{"id":1027,"date":"2024-01-30T12:19:50","date_gmt":"2024-01-30T06:19:50","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=1027"},"modified":"2024-11-26T16:00:00","modified_gmt":"2024-11-26T10:00:00","slug":"%e0%a6%9c%e0%a6%9f%e0%a6%bf%e0%a6%b2-%e0%a6%b8%e0%a6%82%e0%a6%96%e0%a7%8d%e0%a6%af%e0%a6%be%e0%a6%b0-%e0%a6%aa%e0%a7%8b%e0%a6%b2%e0%a6%be%e0%a6%b0-%e0%a6%86%e0%a6%95%e0%a7%83%e0%a6%a4%e0%a6%bf","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/%e0%a6%9c%e0%a6%9f%e0%a6%bf%e0%a6%b2-%e0%a6%b8%e0%a6%82%e0%a6%96%e0%a7%8d%e0%a6%af%e0%a6%be%e0%a6%b0-%e0%a6%aa%e0%a7%8b%e0%a6%b2%e0%a6%be%e0%a6%b0-%e0%a6%86%e0%a6%95%e0%a7%83%e0%a6%a4%e0%a6%bf\/","title":{"rendered":"\u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u09aa\u09cb\u09b2\u09be\u09b0 \u0986\u0995\u09c3\u09a4\u09bf\u00a0  (Polar form of complex number)"},"content":{"rendered":"<h2><span style=\"color: #008000;\">\u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be \u0995\u09be\u0995\u09c7 \u09ac\u09b2\u09c7? \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u09aa\u09cb\u09b2\u09be\u09b0 \u0986\u0995\u09c3\u09a4\u09bf<\/span><\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=ps7rYtaCgSs&amp;t=1s?ref=https%3A%2F%2F10minuteschool.com%2Fcontent%2F%25E0%25A6%259C%25E0%25A6%259F%25E0%25A6%25BF%25E0%25A6%25B2-%25E0%25A6%25B8%25E0%25A6%2582%25E0%25A6%2596%25E0%25A7%258D%25E0%25A6%25AF%25E0%25A6%25BE-complex-numbers%2F&amp;post_id=1025&amp;blog_category_id=4268\" target=\"_blank\" rel=\"noopener\">\u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be<\/a>\u00a0\u09b9\u099a\u09cd\u099b\u09c7 \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u09ac\u09b0\u09cd\u09a7\u09bf\u09a4 \u09b0\u09c2\u09aa, \u09af\u09be\u00a0<span class=\"katex-eq katex-rendered\" data-katex-display=\"false\"><span class=\"katex\"><span class=\"katex-mathml\">(i=\u22121)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u00a0\u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09b8\u09c2\u099a\u09bf\u09a4 \u098f\u0995\u099f\u09bf \u0995\u09be\u09b2\u09cd\u09aa\u09a8\u09bf\u0995 \u098f\u0995\u0995\u09c7\u09b0 \u09b8\u0982\u09af\u09c1\u0995\u09cd\u09a4\u09bf\u09b0 \u09ae\u09be\u09a7\u09cd\u09af\u09ae\u09c7 \u0997\u09a0\u09bf\u09a4\u0964 \u0996\u09cd\u09b0\u09bf\u09b8\u09cd\u099f\u09aa\u09c2\u09b0\u09cd\u09ac\u00a050\u00a0\u0985\u09ac\u09cd\u09a6\u09c7 \u0997\u09cd\u09b0\u09bf\u0995 \u0997\u09a3\u09bf\u09a4\u09ac\u09bf\u09a6 \u0993 \u09aa\u09cd\u09b0\u0995\u09cc\u09b6\u09b2\u09c0 \u0986\u09b2\u09c7\u0995\u099c\u09be\u09a8\u09cd\u09a6\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09b0 \u09b9\u09c7\u09b0\u09a8 \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u09a7\u09be\u09b0\u09a3\u09be \u09a6\u09c7\u09a8\u0964 \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u09af\u09cb\u0997, \u09ac\u09bf\u09af\u09bc\u09cb\u0997, \u0997\u09c1\u09a3 \u098f\u09ac\u0982 \u09ad\u09be\u0997 \u09b8\u09b0\u09cd\u09ac\u09aa\u09cd\u09b0\u09a5\u09ae \u09aa\u09cd\u09b0\u09ac\u09b0\u09cd\u09a4\u09a8 \u0995\u09b0\u09c7\u09a8 \u0987\u09a4\u09be\u09b2\u09bf\u09b0 \u0997\u09a3\u09bf\u09a4\u09ac\u09bf\u09a6 Rafael Bombelli (1526-1572)\u0964 \u09a4\u09bf\u09a8\u09bf \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u0986\u09a6\u09b0\u09cd\u09b6\u09b0\u09c2\u09aa\u00a0<span class=\"katex-eq katex-rendered\" data-katex-display=\"false\"><span class=\"katex\"><span class=\"katex-mathml\">a+ib<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">ib<\/span><\/span><\/span><\/span><\/span>\u00a0\u09ac\u09cd\u09af\u09ac\u09b9\u09be\u09b0 \u0995\u09b0\u09c7\u09a8\u0964<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\cos \\theta =\\frac{x}{r} \\Rightarrow x=r\\cos \\theta\\\\ \\sin \\theta =\\frac{y}{r} \\Rightarrow y=r\\sin \\theta\\\\ \\therefore x+iy=r(\\cos \\theta+i \\sin \\theta)\u00a0<\/span>\n\r\n\r\n\r\n<p><span style=\"font-weight: 400;\">\u098f\u0996\u09be\u09a8\u09c7- <\/span><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span>=<\/span><span style=\"font-weight: 400;\">Modulas (\u09aa\u09b0\u09ae\u09ae\u09be\u09a8)<\/span><\/p>\r\n\r\n\r\n\r\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta<\/span>= Argument (\u09a8\u09a4\u09bf)<\/span><\/p>\r\n\r\n\r\n\r\n<div class=\"wp-block-image\" style=\"text-align: center;\">\r\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" class=\"aligncenter wp-image-1183\" src=\"https:\/\/stage-wp.10minuteschool.com\/wp-content\/uploads\/2021\/10\/image-52.png\" alt=\"\u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be \u0995\u09be\u0995\u09c7 \u09ac\u09b2\u09c7\" width=\"673\" height=\"326\" srcset=\"https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2021\/10\/image-52.png 673w, https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2021\/10\/image-52-300x145.png 300w\" sizes=\"(max-width: 673px) 100vw, 673px\" \/>\r\n<figcaption><a href=\"https:\/\/10minuteschool.com\/academic\/10\/\">\u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0<\/a> \u09aa\u09cb\u09b2\u09be\u09b0 \u09b0\u09c2\u09aa<\/figcaption>\r\n<\/figure>\r\n<\/div>\r\n\r\n\r\n\r\n<p>&nbsp;<\/p>\r\n<ul>\r\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Modulus (<\/b><b>\u09aa\u09b0\u09ae\u09ae\u09be\u09a8<\/b><b>)<\/b><b>: <\/b><\/li>\r\n<\/ul>\r\n\r\n\r\n<span class=\"katex-eq\" data-katex-display=\"false\">z=x+iy<\/span>\n\r\n\r\n\r\n<p><span style=\"font-weight: 400;\">\u0993 \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a5\u09c7\u0995\u09c7 <\/span><span style=\"font-weight: 400;\">z<\/span><span style=\"font-weight: 400;\"> \u098f\u09b0 \u09b8\u09b0\u09be\u09b8\u09b0\u09bf \u09ac\u09be \u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0\u09a4\u09ae \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac\u0995\u09c7 Modulus<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09ac\u09be \u09aa\u09b0\u09ae\u09ae\u09be\u09a8 \u09ac\u09b2\u09c7\u0964<\/span><\/p>\r\n\r\n\r\n\r\n<ul>\r\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Argument<\/b><b> (\u09a8\u09a4\u09bf)<\/b><span style=\"font-weight: 400;\">:<\/span><span style=\"font-weight: 400;\"> \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span><span style=\"font-weight: 400;\"> \u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\">OP<\/span><\/span><span style=\"font-weight: 400;\"> \u09b0\u09c7\u0996\u09be \u09af\u09c7 \u0995\u09cb\u09a3 \u0989\u09ce\u09aa\u09a8\u09cd\u09a8 \u0995\u09b0\u09c7 \u09a4\u09be\u0995\u09c7 Argument\/\u09a8\u09a4\u09bf \u09ac\u09b2\u09c7\u0964 \u098f\u0995\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\">arg(Z)<\/span><\/span><span style=\"font-weight: 400;\"> \u09ac\u09be <span class=\"katex-eq\" data-katex-display=\"false\">amp(Z)<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09b9\u09df\u0964<\/span><\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u0985\u0982\u09b6 \u09b9\u09a4\u09c7 \u0998\u09dc\u09bf\u09b0 \u0995\u09be\u099f\u09be\u09b0 \u09a6\u09bf\u0995\u09c7, \u0998\u09dc\u09bf\u09b0 \u0995\u09be\u099f\u09be\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09a6\u09bf\u0995\u09c7 \u098f\u0995\u09be\u09a7\u09bf\u0995 \u0998\u09c2\u09b0\u09cd\u09a3\u09a8\u09c7 \u0985\u09b8\u0982\u0996\u09cd\u09af \u0986\u09b0\u09cd\u0997\u09c1\u09ae\u09c7\u09a8\u09cd\u099f \u09aa\u09be\u0993\u09df\u09be \u09b8\u09ae\u09cd\u09ad\u09ac\u0964\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n<div class=\"wp-block-image\" style=\"text-align: center;\">\r\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" class=\"aligncenter wp-image-1184\" src=\"https:\/\/stage-wp.10minuteschool.com\/wp-content\/uploads\/2021\/10\/image-53.png\" alt=\"\u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be \u0995\u09be\u0995\u09c7 \u09ac\u09b2\u09c7\" width=\"673\" height=\"326\" srcset=\"https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2021\/10\/image-53.png 673w, https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2021\/10\/image-53-300x145.png 300w\" sizes=\"(max-width: 673px) 100vw, 673px\" \/>\r\n<figcaption><br \/>\u0998\u09dc\u09bf\u09b0 \u0995\u09be\u0981\u099f\u09be\u09b0 \u0998\u09c2\u09b0\u09cd\u09a3\u09a8\u09c7\u09b0 \u09a6\u09bf\u0995 \u0985\u09a8\u09c1\u09af\u09be\u09df\u09c0 \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u0986\u09b0\u09cd\u0997\u09c1\u09ae\u09c7\u09a8\u09cd\u099f<\/figcaption>\r\n<\/figure>\r\n<\/div>\r\n\r\n\r\n\r\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\therefore<\/span><\/span><span style=\"font-weight: 400;\"> Argument \u098f\u09b0 \u09b8\u09be\u09a7\u09be\u09b0\u09a3\u09b0\u09c2\u09aa- <\/span><\/p>\r\n\r\n\r\n<span class=\"katex-eq\" data-katex-display=\"false\">arg (z) =2n\\pi+ \\theta\\\\ \\Rightarrow arg (z) =2n \\pi+\\tan^{-1}\\frac{y}{x}\u00a0<\/span>\n\r\n\r\n\r\n<ul>\r\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>\u09ae\u09c2\u0996\u09cd\u09af \u0986\u09b0\u09cd\u0997\u09c1\u09ae\u09c7\u09a8\u09cd\u099f <\/b><b>(Main argument)<\/b><span style=\"font-weight: 400;\">:<\/span><span style=\"font-weight: 400;\"> \u09af\u09a6\u09bf <span