{"id":1123,"date":"2021-05-03T19:00:45","date_gmt":"2021-05-03T19:00:45","guid":{"rendered":"https:\/\/konrad.burnik.org\/wordpress\/?p=1123"},"modified":"2024-06-30T19:47:32","modified_gmt":"2024-06-30T19:47:32","slug":"split-complex-numbers-as-clifford-algebras","status":"publish","type":"post","link":"https:\/\/konrad.burnik.org\/wordpress\/split-complex-numbers-as-clifford-algebras\/","title":{"rendered":"Split-Complex Numbers as a Clifford Algebra"},"content":{"rendered":"\n


The split-complex numbers are a special case of Clifford Algebra<\/a>. This can be seen as follows. Let \"X = \mathbb{R}^1\". Let \"e_1 = \langle a \rangle\" with \"a \not = 0\" be a basis element of \"\mathbb{R}^1\". Then the Clifford Algebra \"\mathcal{C}(\mathbb{R}^1)\" is defined as follows. The elements of the Clifford algebra are \"\alpha + \beta e_1\". Setting the rule for the Clifford product \"e_1 \vee e_1\" as
<\/a>

(1) <\/span>   <\/span>\"\begin{equation*}e_1 \vee e_1 = 1\end{equation*}\"<\/p>
yields

  <\/span>   <\/span>\"\[(\alpha_1 + \beta_1 e_1) \vee (\alpha_2 + \beta_2 e_1) = (\alpha_1 \alpha_2 + \beta_1 \beta_2) + (\alpha_1 \beta_2 + \beta_1 \alpha_2) e_1\]\"<\/p>
The isomorphism is \"1 \mapsto 1\", \"e_1 \mapsto j\". On the other hand, setting the rule to \"e_1 \vee e_1 = 0\" or \"e_1 \vee e_1 = -1\" in (1<\/a>) yields a correspondence with dual-numbers<\/a> or complex numbers<\/a> respectively. I am planning to expand on this fact in one of the subsequent posts.<\/p>\n\n\n\n

<\/p>\n

0<\/span><\/a><\/div> <\/div>
<\/div><\/div>
<\/div>","protected":false},"excerpt":{"rendered":"

The split-complex numbers are a special case of Clifford Algebra. This can be seen as follows. Let . Let with be a basis element of . Then the Clifford Algebra is defined as follows. The elements of the Clifford algebra are . Setting the rule for the Clifford product as (1)   yields     […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_mi_skip_tracking":false,"_exactmetrics_sitenote_active":false,"_exactmetrics_sitenote_note":"","_exactmetrics_sitenote_category":0,"_s2mail":"yes","footnotes":""},"categories":[9],"tags":[17,13],"class_list":["post-1123","post","type-post","status-publish","format-standard","hentry","category-hyperbolic-numbers","tag-clifford-algebra","tag-split-complex-numbers"],"_links":{"self":[{"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/posts\/1123","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/comments?post=1123"}],"version-history":[{"count":28,"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/posts\/1123\/revisions"}],"predecessor-version":[{"id":1248,"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/posts\/1123\/revisions\/1248"}],"wp:attachment":[{"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/media?parent=1123"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/categories?post=1123"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/tags?post=1123"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}