{"id":6511,"date":"2021-10-17T18:16:42","date_gmt":"2021-10-17T12:46:42","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6511"},"modified":"2022-01-16T17:06:32","modified_gmt":"2022-01-16T11:36:32","slug":"multiplication-of-complex-numbers","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/multiplication-of-complex-numbers\/","title":{"rendered":"Multiplication of Complex Numbers – Properties and Examples"},"content":{"rendered":"

Here you will learn multiplication of complex numbers and its properties with examples.<\/p>\n

Let’s begin –<\/p>\n

Multiplication of Complex Numbers<\/h2>\n

Let \\(z_1\\) = \\(a_1 + ib_1\\) and \\(z_2\\) = \\(a_2 + ib_2\\) be two complex numbers. Then the multiplication of \\(z_1\\) with \\(z_2\\) is denoted by \\(z_1 z_2\\) and is defined as the complex number<\/p>\n

\n

\\((a_1a_2 – b_1b_2)\\) + i\\((a_1b_2 + a_2b_1)\\).<\/p>\n

Thus,  \\(z_1\\)\\(z_2\\) = \\(a_1 + ib_1\\)\\(a_2 + ib_2\\)<\/p>\n

\\(\\implies\\)  \\(z_1\\)\\(z_2\\) = \\((a_1a_2 – b_1b_2)\\) + i\\((a_1b_2 + a_2b_1)\\)<\/p>\n

\\(\\implies\\)  \\((a_1a_2 – b_1b_2)\\) = [\\(Re(z_1) Re(z_2) – Im(z_1) Im(z_2)\\)] + i [\\(Re(z_1) Im(z_2) + Re(z_2) Im(z_1)\\)]<\/p>\n<\/blockquote>\n

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

\\(z_1 z_2\\) = (3 + 2i)(2 – 3i)<\/p>\n

= \\((3 \\times 2 – 2 \\times (-3)) + i(3 \\times -3 + 2 \\times 2)\\) = 12 – 5i<\/p>\n

Note<\/strong> : The product \\(z_1z_2\\) can also be obtained if we actually carry out the multiplication (\\(a_1 + ib_1\\))(\\(a_2 + ib_2\\)) as given below :<\/p>\n

(\\(a_1 + ib_1\\))(\\(a_2 + ib_2\\)) = \\(a_1a_2\\) + \\(ia_1b_2\\) + \\(ib_1a_2\\) + \\(i^2b_1b_2\\)<\/p>\n

= \\((a_1a_2 – b_1b_2)\\) + i\\((a_1b_2 + a_2b_1)\\)           [\\(because\\)  \\(i^2\\) = -1]<\/p>\n

Properties of Multiplication<\/h3>\n

(i) Multiplication is Commutative<\/strong> : For any two complex numbers \\(z_1\\) and \\(z_2\\), we have<\/p>\n

\n

\\(z_1 z_2\\) = \\(z_2 z_1\\)<\/p>\n<\/blockquote>\n

(ii) Multiplication is Associative<\/strong> : For any three complex numbers \\(z_1\\), \\(z_2\\), \\(z_3\\), we have<\/p>\n

\n

(\\(z_1\\) \\(z_2\\)) \\(z_3\\)  = \\(z_1\\) (\\(z_2\\) \\(z_3\\))<\/p>\n<\/blockquote>\n

(iii) Existence of Identity Element for Multiplication<\/strong> : The complex number 1 = 1 + i0 is the identity element for multiplication i.e. for every complex number z, we have<\/p>\n

\n

z.1 = z = 1.z<\/p>\n<\/blockquote>\n

(iv) Existence of Multiplicative Inverse<\/strong> : Corresponding to every non-zero complex number z = a + ib there exists a complex number \\(z_1\\) = x + iy such that<\/p>\n

\n

\\(z.z_1\\) = 1 = \\(z_1.z\\)<\/p>\n<\/blockquote>\n

(v) Multiplications of complex numbers is distributive over addition of complex numbers<\/strong> : For any three complex numbers \\(z_1\\), \\(z_2\\), \\(z_3\\), we have<\/p>\n

\n

(i) \\(z_1(z_2 + z_3)\\) = \\(z_1z_2 + z_1z_3\\)<\/p>\n

(ii) \\((z_2 + z_3)z_1\\) = \\(z_2z_1 + z_3z_1\\)<\/p>\n<\/blockquote>\n\n\n

\n
Next – Division of Complex Numbers<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Addition of Complex Numbers \u2013 Properties & Examples<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn multiplication of complex numbers and its properties with examples. Let’s begin – Multiplication of Complex Numbers Let \\(z_1\\) = \\(a_1 + ib_1\\) and \\(z_2\\) = \\(a_2 + ib_2\\) be two complex numbers. Then the multiplication of \\(z_1\\) with \\(z_2\\) is denoted by \\(z_1 z_2\\) and is defined as the complex number …<\/p>\n

Multiplication of Complex Numbers – Properties and Examples<\/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":[229,242,243],"yoast_head":"\nMultiplication of Complex Numbers - Properties and Examples<\/title>\n<meta name=\"description\" content=\"In this post you will learn multiplication of complex numbers and its properties 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\/multiplication-of-complex-numbers\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Multiplication of Complex Numbers - 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