class=\"katex-eq\" data-katex-display=\"false\">-\\pi&lt;\\theta \\leq \\pi<\/span><\/span><span style=\"font-weight: 400;\"> \u09b9\u09df \u09a4\u09ac\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\">\\theta<\/span><\/span><span style=\"font-weight: 400;\"> \u0995\u09c7 \u09ae\u09c2\u0996\u09cd\u09af \u0986\u09b0\u09cd\u0997\u09c1\u09ae\u09c7\u09a8\u09cd\u099f \u09ac\u09b2\u09be \u09b9\u09df\u0964 \u09b9\u09bf\u09b8\u09be\u09ac\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u09b8\u09ac\u09b8\u09ae\u09df \u09ae\u09c2\u0996\u09cd\u09af \u0986\u09b0\u09cd\u0997\u09c1\u09ae\u09c7\u09a8\u09cd\u099f<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\"> \u09ac\u09cd\u09af\u09ac\u09b9\u09c3\u09a4 \u09b9\u09df\u0964<\/span><\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<p><span style=\"font-weight: 400;\">\u099a\u09bf\u09a4\u09cd\u09b0 \u09b8\u09b9 <span class=\"katex-eq\" data-katex-display=\"false\">P(x, y)<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09ac\u09bf\u09ad\u09bf\u09a8\u09cd\u09a8 \u099a\u09a4\u09c1\u09b0\u09cd\u09ad\u09be\u0997\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be <span class=\"katex-eq\" data-katex-display=\"false\">z=x+iy<\/span>\u00a0\u098f\u09b0 \u0986\u09b0\u09cd\u0997\u09c1\u09ae\u09c7\u09a8\u09cd\u099f \u09a8\u09bf\u09b0\u09cd\u09a3\u09df:<\/span><\/p>\r\n\r\n\r\n\r\n<div class=\"wp-block-image\">\r\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" class=\"aligncenter wp-image-1185\" src=\"https:\/\/stage-wp.10minuteschool.com\/wp-content\/uploads\/2021\/10\/image-54.png\" alt=\"\u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be \u0995\u09be\u0995\u09c7 \u09ac\u09b2\u09c7\" width=\"673\" height=\"611\" srcset=\"https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2021\/10\/image-54.png 673w, https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2021\/10\/image-54-300x272.png 300w\" sizes=\"(max-width: 673px) 100vw, 673px\" \/><\/figure>\r\n<\/div>\r\n\r\n\r\n\r\n<p>&nbsp;<\/p>\r\n<ul>\r\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>\u0985\u09df\u09b2\u09be\u09b0\u09c7\u09b0 \u09b8\u09c2\u09a4\u09cd\u09b0 <\/b><b>(Euler&#8217;s formula)<\/b><b>:<\/b><\/li>\r\n<\/ul>\r\n\r\n\r\n<span class=\"katex-eq\" data-katex-display=\"false\">\\cos \\theta +i \\sin \\theta=e^{i\\theta}\\\\ \\cos \\theta-i \\sin \\theta=e^{-i\\theta}\\\\ (\\cos \\theta + i \\sin \\theta)^n=\\cos n\\theta +i \\sin n \\theta<\/span>\n\r\n\r\n\r\n<ul>\r\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>\u0985\u09a8\u09c1\u09ac\u09a8\u09cd\u09a7\u09c0 \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be <\/b><b>(Conjugate complex numbers)<\/b><span style=\"font-weight: 400;\">: <span class=\"katex-eq\" data-katex-display=\"false\">z=x+iy<\/span><\/span><span style=\"font-weight: 400;\"> \u098f\u09b0 \u0985\u09a8\u09c1\u09ac\u09a8\u09cd\u09a7\u09c0 \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be <span class=\"katex-eq\" data-katex-display=\"false\">\\bar{z}=x-iy<\/span><\/span><span style=\"font-weight: 400;\">\u0964 \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u0985\u0995\u09cd\u09b7 (<span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span><span style=\"font-weight: 400;\">-\u0985\u0995\u09cd\u09b7) \u09a5\u09c7\u0995\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\">z=x+iy<\/span><\/span><span style=\"font-weight: 400;\"> \u0993 <span class=\"katex-eq\" data-katex-display=\"false\">\\bar{z}=x-iy<\/span><\/span><span style=\"font-weight: 400;\"> \u098f\u09b0 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac \u09b8\u09ae\u09be\u09a8\u0964 \u09a4\u09be\u09a6\u09c7\u09b0 \u09aa\u09b0\u09ae\u09ae\u09be\u09a8 \u09b8\u09ae\u09be\u09a8 \u0985\u09b0\u09cd\u09a5\u09be\u09ce, <span class=\"katex-eq\" data-katex-display=\"false\">|z|=|\\bar{z}|=\\sqrt{x^2+y^2}<\/span><\/span><span style=\"font-weight: 400;\"> ,\u09a4\u09be\u09a6\u09c7\u09b0 \u09a8\u09a4\u09bf\u09b0 \u09b8\u09be\u0982\u0996\u09cd\u09af\u09bf\u0995 \u09ae\u09be\u09a8 \u09b8\u09ae\u09be\u09a8 \u0995\u09bf\u09a8\u09cd\u09a4\u09c1 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u099a\u09bf\u09b9\u09cd\u09a8\u09af\u09c1\u0995\u09cd\u09a4 \u0985\u09b0\u09cd\u09a5\u09be\u09ce, <span class=\"katex-eq\" data-katex-display=\"false\">z=x+iy<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u098f\u09b0 \u09a8\u09a4\u09bf <\/span><span style=\"font-weight: 400;\"> \u09b9\u09b2\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\">\\bar{z}=x-iy<\/span><\/span><span style=\"font-weight: 400;\"> \u098f\u09b0 \u09a8\u09a4\u09bf <span class=\"katex-eq\" data-katex-display=\"false\">(-\\theta)<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u09b9\u09ac\u09c7\u0964<\/span><span style=\"font-weight: 400;\"> <span class=\"katex-eq\" data-katex-display=\"false\">\\therefore arg (\\bar{z}) =-arg (z)<\/span><\/span>\u00a0<span style=\"font-weight: 400;\">.