{"id":42065,"date":"2025-10-26T12:23:50","date_gmt":"2025-10-26T06:53:50","guid":{"rendered":"https:\/\/tocxten.com\/?page_id=42065"},"modified":"2025-10-26T15:03:40","modified_gmt":"2025-10-26T09:33:40","slug":"complex-numbers","status":"publish","type":"page","link":"https:\/\/tocxten.com\/index.php\/complex-numbers\/","title":{"rendered":"Complex Numbers"},"content":{"rendered":"\n<p class=\"has-medium-font-size\">In classical mathematics, we deal primarily with real numbers, which are points on a one-dimensional number line. In quantum mechanics and quantum computing, we require complex numbers, which extend this system to two dimensions.<br><br>A complex number z is written as:  <strong>z = a + bi , <\/strong>where a is the real part, b is the imaginary part, and i is the imaginary unit (i\u00b2 = -1).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Geometric Representation<\/h3>\n\n\n\n<p class=\"has-medium-font-size\">Complex numbers can be visualized on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part. The magnitude or modulus of z is <strong>|z| = \u221a(a\u00b2 + b\u00b2)<\/strong> and the argument (angle) is <strong>\u03b8 = tan\u207b\u00b9(b\/a)<\/strong>. Using Euler\u2019s formula, z can be represented as <strong>z = re^(i\u03b8).<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Example<\/h3>\n\n\n\n<p class=\"has-medium-font-size\">Let z\u2081 = 1 + i and z\u2082 = 2 &#8211; i<br><strong>Addition<\/strong>: z\u2081 + z\u2082 = 3<br><strong>Multiplication: <\/strong>z\u2081z\u2082 = 3 + i<br><strong>Magnitude of z\u2081<\/strong>: |z\u2081| = \u221a2<br><strong>Polar form:<\/strong> z\u2081 = \u221a2 e^(i\u03c0\/4)<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Quantum Connection<\/h3>\n\n\n\n<p class=\"has-medium-font-size\">A qubit (quantum bit) is represented as |\u03c8\u27e9 = \u03b1|0\u27e9 + \u03b2|1\u27e9 where \u03b1, \u03b2 \u2208 \u2102 and |\u03b1|\u00b2 + |\u03b2|\u00b2 = 1.<br>This normalization ensures the total probability equals 1.<\/p>\n\n\n\n<p class=\"has-medium-font-size\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In classical mathematics, we deal primarily with real numbers, which are points on a one-dimensional number line. In quantum mechanics and quantum computing, we require complex numbers, which extend this&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":"","_links_to":"","_links_to_target":""},"class_list":["post-42065","page","type-page","status-publish","hentry"],"post_mailing_queue_ids":[],"_links":{"self":[{"href":"https:\/\/tocxten.com\/index.php\/wp-json\/wp\/v2\/pages\/42065","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/tocxten.com\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/tocxten.com\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/tocxten.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/tocxten.com\/index.php\/wp-json\/wp\/v2\/comments?post=42065"}],"version-history":[{"count":3,"href":"https:\/\/tocxten.com\/index.php\/wp-json\/wp\/v2\/pages\/42065\/revisions"}],"predecessor-version":[{"id":42071,"href":"https:\/\/tocxten.com\/index.php\/wp-json\/wp\/v2\/pages\/42065\/revisions\/42071"}],"wp:attachment":[{"href":"https:\/\/tocxten.com\/index.php\/wp-json\/wp\/v2\/media?parent=42065"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}