{"id":6516,"date":"2021-10-17T20:00:17","date_gmt":"2021-10-17T14:30:17","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6516"},"modified":"2021-11-21T18:18:18","modified_gmt":"2021-11-21T12:48:18","slug":"how-to-find-the-conjugate-of-a-complex-number","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/how-to-find-the-conjugate-of-a-complex-number\/","title":{"rendered":"How to Find the Conjugate of a Complex Number"},"content":{"rendered":"

Here you will learn how to find the conjugate of a complex number and properties of conjugate with examples.<\/p>\n

Let’s begin –\u00a0<\/p>\n

How to Find the Conjugate of a Complex Number<\/h2>\n

Let z = a + ib be a complex number. Then the conjugate of z is denoted by \\(\\bar{z}\\) and is equal to a – ib.<\/p>\n

\n

Thus, z = a + ib \\(\\implies\\) \\(\\bar{z}\\) = a – ib<\/p>\n<\/blockquote>\n

It follows from this definition that the conjugate of a complex number is obtained by replacing i by -i.<\/p>\n

For Example<\/span><\/strong> : If z = 3 + 4i, then \\(\\bar{z}\\) = 3 – 4i.<\/p>\n

Properties of Conjugate<\/h3>\n

If \\(z\\), \\(z_1\\), \\(z_2\\) are complex numbers, then<\/p>\n

(i)<\/strong> \\(\\bar{\\bar{z}}\\) = z<\/p>\n

(ii)<\/strong> z + \\(\\bar{z}\\) = 2 Re (z)<\/p>\n

(iii)<\/strong> z – \\(\\bar{z}\\) = 2i Im (z)<\/p>\n

(iv)<\/strong> z = \\(\\bar{z}\\) \\(\\iff\\) z is purely real<\/p>\n

(v)<\/strong> z + \\(\\bar{z}\\) = 0 \\(\\implies\\) z is purely imaginary<\/p>\n

(vi)<\/strong> z\\(\\bar{z}\\) = \\([Re (z)]^2\\) + \\([Im (z)]^2\\)<\/p>\n

(vii)<\/strong> \\(\\bar{z_1 + z_2}\\) =\u00a0 \\(\\bar{z_1}\\) + \\(\\bar{z_2}\\)<\/p>\n

(viii)<\/strong> \\(\\bar{z_1 – z_2}\\) =\u00a0 \\(\\bar{z_1}\\) – \\(\\bar{z_2}\\)<\/p>\n

(ix)<\/strong> \\(\\bar{z_1z_2}\\) =\u00a0 \\(\\bar{z_1}\\) \\(\\bar{z_2}\\)<\/p>\n

(x)<\/strong> \\(\\bar{z_1\\over z_2}\\) =\u00a0 \\(\\bar{z_1}\\over \\bar{z_2}\\)<\/p>\n

Example<\/span><\/strong> : Multiply 3 – 2i by its conjugate.<\/p>\n

Solution<\/span><\/strong> : The conjugate of 3 – 2i is 3 + 2i.<\/p>\n

Hence, required product is = (3 – 2i)(3 + 2i) = \\(9 – 4i^2\\) = 9 + 4 = 13<\/p>\n

Example<\/span><\/strong> : Find the conjugate of \\(1\\over 3 + 4i\\).<\/p>\n

Solution<\/span><\/strong> : Let z = \\(1\\over 3 + 4i\\). Then,<\/p>\n

z = \\(1\\over 3 + 4i\\) \\(\\times\\) \\(3 – 4i\\over 3 – 4i\\) = \\(3 – 4i\\over 9 + 16\\) = \\({3\\over 25} – {4\\over 25}i\\)<\/p>\n

\\(\\therefore\\) Conjugate of z is \\(\\bar{z}\\) = \\({3\\over 25} – {4\\over 25}i\\).\u00a0<\/p>\n\n\n

\n
Next – How to Find the Modulus of a Complex Number<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Division of Complex Numbers<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn how to find the conjugate of a complex number and properties of conjugate with examples. Let’s begin –\u00a0 How to Find the Conjugate of a Complex Number Let z = a + ib be a complex number. Then the conjugate of z is denoted by \\(\\bar{z}\\) and is equal to a …<\/p>\n

How to Find the Conjugate of a Complex Number<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[41],"tags":[229,238,239],"yoast_head":"\nHow to Find the Conjugate of a Complex Number<\/title>\n<meta name=\"description\" content=\"In this post you will learn how to find the conjugate of a complex number with examples.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/how-to-find-the-conjugate-of-a-complex-number\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"How to Find the Conjugate of a Complex Number\" \/>\n<meta property=\"og:description\" content=\"In this post you will learn how to find the conjugate of a complex number with examples.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/how-to-find-the-conjugate-of-a-complex-number\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-10-17T14:30:17+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-11-21T12:48:18+00:00\" \/>\n<meta name=\"author\" content=\"mathemerize\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"mathemerize\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathemerize.com\/how-to-find-the-conjugate-of-a-complex-number\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/how-to-find-the-conjugate-of-a-complex-number\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"How to Find the Conjugate of a Complex Number\",\"datePublished\":\"2021-10-17T14:30:17+00:00\",\"dateModified\":\"2021-11-21T12:48:18+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/how-to-find-the-conjugate-of-a-complex-number\/\"},\"wordCount\":277,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"keywords\":[\"complex numbers\",\"Conjugate of a Complex Number\",\"properties of conjugate of a complex number\"],\"articleSection\":[\"Complex Numbers\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/how-to-find-the-conjugate-of-a-complex-number\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/how-to-find-the-conjugate-of-a-complex-number\/\",\"url\":\"https:\/\/mathemerize.com\/how-to-find-the-conjugate-of-a-complex-number\/\",\"name\":\"How to Find the Conjugate of a Complex Number\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2021-10-17T14:30:17+00:00\",\"dateModified\":\"2021-11-21T12:48:18+00:00\",\"description\":\"In this post you will learn how to find the conjugate of a complex number with examples.\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/how-to-find-the-conjugate-of-a-complex-number\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/how-to-find-the-conjugate-of-a-complex-number\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/how-to-find-the-conjugate-of-a-complex-number\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"How to Find the Conjugate of a Complex Number\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathemerize.com\/#website\",\"url\":\"https:\/\/mathemerize.com\/\",\"name\":\"Mathemerize\",\"description\":\"Maths Tutorials - 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