<\/span><\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<div class=\"wp-block-image\">\r\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" class=\"aligncenter wp-image-1186\" src=\"https:\/\/stage-wp.10minuteschool.com\/wp-content\/uploads\/2021\/10\/image-55.png\" alt=\"\u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be \u0995\u09be\u0995\u09c7 \u09ac\u09b2\u09c7\" width=\"673\" height=\"278\" srcset=\"https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2021\/10\/image-55.png 673w, https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2021\/10\/image-55-300x124.png 300w\" sizes=\"(max-width: 673px) 100vw, 673px\" \/><\/figure>\r\n<\/div>\r\n\r\n\r\n\r\n<p>&nbsp;<\/p>\r\n<p><span style=\"font-weight: 400;\">\u0986\u09ac\u09be\u09b0, <span class=\"katex-eq\" data-katex-display=\"false\">z\\bar{z}=(x+iy)(x-iy)=x^2+y^2=(\\sqrt{x^2+y^2})^2=|z|^2<\/span><\/span><\/p>\r\n\r\n\r\n\r\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\therefore z\\bar{z}=|z|^2<\/span>, \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u0995\u09cb\u09a8\u09cb \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be \u098f\u09ac\u0982 \u09a4\u09be\u09b0 \u0985\u09a8\u09c1\u09ac\u09a8\u09cd\u09a7\u09c0 \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u0997\u09c1\u09a3\u09ab\u09b2 \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u09aa\u09b0\u09ae\u09ae\u09be\u09a8\u09c7\u09b0<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09ac\u09b0\u09cd\u0997\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8\u0964<\/span><\/p>\r\n\r\n\r\n\r\n<h2><span style=\"color: #008000;\"><b>\u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u09a7\u09b0\u09cd\u09ae (Properties of Complex Number)<\/b><span style=\"font-weight: 400;\">:<\/span><\/span><\/h2>\r\n\r\n\r\n\r\n<ul>\r\n<li><b style=\"font-weight: 400;\"> <span class=\"katex-eq\" data-katex-display=\"false\">x+iy=0<\/span><\/b> <b style=\"font-weight: 400;\">\u09b9\u09b2\u09c7, <\/b><span class=\"katex-eq\" data-katex-display=\"false\">x=0,\u00a0y=0<\/span><\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<p><b>\u09aa\u09cd\u09b0\u09ae\u09be\u09a3<\/b><span style=\"font-weight: 400;\">: <span class=\"katex-eq\" data-katex-display=\"false\">x+iy=0 \\Rightarrow x=-iy \\Rightarrow x^2=(-iy)^2=i^2y^2=-1.y^2 \\Rightarrow x^2+y^2=0\u00a0<\/span><\/span><\/p>\r\n\r\n\r\n\r\n<p><span style=\"font-weight: 400;\">\u09af\u09c7\u09b9\u09c7\u09a4\u09c1 \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09b8\u0982\u0996\u09cd\u09af\u09be (<\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\"> \u0993 <\/span><span style=\"font-weight: 400;\">y<\/span><span style=\"font-weight: 400;\">) \u09aa\u09c3\u09a5\u0995\u09ad\u09be\u09ac\u09c7 \u09b6\u09c2\u09a8\u09cd\u09af \u09a8\u09be \u09b9\u09b2\u09c7 \u09a4\u09be\u09a6\u09c7\u09b0 \u09ac\u09b0\u09cd\u0997\u09c7\u09b0 \u09b8\u09ae\u09b7\u09cd\u099f\u09bf \u09b6\u09c2\u09a8\u09cd\u09af \u09b9\u09a4\u09c7 \u09aa\u09be\u09b0\u09c7 \u09a8\u09be,<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09b8\u09c1\u09a4\u09b0\u09be\u0982 <span class=\"katex-eq\" data-katex-display=\"false\"> x=0,\u00a0y=0<\/span><\/span><\/p>\r\n\r\n\r\n\r\n<p><strong><b style=\"font-weight: 400;\">\u09a6\u09c1\u0987\u099f\u09bf \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be <\/b><span class=\"katex-eq\" data-katex-display=\"false\">x_1+iy_1<\/span><b style=\"font-weight: 400;\">\u00a0\u0993 <\/b><span class=\"katex-eq\" data-katex-display=\"false\">x_2+iy_2<\/span><b style=\"font-weight: 400;\"> \u09b8\u09ae\u09be\u09a8 \u09b9\u09ac\u09c7 \u09af\u09a6\u09bf \u0993 \u0995\u09c7\u09ac\u09b2 \u09af\u09a6\u09bf <span class=\"katex-eq\" data-katex-display=\"false\">x_1=x_2<\/span> \u0993 <\/b><span class=\"katex-eq\" data-katex-display=\"false\">y_1=y_2<\/span><b style=\"font-weight: 400;\">\u00a0\u09b9\u09df\u0964<\/b><\/strong><\/p>\r\n\r\n\r\n\r\n<p><b>\u09aa\u09cd\u09b0\u09ae\u09be\u09a3<\/b><span style=\"font-weight: 400;\">: <\/span><\/p>\r\n\r\n\r\n<span class=\"katex-eq\" data-katex-display=\"false\">x_1+iy_1=x_2+iy_2 \\Rightarrow x_1-x_2=i(y_2-y_1) \\Rightarrow x_1-x_2+i(y_1-y_2)=0\\\\ \\therefore x_1-x_2=0 \\Rightarrow x_1=x_2\\; \u098f\u09ac\u0982\\; y_1-y_2=0 \\Rightarrow y_1=y_2<\/span>\n\r\n\r\n\r\n<ul>\r\n<li><strong><b style=\"font-weight: 400;\">\u09a6\u09c1\u0987\u099f\u09bf \u0985\u09a8\u09c1\u09ac\u09a8\u09cd\u09a7\u09c0 \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be <\/b><span class=\"katex-eq\" data-katex-display=\"false\">&gt;x+iy<\/span> <b style=\"font-weight: 400;\">\u0993 <\/b><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">x-iy<\/span><\/span><b style=\"font-weight: 400;\"> \u098f\u09b0 \u09b8\u09ae\u09b7\u09cd\u099f\u09bf\u00a0 \u098f\u09ac\u0982 \u0997\u09c1\u09a3\u09ab\u09b2 \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09b8\u0982\u0996\u09cd\u09af\u09be\u0964\u00a0<\/b><\/strong><\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<p><b>\u09aa\u09cd\u09b0\u09ae\u09be\u09a3<\/b><span style=\"font-weight: 400;\">:<\/span><span style=\"font-weight: 400;\"> \u09b8\u09ae\u09b7\u09cd\u099f\u09bf <span class=\"katex-eq\" data-katex-display=\"false\">=x+iy+x-iy=2x<\/span><\/span><span style=\"font-weight: 400;\">, \u09af\u09be \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09b8\u0982\u0996\u09cd\u09af\u09be\u0964<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n<p><span style=\"font-weight: 400;\">\u0997\u09c1\u09a3\u09ab\u09b2 <\/span><span style=\"font-weight: 400;\">=<span class=\"katex-eq\" data-katex-display=\"false\">(x+iy)(x-iy)=x^2-(iy)^2=x^2-i^2y^2=x^2-(-1).y^2=x^2+y^2<\/span><\/span><span style=\"font-weight: 400;\">, \u09af\u09be \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09b8\u0982\u0996\u09cd\u09af\u09be\u0964<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n<ul>\r\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>\u0985\u09a8\u09c1\u09ac\u09a8\u09cd\u09a7\u09c0 \u09a8\u09df \u098f\u09b0\u09c2\u09aa \u09a6\u09c1\u0987\u099f\u09bf \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be<\/b><b> <span class=\"katex-eq\" data-katex-display=\"false\">x_1+iy_1<\/span><\/b><b> \u0993 <span class=\"katex-eq\" data-katex-display=\"false\">x_2+iy_2\u00a0(y_1\\neq y_2)<\/span> <\/b><b>\u098f\u09b0 \u09b8\u09ae\u09b7\u09cd\u099f\u09bf, \u09ac\u09bf\u09df\u09cb\u0997\u09ab\u09b2, \u0997\u09c1\u09a3\u09ab\u09b2 \u0993 \u09ad\u09be\u0997\u09ab\u09b2 \u09aa\u09cd\u09b0\u09a4\u09cd\u09af\u09c7\u0995\u099f\u09bf\u0987 \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be \u09b9\u09ac\u09c7\u0964<\/b><\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<p><b>\u09aa\u09cd\u09b0\u09ae\u09be\u09a3<\/b><span style=\"font-weight: 400;\">: \u09b8\u09ae\u09b7\u09cd\u099f\u09bf <span class=\"katex-eq\" data-katex-display=\"false\">=x_1+iy_1+x_2+iy_2=(x_1+x_2)+i(y_1+y_2)<\/span><\/span><span style=\"font-weight: 400;\">, \u09af\u09be \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u0964<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n<p><span style=\"font-weight: 400;\">\u09ac\u09bf\u09df\u09cb\u0997\u09ab\u09b2 <span class=\"katex-eq\" data-katex-display=\"false\">=x_1+iy_1-(x_2+iy_2)=(x_1-x_2)+i(y_1-y_2)<\/span><\/span><span style=\"font-weight: 400;\">, \u09af\u09be \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u0964<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n<p><span style=\"font-weight: 400;\">\u0997\u09c1\u09a3\u09ab\u09b2 <\/span><\/p>\r\n\r\n\r\n<span class=\"katex-eq\" data-katex-display=\"false\">=(x_1+iy_1)(x_2+iy_2)=x_1x_2+ix_1y_2+ix_2y_1+i^2y_1y_2\\\\ =x_1x_2+i(x_1y_2+x_2y_1)-y_1y_2=(x_1x_2-y_1y_2)+i(x_1y_2+x_2y_1)<\/span>\n\r\n\r\n\r\n<p><span style=\"font-weight: 400;\">\u00a0\u09af\u09be \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u0964<\/span><\/p>\r\n\r\n\r\n\r\n<p><span style=\"font-weight: 400;\">\u09ad\u09be\u0997\u09ab\u09b2 <span class=\"katex-eq\" data-katex-display=\"false\">=\\frac{(x_1+iy_1)}{(x_2+iy_2)}=\\frac{(x_1+iy_1)(x_2-iy_2)}{(x_2+iy_2)(x_2-iy_2)}<\/span><\/span><span style=\"font-weight: 400;\">\u00a0[\u09a8\u09cb\u099f: \u09b9\u09b0\u09c7\u09b0 \u0985\u09a8\u09c1\u09ac\u09a8\u09cd\u09a7\u09c0 \u09a6\u09cd\u09ac\u09be\u09b0\u09be<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\"> \u09b2\u09ac \u0993 \u09b9\u09b0\u0995\u09c7 \u0997\u09c1\u09a3 \u09a6\u09bf\u09a4\u09c7 \u09b9\u09df]<\/span><\/p>\r\n\r\n\r\n\r\n<p>&nbsp;<\/p>\r\n\r\n\r\n<span class=\"katex-eq\" data-katex-display=\"false\">=\\frac{x_1x_2-ix_1y_2+ix_2y_1-i^2y_1y_2}{x_2^2-i^2y_2^2}=\\frac{x_1x_2+y_1y_2+i(x_2y_1-x_1y_2)}{x_2^2+y_2^2}\\\\ =\\frac{x_1x_2+y_1y_2}{x_2^2+y_2^2}+i\\frac{x_2y_1-x_1y_2}{x_2^2+y_2^2}<\/span>\n\r\n\r\n\r\n<p>,<span style=\"font-weight: 400;\"> \u09af\u09be \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u0964<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n<p><b>\u09ac\u09bf.