{"id":3171,"date":"2024-08-26T22:30:26","date_gmt":"2024-08-26T22:30:26","guid":{"rendered":"https:\/\/workhouse.sweetdishy.com\/?p=3171"},"modified":"2024-08-26T22:30:27","modified_gmt":"2024-08-26T22:30:27","slug":"operations-involving-cartesian-complex-numbers","status":"publish","type":"post","link":"https:\/\/workhouse.sweetdishy.com\/index.php\/2024\/08\/26\/operations-involving-cartesian-complex-numbers\/","title":{"rendered":"Operations Involving Cartesian Complex Numbers"},"content":{"rendered":"\n

(a) <\/a>Addition and subtraction<\/p>\n\n\n\n

\"image\"\/<\/figure>\n\n\n\n

and \"image\"<\/p>\n\n\n\n

Thus, \"image\"<\/p>\n\n\n\n

and \"image\"<\/p>\n\n\n\n

(b) <\/a>Multiplication<\/p>\n\n\n\n

\"image\"\/<\/figure>\n\n\n\n

But j<\/em>2<\/sup>=\u20131, thus,<\/p>\n\n\n\n

\"image\"\/<\/figure>\n\n\n\n

For example,<\/p>\n\n\n\n

\"image\"\/<\/figure>\n\n\n\n

(c) <\/a>Complex conjugate<\/p>\n\n\n\n

The complex conjugate<\/em> of (a<\/em>+jb<\/em>) is (a<\/em> \u2013jb<\/em>). For example, the conjugate of (3 \u2013j<\/em>2) is (3+j<\/em>2). The product of a complex number and its complex conjugate is always a real number, and this is an important property used when dividing complex numbers. Thus,<\/p>\n\n\n\n

\"image\"\/<\/figure>\n\n\n\n

For example, \"image\"<\/p>\n\n\n\n

and \"image\"<\/p>\n\n\n\n

(d) <\/a>Division<\/p>\n\n\n\n

The expression of one complex number divided by another, in the form a<\/em>+jb<\/em>, is accomplished by multiplying the numerator and denominator by the complex conjugate of the denominator. This has the effect of making the denominator a real number. For example,<\/p>\n\n\n\n

\"image\"\/<\/figure>\n\n\n\n

The elimination of the imaginary part of the denominator by multiplying both the numerator and denominator by the conjugate of the denominator is often termed rationalizing<\/em>.<\/p>\n\n\n\n

Example 7.1<\/h5>\n\n\n\n

In an electrical circuit the total impedance ZT<\/sub><\/em> is given by:<\/p>\n\n\n\n

\"image\"\/<\/figure>\n\n\n\n

Determine ZT<\/sub><\/em> in (a<\/em>+jb<\/em>) form, correct to two decimal places, when Z<\/em>1<\/sub>=5 \u2013j<\/em>3, Z<\/em>2<\/sub>=4+j<\/em>7 and Z<\/em>3<\/sub>=3.9 \u2013j<\/em>6.7.<\/p>\n\n\n\n

Solution<\/h5>\n\n\n\n
\"image\"\/<\/figure>\n\n\n\n
\"image\"\/<\/figure>\n\n\n\n

Thus, \"image\"<\/p>\n\n\n\n

=8.65 \u2013<\/strong>j<\/strong><\/em>6.26<\/strong>, correct to two decimal places.<\/p>\n\n\n\n

Example 7.2<\/h5>\n\n\n\n

Given Z<\/em>1<\/sub>=3+j<\/em>4 and Z<\/em>2<\/sub>=2 \u2013j<\/em>5 determine in Cartesian form correct to three decimal places:<\/p>\n\n\n\n

\"image\"\/<\/figure>\n\n\n\n
Solution<\/h5>\n\n\n\n
\"image\"\/<\/figure>\n\n\n\n
\"image\"\/<\/figure>\n\n\n\n
\"image\"\/<\/figure>\n\n\n\n
\"image\"\/<\/figure>\n","protected":false},"excerpt":{"rendered":"

(a) Addition and subtraction and  Thus,  and  (b) Multiplication But j2=\u20131, thus, For example, (c) Complex conjugate The complex conjugate of (a+jb) is (a \u2013jb). For example, the conjugate of (3 \u2013j2) is (3+j2). The product of a complex number and its complex conjugate is always a real number, and this is an important property used when dividing complex numbers. Thus, For […]<\/p>\n","protected":false},"author":1,"featured_media":3168,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[428],"tags":[],"class_list":["post-3171","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-complex-numbers"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/08\/mathematics.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3171","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/comments?post=3171"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3171\/revisions"}],"predecessor-version":[{"id":3172,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3171\/revisions\/3172"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/3168"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=3171"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=3171"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=3171"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}