{"id":443,"date":"2019-01-07T12:52:30","date_gmt":"2019-01-07T17:52:30","guid":{"rendered":"http:\/\/josmfs.net\/?p=443"},"modified":"2021-06-12T15:40:25","modified_gmt":"2021-06-12T19:40:25","slug":"complex-numbers-geometric-viewpoint","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2019\/01\/07\/complex-numbers-geometric-viewpoint\/","title":{"rendered":"Complex Numbers \u2013 Geometric Viewpoint"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft  wp-image-2247\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2021\/06\/Geometry-of-Complex-Numbers-Fig2.jpg\" alt=\"\" width=\"202\" height=\"147\" srcset=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2021\/06\/Geometry-of-Complex-Numbers-Fig2.jpg 403w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2021\/06\/Geometry-of-Complex-Numbers-Fig2-300x219.jpg 300w\" sizes=\"auto, (max-width: 202px) 100vw, 202px\" \/>This may be a futile attempt at an elementary introduction to complex variables by emphasizing their geometric properties. The elementary part is probably undermined by an initial discussion of field extensions and a necessary reference to trigonometry. Hopefully, the suppression of the explicit use of complex powers of Euler\u2019s constant e until the very end will allow the geometric ideas to have center stage. A primary goal of the essay is to realize that complex polynomials involve sums of circles in the plane. The image of real polynomials as wavy curves in the plane is misleading for an understanding of complex behavior. See <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2019\/01\/Geometry-of-Complex-Numbers-190107.pdf\">Complex Numbers \u2013 Geometric Viewpoint<\/a>.<\/p>\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>This may be a futile attempt at an elementary introduction to complex variables by emphasizing their geometric properties. The elementary part is probably undermined by an initial discussion of field extensions and a necessary reference to trigonometry. Hopefully, the suppression of the explicit use of complex powers of Euler\u2019s constant e until the very end [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[32,51,52,53,27,41,31],"class_list":["post-443","post","type-post","status-publish","format-standard","hentry","category-math-inquiries","tag-calculus","tag-complex-numbers","tag-fields","tag-leonhard-euler","tag-logarithm","tag-trigonometry","tag-vectors"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/443","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/comments?post=443"}],"version-history":[{"count":3,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/443\/revisions"}],"predecessor-version":[{"id":2248,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/443\/revisions\/2248"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=443"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=443"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=443"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}