ negative\\ then\\ (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^n  =   cos\u2061\\ n \u03b8 +  i sin\u2061\\ n \u03b8\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^{-n}  =   cos\u2061\\ n \u03b8-  i sin\u2061\\ n \u03b8\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {ROOTS\\ OF\\ COMPLEX\\ NUMBERS }}\\]\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[If\\  \u03c9\\  is\\ cube\\ roots\\ of\\ unity,\\   then\\  \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[( i )\\ \u03c9 ^3\\ =\\  1\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(ii)\\  1\\ +\\ \u03c9\\  +\\ \u03c9 ^2\\ =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Working\\ rule\\ to\\ find\\ the\\ n^{th}\\ roots\\ of\\ a\\ complex numbers}:\\ \\hspace{18cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1)\\ Write\\ the\\ given\\ complex\\ number\\ in\\ Polar\\ form\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2)\\ Add\\ \u20182k\u03c0\u2019\\ to\\ the\\ argument\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3)\\ Apply\\ De-Moivre\u2019s\\ theorem\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4)\\ Put\\  k = 0,1,\u2026\u2026. up to\\ (n-1)\\].<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{x^n\\ -\\ 1\\  =\\ 0}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ x^n\\   =\\ 1\\  \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  =\\ (cos\\ 0\\ +\\ i\\ sin\\ 0)^\\frac{1}{n}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  =\\ (cos\\ (0\\ + 2k\u03c0) +\\ i\\ sin\\ (0\\ + 2k\u03c0))^\\frac{1}{n}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  =\\ (cos\\ 2k\u03c0\\ +\\ i\\ sin\\ 2k\u03c0)^\\frac{1}{n}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  =\\ cos\\ (\\frac{2k\u03c0}{n})\\ +\\ i\\ sin\\ (\\frac{2k\u03c0}{n})\\ where\\ k\\ =\\ 0,\\ 1,\\ 2,\\ 3,\\ &#8230;&#8230;..\\ n-1\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{x^n\\ +\\ 1\\  =\\ 0}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ x^n\\   =\\ -\\ 1\\  \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  =\\ (cos\\ \u03c0\\ +\\ i\\ sin\\ \u03c0)^\\frac{1}{n}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  =\\ (cos\\ (\u03c0\\ + 2k\u03c0) +\\ i\\ sin\\ (\u03c0\\ + 2k\u03c0))^\\frac{1}{n}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  =\\ cos\\ (\\frac{\u03c0\\ + 2k\u03c0}{n})\\ +\\ i\\ sin\\ (\\frac{\u03c0\\ + 2k\u03c0}{n})\\ where\\ k\\ =\\ 0,\\ 1,\\ 2,\\ 3,\\ &#8230;&#8230;..\\ n-1\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"autorelaxed\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"4869133702\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || 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