{"id":4235,"date":"2022-06-10T18:53:42","date_gmt":"2022-06-10T18:53:42","guid":{"rendered":"https:\/\/mdr.foobrdigital.com\/?p=4235"},"modified":"2022-06-10T18:53:42","modified_gmt":"2022-06-10T18:53:42","slug":"operations-on-complex-numbers","status":"publish","type":"post","link":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/2022\/06\/10\/operations-on-complex-numbers\/","title":{"rendered":"Operations on Complex Numbers"},"content":{"rendered":"\n<p>The various operations of addition, subtraction,\u00a0multiplication, division of natural numbers can also be performed for complex numbers also. The details of the various arithmetic operations of complex numbers are as follows.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Addition of Complex&nbsp;Numbers<\/h3>\n\n\n\n<p>Th addition of complex&nbsp;numbers is similar to the addition of natural numbers. Here in complex numbers, the real part is added to the real part and the imaginary part&nbsp;is added to the imaginary part. For two complex numbers of the form&nbsp;z1=a+idz1=a+id&nbsp;and&nbsp;z2=c+idz2=c+id, the sum of complex numbers&nbsp;z1+z2=(a+c)+i(b+d)z1+z2=(a+c)+i(b+d).&nbsp; The complex numbers follow all the following properties of addition.<\/p>\n\n\n\n<ul><li><strong>Closure Law:<\/strong>&nbsp;The sum of two complex numbers is also a complex number. For two complex numbers&nbsp;z1z1&nbsp;and&nbsp;z2z2, the sum of&nbsp;z1+z2z1+z2&nbsp;is also a complex number.<\/li><li><strong>Commutative Law:<\/strong>&nbsp;For two complex numbers&nbsp;z1z1,&nbsp;z2z2&nbsp;we have&nbsp;z1+z2=z2+z1z1+z2=z2+z1.<\/li><li><strong>Associative Law:<\/strong>&nbsp;For the given three complex numbers&nbsp;z1,z2,z3z1,z2,z3&nbsp;we have&nbsp;z1+(z2+z3)=(z1+z2)+z3z1+(z2+z3)=(z1+z2)+z3.<\/li><li><strong>Additive Identity:<\/strong>&nbsp;For a complex number z = a + ib, there exists 0 = 0 + i0, such that z + 0 = 0 + z = 0.&nbsp;<\/li><li><strong>Additive Inverse:<\/strong>&nbsp;For the complex number z = a + ib, there exists a complex number -z = -a -ib such that z + (-z) = (-z) + z = 0.&nbsp; Here -z is the additive inverse.<\/li><\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Subtraction of Complex Numbers<\/h3>\n\n\n\n<p>The subtraction of complex numbers follows a similar process of subtraction of natural numbers. Here for any two complex numbers, the subtraction is separately performed across the real part and then the subtraction is performed across the imaginary part. For the complex numbers&nbsp;z1z1&nbsp;= a + ib,&nbsp;z2=c+idz2=c+id, we have&nbsp;z1\u2212z2z1\u2212z2&nbsp;= (a &#8211; c) + i(b &#8211; d)<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Multiplication of Complex Numbers<\/h3>\n\n\n\n<p>The\u00a0multiplication of complex numbers\u00a0is slightly different from the multiplication of natural numbers. Here we need to use the formula of\u00a0i2=\u22121i2=\u22121.\u00a0 For the two complex numbers\u00a0z1z1\u00a0= a + ib,\u00a0z2z2\u00a0= c + id, the product is\u00a0z1.z2z1.z2\u00a0= (ca &#8211; bd) + i(ad + bc).\u00a0 \u00a0<\/p>\n\n\n\n<p>The multiplication of complex numbers is polar form is slightly different from the above mentioned form&nbsp;of multiplication. Here the absolute values of the two complex numbers are multiplied and their arguments are added to obtain the product of the complex numbers. For the complex numbers&nbsp;z1=r1(Cos\u03b81+iSin\u03b81)z1=r1(Cos\u03b81+iSin\u03b81),&nbsp;and&nbsp;<em>z<\/em><sub>2<\/sub>&nbsp;=&nbsp;z2=r1(Cos\u03b82+iSin\u03b82)z2=r1(Cos\u03b82+iSin\u03b82), the product of the complex numbers is&nbsp;z1.z2=r1.r2(Cos(\u03b81+\u03b82)+iSin(\u03b81+\u03b82))z1.z2=r1.r2(Cos(\u03b81+\u03b82)+iSin(\u03b81+\u03b82)).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Division of Complex Numbers<\/h3>\n\n\n\n<p>The division of complex numbers makes use of the formula of reciprocal of a complex number. For the two complex numbers&nbsp;z1z1&nbsp;= a + ib,&nbsp;z2z2&nbsp;= c + id, we have the division as&nbsp;z1z2=(a+ib)\u00d71(c+id)=(a+ib)\u00d7(c\u2212id)(c2+d2)z1z2=(a+ib)\u00d71(c+id)=(a+ib)\u00d7(c\u2212id)(c2+d2).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The various operations of addition, subtraction,\u00a0multiplication, division of natural numbers can also be performed for complex numbers also. The details of the various arithmetic operations of complex numbers are as follows. Addition of Complex&nbsp;Numbers Th addition of complex&nbsp;numbers is similar to the addition of natural numbers. Here in complex numbers, the real part is added [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[653],"tags":[],"_links":{"self":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/4235"}],"collection":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/comments?post=4235"}],"version-history":[{"count":0,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/4235\/revisions"}],"wp:attachment":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/media?parent=4235"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/categories?post=4235"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/tags?post=4235"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}