\u09a6\u09cd\u09b0<\/b><span style=\"font-weight: 400;\">.: \u0995\u09cb\u09a8\u09cb \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u0995\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\">A+iB<\/span>\u00a0\u0986\u0995\u09be\u09b0\u09c7 \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09a4\u09c7 \u0989\u09aa\u09b0\u09cd\u09af\u09c1\u0995\u09cd\u09a4 \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf \u0985\u09a8\u09c1\u09b8\u09b0\u09a3 \u0995\u09b0\u09a4\u09c7 \u09b9\u09df\u0964<\/span><\/p>\r\n\r\n\r\n\r\n<ul>\r\n<li><strong><span class=\"katex-eq\" data-katex-display=\"false\">z=x+iy<\/span> <b style=\"font-weight: 400;\">\u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u098f\u09ac\u0982 \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u09aa\u09c2\u09b0\u09cd\u09a3\u09b8\u0982\u0996\u09cd\u09af\u09be \u09b9\u09b2\u09c7 <\/b><span class=\"katex-eq\" data-katex-display=\"false\">Z^n<\/span><b style=\"font-weight: 400;\">\u00a0\u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be \u09b9\u09ac\u09c7\u0964<\/b><\/strong><\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<p><b>\u09aa\u09cd\u09b0\u09ae\u09be\u09a3<\/b><span style=\"font-weight: 400;\">:<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">z^2=(x+iy)^2=x^2-y^2+i2xy<\/span><span style=\"font-weight: 400;\">\u00a0\u09af\u09be \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u0964<\/span><\/p>\r\n\r\n\r\n<span class=\"katex-eq\" data-katex-display=\"false\">z^3=(x+iy)^3=x^3+3x^2iy+3x(iy)^2+(iy)^3=x^3+i3x^2y+3xi^2y^2+i^3y^3\\\\ =x^3-3xy^2+i(3x^2y-y^3)<\/span>\n\r\n\r\n\r\n<p>, \u09af\u09be \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u0964<\/p>\r\n\r\n\r\n\r\n<p><span style=\"font-weight: 400;\">\u0985\u09a8\u09c1\u09b0\u09c2\u09aa\u09ad\u09be\u09ac\u09c7 \u0985\u0997\u09cd\u09b0\u09b8\u09b0 \u09b9\u09b2\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\">Z^n<\/span><\/span><span style=\"font-weight: 400;\"> \u098f\u0995\u099f\u09bf \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be \u09a6\u09c7\u0996\u09be\u09a8\u09cb \u09af\u09be\u09ac\u09c7\u0964<\/span><\/p>\r\n\r\n\r\n\r\n<h2><span style=\"color: #008000;\"><b>\u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u09aa\u09b0\u09ae\u09ae\u09be\u09a8 (\u09ae\u09a1\u09c1\u09b2\u09be\u09b8) \u098f\u09ac\u0982 \u09a8\u09a4\u09bf (\u0986\u09b0\u09cd\u0997\u09c1\u09ae\u09c7\u09a8\u09cd\u099f) \u09b8\u0982\u0995\u09cd\u09b0\u09be\u09a8\u09cd\u09a4 \u09a7\u09b0\u09cd\u09ae <\/b><b>(Absolute value (modulus) and slope (argument) of complex numbers)<\/b><\/span><\/h2>\r\n\r\n\r\n\r\n<ul>\r\n<li><span style=\"font-weight: 400;\"><strong><span class=\"katex-eq\" data-katex-display=\"false\">z=x+iy<\/span><\/strong><\/span><b><strong>\u00a0\u09b9\u09b2\u09c7, <\/strong><\/b><\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<p><b>(i) <span class=\"katex-eq\" data-katex-display=\"false\">|z|=|\\bar{z}|=|-z|=|-\\bar{z}|<\/span><\/b><\/p>\r\n\r\n\r\n\r\n<p><b>(ii) <span class=\"katex-eq\" data-katex-display=\"false\">|z|^2=|\\bar{z}|^2=|-z|^2=|-\\bar{z}|^2=z\\bar{z}<\/span><\/b><\/p>\r\n\r\n\r\n\r\n<p><b>\u09aa\u09cd\u09b0\u09ae\u09be\u09a3<\/b><span style=\"font-weight: 400;\">:<\/span><\/p>\r\n\r\n\r\n\r\n<p><b>(a)<\/b> <span class=\"katex-eq\" data-katex-display=\"false\">z=x+iy<\/span><span style=\"font-weight: 400;\"> \u09a4\u09be\u09b9\u09b2\u09c7, <span class=\"katex-eq\" data-katex-display=\"false\">\\bar{z}=x-iy,\u00a0-z=-x-iy<\/span><\/span><span style=\"font-weight: 400;\"> \u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\">-\\bar{z}=-x+iy<\/span><\/span><\/p>\r\n\r\n\r\n\r\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\therefore |z|=\\sqrt{x^2+y^2}<\/span> \u098f\u09ac\u0982 <\/span><\/p>\r\n\r\n\r\n<span class=\"katex-eq\" data-katex-display=\"false\">|\\bar{z}|=\\sqrt{x^2+(-y)^2}=\\sqrt{x^2+y^2}\\\\ |-z|=\\sqrt{(-x)^2+(-y)^2}=\\sqrt{x^2+y^2}\\; \u098f\u09ac\u0982\\; |-\\bar{z}|=\\sqrt{(-x)^2+y^2}=\\sqrt{x^2+y^2}\\\\ \\therefore |z|=|\\bar{z}|=|-z|=|-\\bar{z}|\u00a0<\/span>\n<p>&nbsp;<\/p>\n\r\n\r\n\r\n<p><b>(b) <\/b><span class=\"katex-eq\" data-katex-display=\"false\">|z|^2=x^2+y^2,|\\bar{z}|^2=x^2+y^2,|-z|^2=x^2+y^2,|-\\bar{z}|^2=x^2+y^2<\/span> <span style=\"font-weight: 