{"id":6514,"date":"2021-10-17T19:05:42","date_gmt":"2021-10-17T13:35:42","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6514"},"modified":"2021-11-21T18:19:25","modified_gmt":"2021-11-21T12:49:25","slug":"division-of-complex-numbers","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/division-of-complex-numbers\/","title":{"rendered":"Division of Complex Numbers"},"content":{"rendered":"

Here you will learn what is the division of complex numbers with examples.<\/p>\n

Let’s begin –<\/p>\n

Division of Complex Numbers<\/h2>\n

The division of a complex number \\(z_1\\) by a non-zero complex number \\(z_2\\) is defined as the multiplication of \\(z_1\\) by the multiplicative inverse of \\(z_2\\) and is denoted by \\(z_1\\over z_2\\).<\/p>\n

\n

Thus, \\(z_1\\over z_2\\) = \\(z_1.{z_2}^{-1}\\) = \\(z_1\\).(\\(1\\over z_2\\))<\/p>\n<\/blockquote>\n

How to Find Muliplicative Inverse :<\/strong><\/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

How to Divide Two Complex Numbers :<\/strong><\/p>\n

Let \\(z_1\\) = \\(a_1 + ib_1\\) and \\(z_2\\) = \\(a_2 + ib_2\\). Then<\/p>\n

\\(z_1\\over z_2\\) = (\\(a_1 + ib_1\\)){\\(a_2\\over {a_2}^2 + {b_2}^2\\) + \\(i{(-b_2)\\over {a_2}^2 + {b_2}^2}\\)}<\/p>\n

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

By definition of multiplication,<\/p>\n

\n

\\(z_1\\over z_2\\) = (\\(a_1a_2 + b_1b_2\\over {a_2}^2 + {b_2}^2\\)) + i(\\(a_2b_1 – a_1b_2\\over {a_2}^2 + {b_2}^2\\))<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : If \\(z_1\\) = 2 + 3i and \\(z_2\\) = 1 + 2i, then find \\(z_1\\over z_2\\).<\/p>\n

Solution<\/span><\/strong> : We have \\(z_1\\) = 2 + 3i and \\(z_2\\) = 1 + 2i,<\/p>\n

\\(\\implies\\) \\(1\\over z_2\\) = \\(1\\over 1 + 2i\\) = \\({1\\over 5} – {2\\over 5}i\\)<\/p>\n

Now, <\/p>\n

\\(z_1\\over z_2\\) = \\(z_1\\).\\(1\\over z_2\\) = (2 + 3i)( \\({1\\over 5} – {2\\over 5}i\\))<\/p>\n

= (\\({2\\over 5} + {6\\over 5}\\)) + i(\\({-4\\over 5} + {3\\over 5}\\)) = \\(8\\over 5\\) – \\(1\\over 5\\)i<\/p>\n\n\n

\n
Next – How to Find the Conjugate of a Complex Number<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Multiplication of Complex Numbers \u2013 Properties & Examples<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what is the division of complex numbers with examples. Let’s begin – Division of Complex Numbers The division of a complex number \\(z_1\\) by a non-zero complex number \\(z_2\\) is defined as the multiplication of \\(z_1\\) by the multiplicative inverse of \\(z_2\\) and is denoted by \\(z_1\\over z_2\\). Thus, \\(z_1\\over z_2\\) …<\/p>\n

Division of Complex Numbers<\/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":[229,240,241],"yoast_head":"\nDivision of Complex Numbers - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is the division of complex numbers 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\/division-of-complex-numbers\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Division of Complex Numbers - 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