{"id":4233,"date":"2022-06-10T18:52:35","date_gmt":"2022-06-10T18:52:35","guid":{"rendered":"https:\/\/mdr.foobrdigital.com\/?p=4233"},"modified":"2022-06-10T18:52:35","modified_gmt":"2022-06-10T18:52:35","slug":"ordering-of-complex-numbers","status":"publish","type":"post","link":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/2022\/06\/10\/ordering-of-complex-numbers\/","title":{"rendered":"Ordering of Complex Numbers"},"content":{"rendered":"\n<p>The ordering of complex numbers is not possible. Real numbers and other related number systems can be ordered, but complex numbers cannot be ordered. The complex numbers do not have the structure of an ordered field, and there is no ordering of the complex numbers that are compatible with addition and multiplication. Also, the non-trivial sum of squares in an ordered field is a number&nbsp;\u22600\u22600, but in a complex number, the non-trivial sum of squares is equal to i<sup>2<\/sup>&nbsp;+ 1<sup>2<\/sup>&nbsp;= 0. The complex numbers can be measured and represented in a two-dimensional&nbsp;argrand plane by&nbsp;their magnitude, which is its&nbsp;distance from the origin.<\/p>\n\n\n\n<p><strong>Euler&#8217;s Formula:<\/strong>\u00a0As per\u00a0Euler&#8217;s formula\u00a0for any real value\u00a0\u03b8 we have e<sup>i\u03b8<\/sup>\u00a0= Cos\u03b8\u00a0+ iSin\u03b8, and it represents\u00a0the complex number in the coordinate plane where Cos\u03b8\u00a0is the real part and is represented with respect to the x-axis,\u00a0Sin\u03b8\u00a0is the imaginary part that is represented with respect to the y-axis,\u00a0\u03b8\u00a0is the angle made with respect to the x-axis and the imaginary line, which is connecting the origin and the complex number. As per Euler&#8217;s formula and for the functional representation of x and y we have\u00a0e<sup>x + iy<\/sup>\u00a0= e<sup>x<\/sup>(cosy + isiny) = e<sup>x<\/sup>cosy + ie<sup>x<\/sup>Siny. This decomposes the exponential function into its real and imaginary parts.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The ordering of complex numbers is not possible. Real numbers and other related number systems can be ordered, but complex numbers cannot be ordered. The complex numbers do not have the structure of an ordered field, and there is no ordering of the complex numbers that are compatible with addition and multiplication. Also, the non-trivial [&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\/4233"}],"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=4233"}],"version-history":[{"count":0,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/4233\/revisions"}],"wp:attachment":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/media?parent=4233"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/categories?post=4233"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/tags?post=4233"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}