{"id":6522,"date":"2021-10-17T22:44:54","date_gmt":"2021-10-17T17:14:54","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6522"},"modified":"2021-11-21T18:09:25","modified_gmt":"2021-11-21T12:39:25","slug":"how-to-find-square-root-of-complex-number","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/how-to-find-square-root-of-complex-number\/","title":{"rendered":"How to Find Square Root of Complex Number"},"content":{"rendered":"

Here you will learn what is square root and how to find square root of complex number with examples.<\/p>\n

Let’s begin –<\/p>\n

How to Find Square Root of Complex Number<\/h2>\n

Let a + ib be a complex number such that \\(\\sqrt{a + ib}\\) = x + iy, where x and y are real numbers.<\/p>\n

Then,<\/p>\n

\n

\\(\\sqrt{a + ib}\\) = x + iy<\/p>\n

\\(\\implies\\) (a + ib) = \\((x + iy)^2\\)<\/p>\n

\\(\\implies\\) a + ib = \\((x^2 – y^2)\\) + 2ixy<\/p>\n<\/blockquote>\n

On equating real and imaginary parts, we get<\/p>\n

\n

\\(x^2 – y^2\\) = a                  ………….(i)<\/p>\n

and,  2xy = b                        …………..(ii)<\/p>\n

Now, \\((x^2 + y^2)^2\\) = \\((x^2 – y^2)\\) + \\(4x^2y^2\\)<\/p>\n

\\(\\implies\\) \\((x^2 + y^2)^2\\) = \\(a^2 + b^2\\)<\/p>\n

\\(\\implies\\) \\((x^2 + y^2)\\) = \\(\\sqrt{a^2 + b^2}\\)               ………..(iii)<\/p>\n<\/blockquote>\n

Solving the equations (i) and (iii), we get<\/p>\n

\n

\\(x^2\\) = \\((1\\over 2)\\){\\(\\sqrt{a^2 + b^2} + a\\)}   and  \\(y^2\\) = \\((1\\over 2)\\){\\(\\sqrt{a^2 + b^2} – a\\)}<\/p>\n

\\(\\implies\\)  x = \\(\\pm\\) \\(\\sqrt{{(1\\over 2)}{\\sqrt{a^2 + b^2} + a}}\\)  and  y = \\(\\pm\\) \\(i\\sqrt{{(1\\over 2)}{\\sqrt{a^2 + b^2} – a}}\\)<\/p>\n<\/blockquote>\n

If b is positive, then by equation (ii), x and y are of the same sign.<\/p>\n

\n

Hence, \\(\\sqrt{a + ib}\\) = \\(\\pm\\) [\\(\\sqrt{{(1\\over 2)}{\\sqrt{a^2 + b^2} + a}}\\) + \\(i\\sqrt{{(1\\over 2)}{\\sqrt{a^2 + b^2} – a}}\\)]<\/p>\n<\/blockquote>\n

If b is negative, then by equation (ii), x and y are of the different signs.<\/p>\n

\n

Hence, \\(\\sqrt{a + ib}\\) = \\(\\pm\\) [\\(\\sqrt{{(1\\over 2)}{\\sqrt{a^2 + b^2} + a}}\\) – \\(i\\sqrt{{(1\\over 2)}{\\sqrt{a^2 + b^2} – a}}\\)]<\/p>\n<\/blockquote>\n

Example<\/span><\/strong> : Find the square root of 7 – 24i.<\/p>\n

Solution<\/span><\/strong> : Let \\(\\sqrt{7 – 24i}\\) = x + iy. Then,<\/p>\n

(7 – 24i) = \\((x + iy)^2\\)<\/p>\n

\\(\\implies\\) 7 – 24i = \\((x^2 – y^2)\\) + 2ixy<\/p>\n

On equating real and imaginary parts, we get<\/p>\n

\\(x^2 – y^2\\) = 7                  ……….(i)<\/p>\n

and, 2xy = -24                     …………..(ii)<\/p>\n

Now,  \\((x^2 + y^2)^2\\) = \\((x^2 – y^2)^2\\) + \\(4x^2y^2\\)<\/p>\n

\\(\\implies\\) \\((x^2 + y^2)^2\\) = 49 + 576 = 625<\/p>\n

\\(\\implies\\) \\(x^2 + y^2\\) = 25            ………..(iii)<\/p>\n

On Solving equation (i) and (iii), we get<\/p>\n

\\(x^2\\) = 16   and  \\(y^2\\) = 9 <\/p>\n

\\(\\implies\\) x = \\(\\pm 4\\)   and  y = \\(\\pm 3\\)<\/p>\n

From (ii), 2xy is negative. So, x and y are of opposite signs.<\/p>\n

\\(\\therefore\\)   (x = 4 and y = -3)  or,  (x = -4 and y = 3)<\/p>\n

Hence,  \\(\\sqrt{7 – 24i}\\) = \\(\\pm\\) (4 – 3i)<\/p>\n\n\n

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

Here you will learn what is square root and how to find square root of complex number with examples. Let’s begin – How to Find Square Root of Complex Number Let a + ib be a complex number such that \\(\\sqrt{a + ib}\\) = x + iy, where x and y are real numbers. Then, …<\/p>\n

How to Find Square Root of 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":[229,233,234],"yoast_head":"\nHow to Find Square Root of Complex Number<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is square root and how to find square root of 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-square-root-of-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 Square Root of Complex Number\" \/>\n<meta property=\"og:description\" content=\"In this post you will learn what is square root and how to find square root of complex number with examples.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/how-to-find-square-root-of-complex-number\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-10-17T17:14:54+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-11-21T12:39:25+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-square-root-of-complex-number\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/how-to-find-square-root-of-complex-number\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"How to Find Square Root of Complex Number\",\"datePublished\":\"2021-10-17T17:14:54+00:00\",\"dateModified\":\"2021-11-21T12:39:25+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/how-to-find-square-root-of-complex-number\/\"},\"wordCount\":425,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"keywords\":[\"complex numbers\",\"square root of complex number\",\"square root of complex number formula\"],\"articleSection\":[\"Complex Numbers\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/how-to-find-square-root-of-complex-number\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/how-to-find-square-root-of-complex-number\/\",\"url\":\"https:\/\/mathemerize.com\/how-to-find-square-root-of-complex-number\/\",\"name\":\"How to Find Square Root of Complex Number\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2021-10-17T17:14:54+00:00\",\"dateModified\":\"2021-11-21T12:39:25+00:00\",\"description\":\"In this post you will learn what is square root and how to find square root of complex number with examples.\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/how-to-find-square-root-of-complex-number\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/how-to-find-square-root-of-complex-number\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/how-to-find-square-root-of-complex-number\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"How to Find Square Root of Complex Number\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathemerize.com\/#website\",\"url\":\"https:\/\/mathemerize.com\/\",\"name\":\"Mathemerize\",\"description\":\"Maths Tutorials - 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