400;\">\u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\">z\\bar{z}=(x+iy)(x-iy)=x^2+y^2<\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n<span class=\"katex-eq\" data-katex-display=\"false\">\\therefore |z|^2=|\\bar{z}|^2=|-z|^2=|-\\bar{z}|^2=z\\bar{z}\u00a0<\/span>\n\r\n\r\n\r\n<ul>\r\n<li><strong><span class=\"katex-eq\" data-katex-display=\"false\">z_1<\/span><b style=\"font-weight: 400;\"> \u0993 <\/b><span class=\"katex-eq\" data-katex-display=\"false\">z_2<\/span><b style=\"font-weight: 400;\"> \u09a6\u09c1\u0987\u099f\u09bf \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be \u09b9\u09b2\u09c7, <span class=\"katex-eq\" data-katex-display=\"false\">|z_1+z_2|\\leq |z_1|+|z_2|<\/span><\/b><\/strong><\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<p><b>\u09aa\u09cd\u09b0\u09ae\u09be\u09a3<\/b><span style=\"font-weight: 400;\">: <\/span><\/p>\r\n\r\n\r\n<span class=\"katex-eq\" data-katex-display=\"false\">|z_1+z_2|^2=(z_1+z_2)\\overline{(z_1+z_2)} [\\because |z|^2=z\\bar{z}]\\\\ =(z_1+z_2)(\\overline{z_1}+\\overline{z_2}) [\\because \\overline {z_1+z_2}=\\bar{z_1}+\\bar{z_2}]\\\\ =z_1\\bar{z_1}+z_1\\bar{z_2}+z_2\\bar{z_1}+z_2\\bar{z_2}\\\\ =\\overline{z_1^2}+\\overline{z_2^2}+2Re(z_1\\overline{z_2}) [\\because z_1\\overline{z_2}+z_2\\overline{z_1}=2Re(z_1\\overline{z_2})]\\\\ \\therefore |z_1+z_2|^2 \\leq |z_1|^2+|z_2|^2+2|z_1\\overline{z_2}| [\\because Re(z)\\leq |z|]\\\\ =|z_1+z_2|^2 \\leq |z_1|^2+|z_2|^2+2|z_1||\\overline{z_2}| [\\because |z_1\\overline{z_2}|=|z_1||z_2|]\\\\ =|z_1|^2+|z_2|^2+2|z_1||z_2| [\\because |\\overline{z_2}|=|z_2|]\\\\ =(|z_1|+|z_2|)^2\\\\ \\therefore |z_1+z_2| \\leq |z_1|+|z_2|\u00a0<\/span>\n\r\n\r\n\r\n<p><span style=\"font-weight: 400;\">\u0985\u09a8\u09c1\u09b0\u09c2\u09aa\u09c7, \u09a8\u09bf\u09ae\u09cd\u09a8\u09b2\u09bf\u0996\u09bf\u09a4 \u0985\u09b8\u09ae\u09a4\u09be\u0997\u09c1\u09b2\u09cb\u0993 \u09aa\u09cd\u09b0\u09ae\u09be\u09a3 \u0995\u09b0\u09be \u09af\u09be\u09df\u0964<\/span><\/p>\r\n\r\n\r\n<span class=\"katex-eq\" data-katex-display=\"false\">(a) |z_1+z_2+z_3| \\leq |z_1|+|z_2|+|z_3|\\\\ (b) |z_1+z_2+...+z_n| \\leq |z_1|+|z_2|+...+|z_n|\\\\ (c) |z_1-z_2|\\leq |z_1|+|z_2|\u00a0<\/span>\n\r\n\r\n\r\n<p><b>\u09aa\u09cd\u09b0\u09ae\u09be\u09a3<\/b><span style=\"font-weight: 400;\">: <\/span><\/p>\r\n\r\n\r\n<span class=\"katex-eq\" data-katex-display=\"false\">|z_1-z_2|^2=(z_1-z_2)\\overline{(z1-z2)} [\\because |z|^2=z\\bar{z}]\\\\ =(z_1-z_2)(\\overline{z_1}-\\overline{z_2}) [\\because \\overline{z_1+z_2}=\\bar{z_1}+\\bar{z_2}]\\\\ =z_1\\bar{z_1}-z_1\\bar{z_2}-z_2\\bar{z_1}+z_2\\bar{z_2}\\\\ =|z_1|^2-(z_1\\overline{z_2}+z_2\\overline{z_1})+|z_2|^2 [\\because |z|^2=z\\bar{z}]\\\\ =|z_1|^2-(z_1\\overline{z_2}+\\overline{z_1\\overline{z_2}})+|z_2|^2 [\\because \\bar{\\bar{z}}=z]\\\\ =|z_1|^2-2Re(z_1\\bar{z_2})+|z_2|^2 [\\because -x \\leq \\sqrt{x^2+y^2} \\Rightarrow -Re(z) \\leq |z|]\\\\ \\therefore |z_1-z_2|^2\\leq |z_1|^2+|z_2|^2+2|z_1\\overline{z_2}|\\\\ = |z_1|^2+|z_2|^2+2|z_1||\\overline{z_2}| [\\because |z_1\\overline{z_2}|=|z_1||\\overline{z_2}|]\\\\ = |z_1|^2+|z_2|^2+2|z_1||z_2| [\\because |\\overline{z_2}|=|z_2|]\\\\ =(|z_1|+|z_2|)^2\\\\ \\therefore |z_1-z_2| \\leq |z_1|+|z_2|<\/span>\n<p>&nbsp;<\/p>\n<hr \/>\n<p>&nbsp;<\/p>\n<p><em><strong>\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<ul>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-25-online-batch-2-bangla-english-ict\/\">HSC 25 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a \u09e8.\u09e6 (\u09ac\u09be\u0982\u09b2\u09be, \u0987\u0982\u09b0\u09c7\u099c\u09bf, \u09a4\u09a5\u09cd\u09af \u0993 \u09af\u09cb\u0997\u09be\u09af\u09cb\u0997 \u09aa\u09cd\u09b0\u09af\u09c1\u0995\u09cd\u09a4\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-26-online-batch-bangla-english-ict\/\">HSC 26 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a (\u09ac\u09be\u0982\u09b2\u09be, \u0987\u0982\u09b0\u09c7\u099c\u09bf, \u09a4\u09a5\u09cd\u09af \u0993 \u09af\u09cb\u0997\u09be\u09af\u09cb\u0997 \u09aa\u09cd\u09b0\u09af\u09c1\u0995\u09cd\u09a4\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-2025-online-batch\/\">HSC 25 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a (\u09ab\u09bf\u099c\u09bf\u0995\u09cd\u09b8, \u0995\u09c7\u09ae\u09bf\u09b8\u09cd\u099f\u09cd\u09b0\u09bf, \u09ae\u09cd\u09af\u09be\u09a5, \u09ac\u09be\u09df\u09cb\u09b2\u099c\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-2026-online-batch\/\">HSC 26 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a (\u09ab\u09bf\u099c\u09bf\u0995\u09cd\u09b8, \u0995\u09c7\u09ae\u09bf\u09b8\u09cd\u099f\u09cd\u09b0\u09bf, \u09ae\u09cd\u09af\u09be\u09a5, \u09ac\u09be\u09df\u09cb\u09b2\u099c\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/medical-admission-course\/\">\u09ae\u09c7\u09a1\u09bf\u0995\u09c7\u09b2 \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 &#8211; \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/dhaka-university-a-unit-admission-course\/\">\u09a2\u09be\u0995\u09be \u09ad\u09be\u09b0\u09cd\u09b8\u09bf\u099f\u09bf A Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 &#8211; \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/dhaka-university-b-unit-admission-course\/\">\u09a2\u09be\u0995\u09be \u09ad\u09be\u09b0\u09cd\u09b8\u09bf\u099f\u09bf B Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 &#8211; \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/buet-ques-solve\/\">\u09ac\u09c1\u09df\u09c7\u099f \u0995\u09cb\u09b6\u09cd\u099a\u09c7\u09a8 \u09b8\u09b2\u09ad \u0995\u09cb\u09b0\u09cd\u09b8<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/gst-a-unit-admission-course\/\">\u0997\u09c1\u099a\u09cd\u099b A Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 &#8211; \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/gst-b-unit-admission-course\/\">\u0997\u09c1\u099a\u09cd\u099b B Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 &#8211; \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<\/ul>\n<hr \/>\n<p>&nbsp;<\/p>\n<p><em><strong>\u0986\u09ae\u09be\u09a6\u09c7\u09b0 \u09b8\u09cd\u0995\u09bf\u09b2 \u09a1\u09c7\u09ad\u09c7\u09b2\u09aa\u09ae\u09c7\u09a8\u09cd\u099f \u0995\u09cb\u09b0\u09cd\u09b8\u09b8\u09ae\u09c2\u09b9\u0983<\/strong><\/em><\/p>\n<ul>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/study-abroad-complete-guideline\/\">\u09ac\u09bf\u09a6\u09c7\u09b6\u09c7 \u0989\u099a\u09cd\u099a\u09b6\u09bf\u0995\u09cd\u09b7\u09be: Study Abroad Complete Guideline<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/student-hacks\/\">Student Hacks<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/ielts-course\/\">IELTS Course by Munzereen Shahid<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/english-grammar-course\/\">Complete English Grammar Course<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/ms-bundle\/\"> Microsoft Office 3 in 1 Bundle<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/ghore-boshe-freelancing\/\">\u0998\u09b0\u09c7 \u09ac\u09b8\u09c7 Freelancing<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/facebook-marketing\/\">Facebook Marketing<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/adobe-4-in-1-bundle\/\">Adobe 4 in 1 Bundle<\/a><\/span><\/li>\n<\/ul>\n<hr \/>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\"><em>\u09e7\u09e6 \u09ae\u09bf\u09a8\u09bf\u099f \u09b8\u09cd\u0995\u09c1\u09b2\u09c7\u09b0 \u0995\u09cd\u09b2\u09be\u09b8\u0997\u09c1\u09b2\u09cb \u0985\u09a8\u09c1\u09b8\u09b0\u09a3 \u0995\u09b0\u09a4\u09c7 \u09ad\u09bf\u099c\u09bf\u099f: <span style=\"color: #993300;\"><strong><a style=\"color: #993300;\" href=\"https:\/\/10minuteschool.com\/?ref=https%3A%2F%2Fblog.10minuteschool.com%2Fwordpress%2F&amp;post_id=78178&amp;blog_category_id=700\">www.10minuteschool.com<\/a><\/strong><\/span><\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u098f\u0996\u09be\u09a8\u09c7- =Modulas (\u09aa\u09b0\u09ae\u09ae\u09be\u09a8) = Argument (\u09a8\u09a4\u09bf) &nbsp; Modulus (\u09aa\u09b0\u09ae\u09ae\u09be\u09a8): \u0993 \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a5\u09c7\u0995\u09c7 z \u098f\u09b0 \u09b8\u09b0\u09be\u09b8\u09b0\u09bf \u09ac\u09be \u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0\u09a4\u09ae \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac\u0995\u09c7 Modulus\u00a0\u09ac\u09be \u09aa\u09b0\u09ae\u09ae\u09be\u09a8 \u09ac\u09b2\u09c7\u0964 Argument (\u09a8\u09a4\u09bf): \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 \u09b0\u09c7\u0996\u09be \u09af\u09c7 \u0995\u09cb\u09a3 \u0989\u09ce\u09aa\u09a8\u09cd\u09a8 \u0995\u09b0\u09c7 \u09a4\u09be\u0995\u09c7 Argument\/\u09a8\u09a4\u09bf \u09ac\u09b2\u09c7\u0964 \u098f\u0995\u09c7 \u09ac\u09be \u00a0\u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09b9\u09df\u0964<\/p>\n<p> <a class=\"redmore\" href=\"https:\/\/10minuteschool.com\/content\/%e0%a6%9c%e0%a6%9f%e0%a6%bf%e0%a6%b2-%e0%a6%b8%e0%a6%82%e0%a6%96%e0%a7%8d%e0%a6%af%e0%a6%be%e0%a6%b0-%e0%a6%aa%e0%a7%8b%e0%a6%b2%e0%a6%be%e0%a6%b0-%e0%a6%86%e0%a6%95%e0%a7%83%e0%a6%a4%e0%a6%bf\/\">Read 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