{"id":6518,"date":"2021-10-17T21:23:30","date_gmt":"2021-10-17T15:53:30","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6518"},"modified":"2021-11-21T18:14:32","modified_gmt":"2021-11-21T12:44:32","slug":"how-to-find-the-modulus-of-a-complex-number","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/how-to-find-the-modulus-of-a-complex-number\/","title":{"rendered":"How to Find the Modulus of a Complex Number"},"content":{"rendered":"

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

Let’s begin – <\/p>\n

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

The modulus of a complex number z = a + ib is denoted by | z | and is defined as<\/p>\n

\n

| z | = \\(\\sqrt{a^2 + b^2}\\) = \\(\\sqrt{[Re (z)]^2 + [Im (z)]^2}\\)<\/p>\n<\/blockquote>\n

Clearly, | z | \\(\\ge\\) 0 for all z \\(\\in\\) C.<\/p>\n

Example<\/strong><\/span> : If \\(z_1\\) = 3 – 4i, \\(z_2\\) = -5 + 2i and \\(z_3\\) = 1 + \\(\\sqrt{-3}\\), then find modulus of \\(z_1\\), \\(z_2\\) and \\(z_3\\).<\/p>\n

Solution<\/span><\/strong> : We have, \\(z_1\\) = 3 – 4i, \\(z_2\\) = -5 + 2i and \\(z_3\\) = 1 + \\(\\sqrt{-3}\\)<\/p>\n

| \\(z_1\\) | = | 3 – 4i | = \\(\\sqrt{3^2 + (-4)^2}\\) = 5,<\/p>\n

| \\(z_2\\) | = | 5 + 2i | = \\(\\sqrt{(-5)^2 + 2^2}\\) = \\(\\sqrt{29}\\)<\/p>\n

and, | \\(z_3\\) | = | 1 + \\(\\sqrt{-3}\\) | = \\(\\sqrt{1^2 + (\\sqrt{3})^2}\\) = 2<\/p>\n

Remark<\/strong> : In the set C of all complex numbers, the order relation is not defined. As such \\(z_1\\) > \\(z_2\\) or \\(z_1\\) < \\(z_2\\) has no meaning but | \\(z_1\\) | > | \\(z_2\\) | or | \\(z_1\\) | < | \\(z_2\\) | has got its meaning since | \\(z_1\\) | and | \\(z_2\\) | are real numbers.<\/p>\n

Properties of Modulus<\/h3>\n

If z, \\(z_1\\), \\(z_2\\) \\(\\in\\) C, then <\/p>\n

(i)<\/strong> | z | = 0 \\(\\iff\\) z = 0  i.e.  Re (z) = Im (z) = 0<\/p>\n

(ii)<\/strong> | z | = | \\(\\bar{z}\\) | = | -z |<\/p>\n

(iii)<\/strong> – | z | \\(\\le\\) Re (z) \\(\\le\\) | z |  ;  – | z | \\(\\le\\) Im (z) \\(\\le\\) | z | <\/p>\n

(iv)<\/strong> \\(z\\bar{z}\\) = \\(| z |^2\\)<\/p>\n

(v)<\/strong> | \\(z_1 z_2\\) | = | \\(z_1\\) | | \\(z_2\\) |<\/p>\n

(vi)<\/strong> | \\(z_1\\over z_2\\) | = \\( | z_1 | \\over | z_2 |\\) ,  \\(z_2\\) \\(\\ne\\) 0<\/p>\n

(vii)<\/strong>  \\( | z_1 + z_2 |^2\\) = \\( | z_1 |^2\\) + \\( | z_2 |^2\\) + \\( 2 Re (z_1\\bar{z_2})\\)<\/p>\n

(viii)<\/strong>  \\( | z_1 – z_2 |^2\\) = \\( | z_1 |^2\\) + \\( | z_2 |^2\\) – \\( 2 Re (z_1\\bar{z_2})\\)<\/p>\n

(ix)<\/strong>  \\( | z_1 + z_2 |^2\\) + \\( | z_1 – z_2 |^2\\) = 2(\\( | z_1 |^2\\) + \\( | z_2 |^2\\))<\/p>\n

(x)<\/strong> \\( | az_1 – bz_2 |^2\\) + \\( | bz_1 + az_2 |^2\\) = \\(a^2 + b^2\\) (\\( | z_1 |^2\\) + \\( | z_2 |^2\\)), where a. b \\(\\in\\) R.<\/p>\n\n\n

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

Here you will learn how to find the modulus of a complex number and properties of modulus with examples. Let’s begin –  How to Find the Modulus of a Complex Number The modulus of a complex number z = a + ib is denoted by | z | and is defined as | z | …<\/p>\n

How to Find the Modulus 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":[229,236,237],"yoast_head":"\nHow to Find the Modulus of a Complex Number<\/title>\n<meta name=\"description\" content=\"In this post you will learn how to find the modulus 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-modulus-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 Modulus of a Complex Number\" \/>\n<meta property=\"og:description\" content=\"In this post you will learn how to find the modulus of a complex number with examples.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/how-to-find-the-modulus-of-a-complex-number\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-10-17T15:53:30+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-11-21T12:44:32+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-modulus-of-a-complex-number\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/how-to-find-the-modulus-of-a-complex-number\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"How to Find the Modulus of a Complex Number\",\"datePublished\":\"2021-10-17T15:53:30+00:00\",\"dateModified\":\"2021-11-21T12:44:32+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/how-to-find-the-modulus-of-a-complex-number\/\"},\"wordCount\":304,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"keywords\":[\"complex numbers\",\"modulus of a complex number\",\"properties of modulus of a complex number\"],\"articleSection\":[\"Complex Numbers\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/how-to-find-the-modulus-of-a-complex-number\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/how-to-find-the-modulus-of-a-complex-number\/\",\"url\":\"https:\/\/mathemerize.com\/how-to-find-the-modulus-of-a-complex-number\/\",\"name\":\"How to Find the Modulus of a Complex Number\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2021-10-17T15:53:30+00:00\",\"dateModified\":\"2021-11-21T12:44:32+00:00\",\"description\":\"In this post you will learn how to find the modulus of a complex number with examples.\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/how-to-find-the-modulus-of-a-complex-number\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/how-to-find-the-modulus-of-a-complex-number\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/how-to-find-the-modulus-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 Modulus of a Complex Number\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathemerize.com\/#website\",\"url\":\"https:\/\/mathemerize.com\/\",\"name\":\"Mathemerize\",\"description\":\"Maths Tutorials - 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