{"id":6520,"date":"2021-10-17T22:02:30","date_gmt":"2021-10-17T16:32:30","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6520"},"modified":"2021-11-21T18:12:29","modified_gmt":"2021-11-21T12:42:29","slug":"how-to-find-the-reciprocal-of-a-complex-number","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/how-to-find-the-reciprocal-of-a-complex-number\/","title":{"rendered":"How to Find the Reciprocal of a Complex Number"},"content":{"rendered":"

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

Let’s begin – <\/p>\n

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

The reciprocal is also called multiplicative inverse.<\/p>\n

Let z = a + ib be a non-zero complex number. Then<\/p>\n

\\(1\\over z\\) = \\(1\\over a + ib\\)<\/p>\n

Multiply numerator and denominator by conjugate of denominator,<\/p>\n

\\(1\\over z\\) = \\(1\\over a + ib\\) \\(\\times\\) \\(a – ib\\over a – ib\\)<\/p>\n

\\(\\implies\\) \\(1\\over z\\) = \\(a – ib\\over a^2 – i^2b^2\\) = \\(a – ib\\over a^2 + b^2\\)<\/p>\n

\\(\\implies\\) \\(1\\over z\\) = \\(a\\over a^2 + b^2\\) + \\(i(-b)\\over a^2 + b^2\\)<\/p>\n

Clearly, \\(1\\over z\\) is equal to the multiplicatve inverse of z.<\/p>\n

Also, \\(1\\over z\\) = \\(a – ib\\over a^2 + b^2\\) = \\( \\bar{z}\\over | z |^2\\)<\/p>\n

Thus, the multiplicative inverse of a non-zero complex number is same as its reciprocal and is given by<\/p>\n

\n

\\(Re (z)\\over | z |^2\\) + \\(i{(-Im (z))\\over | z |^2}\\) = \\( \\bar{z}\\over | z |^2\\)<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : Find the reciprocal or multiplicative inverse of the following complex numbers.<\/p>\n

(i) 3 + 2i<\/p>\n

(ii) \\((2 + \\sqrt{3}i)^2\\)<\/p>\n

Solution<\/span><\/strong> :<\/p>\n

(i) Let z = 3 + 2i. Then,<\/p>\n

\\(1\\over z\\) = \\(1\\over 3 + 2i\\)<\/p>\n

= \\(3 – 2i\\over (3 + 2i)(3 – 2i)\\) = \\(3 – 2i\\over 9 – 4i^2\\)<\/p>\n

= \\({3\\over 13} – {2\\over 13}i\\)<\/p>\n

(ii) Let z = \\((2 + \\sqrt{3}i)^2\\). Then,<\/p>\n

z = \\(4 + 3i^2 + 4\\sqrt{3}i\\)<\/p>\n

= \\(4 – 3 + 4\\sqrt{3}i\\) = \\(1 + 4\\sqrt{3}i\\)<\/p>\n

\\(\\therefore\\) \\(1\\over z\\) = \\(1\\over 1 + 4\\sqrt{3}i\\)<\/p>\n

Multiply numerator and denominator by conjugate of denominator<\/p>\n

\\(\\implies\\) \\(1\\over z\\) = \\(1 – 4\\sqrt{3}i\\over (1 + 4\\sqrt{3}i)(1 + 4\\sqrt{3}i)\\)<\/p>\n

= \\(1 – 4\\sqrt{3}i\\over 1 + 48\\) = \\(1\\over 49\\) – \\(4\\sqrt{3}i\\over 49\\)<\/p>\n\n\n

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

Here you will learn how to find the reciprocal of a complex number with examples. Let’s begin –  How to Find the Reciprocal of a Complex Number The reciprocal is also called multiplicative inverse. Let z = a + ib be a non-zero complex number. Then \\(1\\over z\\) = \\(1\\over a + ib\\) Multiply numerator …<\/p>\n

How to Find the Reciprocal 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":[229,235],"yoast_head":"\nHow to Find the Reciprocal of a Complex Number<\/title>\n<meta name=\"description\" content=\"In this post you will learn how to find the reciprocal 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-reciprocal-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 Reciprocal of a Complex Number\" \/>\n<meta property=\"og:description\" content=\"In this post you will learn how to find the reciprocal of a complex number with examples.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/how-to-find-the-reciprocal-of-a-complex-number\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-10-17T16:32:30+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-11-21T12:42:29+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-reciprocal-of-a-complex-number\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/how-to-find-the-reciprocal-of-a-complex-number\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"How to Find the Reciprocal of a Complex Number\",\"datePublished\":\"2021-10-17T16:32:30+00:00\",\"dateModified\":\"2021-11-21T12:42:29+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/how-to-find-the-reciprocal-of-a-complex-number\/\"},\"wordCount\":276,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"keywords\":[\"complex numbers\",\"reciprocal of a complex number\"],\"articleSection\":[\"Complex Numbers\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/how-to-find-the-reciprocal-of-a-complex-number\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/how-to-find-the-reciprocal-of-a-complex-number\/\",\"url\":\"https:\/\/mathemerize.com\/how-to-find-the-reciprocal-of-a-complex-number\/\",\"name\":\"How to Find the Reciprocal of a Complex Number\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2021-10-17T16:32:30+00:00\",\"dateModified\":\"2021-11-21T12:42:29+00:00\",\"description\":\"In this post you will learn how to find the reciprocal of a complex number with examples.\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/how-to-find-the-reciprocal-of-a-complex-number\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/how-to-find-the-reciprocal-of-a-complex-number\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/how-to-find-the-reciprocal-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 Reciprocal of a Complex Number\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathemerize.com\/#website\",\"url\":\"https:\/\/mathemerize.com\/\",\"name\":\"Mathemerize\",\"description\":\"Maths Tutorials - 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