{"id":910,"date":"2020-09-24T22:23:02","date_gmt":"2020-09-24T22:23:02","guid":{"rendered":"https:\/\/konrad.burnik.org\/wordpress\/?p=910"},"modified":"2020-12-19T20:40:35","modified_gmt":"2020-12-19T20:40:35","slug":"split-complex-numbers-revisited","status":"publish","type":"post","link":"https:\/\/konrad.burnik.org\/wordpress\/split-complex-numbers-revisited\/","title":{"rendered":"Split-complex numbers revisited"},"content":{"rendered":"\n

<\/p>\n\n\n\n

A split-complex number<\/em> is an algebraic expression of the form \"x + j y\" where \"x, y \in \mathbb{R}\" and \"j^2 = + 1\" where \"j \not = \pm 1\". We can write a split-complex number simply as a pair of real numbers \"(x,y)\". Adding two split-complex numbers when represented as pairs is done component-wise. Hence, given \"(a, b)\" and \"(c, d)\" addition is defined as<\/p>\n\n\n\n

  <\/span>   <\/span>\"\[(a,b) + (c,d) = (a+c, b+d).\]\"<\/p><\/p>\n\n\n\n

However, the product of two split-complex numbers is not defined component-wise since

  <\/span>   <\/span>\"\[(a,b) \cdot (c,d) = (ac + bd, ad + bc).\]\"<\/p><\/p>\n\n\n\n

One nice property of split-complex numbers is that we can change their representation in order to make multiplication component-wise as well.<\/p>\n\n\n\n

Let

  <\/span>   <\/span>\"\begin{align*}\iota = \frac{1 + j}{2} \ \mbox{ and } \ \iota^{*} = \frac{1 - j}{2}.\end{align*}\"<\/p><\/p>\n\n\n\n

Now each hyperbolic number \"z = a + j b\" can uniquely be written in terms of
\"{\iota, \iota^* }\" as follows.

  <\/span>   <\/span>\"\[z = (a + b) \iota + (a - b) \iota^{*}\]\"<\/p><\/p>\n\n\n\n

From here, the multiplication of two numbers \"\tilde{z_1} = a \iota + b \iota^{*}\" and \"\tilde{z_2} = c \iota + d\iota^{*}\" is now given component-wise by

  <\/span>   <\/span>\"\[ \tilde{z_1} \tilde{z_2} = a c \iota + b d \iota^{*}\]\"<\/p><\/p>\n\n\n\n

In this sense the transformed split-complex numbers algebraically behave just like two copies of real numbers since the algebraic operations are performed in each copy independently. More on this alternative representation of split-complex numbers in subsequent posts.<\/p>\n

0<\/span><\/a><\/div> <\/div>
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A split-complex number is an algebraic expression of the form where and where . We can write a split-complex number simply as a pair of real numbers . Adding two split-complex numbers when represented as pairs is done component-wise. Hence, given and addition is defined as     However, the product of two split-complex numbers […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","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":[],"class_list":["post-910","post","type-post","status-publish","format-standard","hentry","category-hyperbolic-numbers"],"_links":{"self":[{"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/posts\/910","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=910"}],"version-history":[{"count":27,"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/posts\/910\/revisions"}],"predecessor-version":[{"id":966,"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/posts\/910\/revisions\/966"}],"wp:attachment":[{"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/media?parent=910"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/categories?post=910"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/tags?post=910"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}