Biquaternion - Wikipedia

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Biquaternion</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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href="https://es.wikipedia.org/wiki/Bicuaterni%C3%B3n" title="Bicuaternión – Spanish" lang="es" hreflang="es" data-title="Bicuaternión" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Biquaternion" title="Biquaternion – French" lang="fr" hreflang="fr" data-title="Biquaternion" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bikuaternion" title="Bikuaternion – Indonesian" lang="id" hreflang="id" data-title="Bikuaternion" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Biquaternium" title="Biquaternium – Latin" lang="la" hreflang="la" data-title="Biquaternium" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Bikwaterniony" title="Bikwaterniony – Polish" lang="pl" hreflang="pl" data-title="Bikwaterniony" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%91%D0%B8%D0%BA%D0%B2%D0%B0%D1%82%D0%B5%D1%80%D0%BD%D0%B8%D0%BE%D0%BD" title="Бикватернион – Russian" lang="ru" hreflang="ru" data-title="Бикватернион" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Bikvaternion" title="Bikvaternion – Slovenian" lang="sl" hreflang="sl" data-title="Bikvaternion" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Bid%C3%B6rdey" title="Bidördey – Turkish" lang="tr" hreflang="tr" data-title="Bidördey" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%91%D1%96%D0%BA%D0%B2%D0%B0%D1%82%D0%B5%D1%80%D0%BD%D1%96%D0%BE%D0%BD%D0%B8" title="Бікватерніони – Ukrainian" lang="uk" hreflang="uk" data-title="Бікватерніони" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Quaternions with complex number coefficients</div> <p>In <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>, the <b>biquaternions</b> are the numbers <span class="texhtml"><i>w</i> + <i>x</i> <b>i</b> + <i>y</i> <b>j</b> + <i>z</i> <b>k</b></span>, where <span class="texhtml"><i>w</i>, <i>x</i>, <i>y</i></span>, and <span class="texhtml mvar" style="font-style:italic;">z</span> are <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, or variants thereof, and the elements of <span class="texhtml">{<b>1</b>, <b>i</b>, <b>j</b>, <b>k</b>}</span> multiply as in the <a href="/wiki/Quaternion_group" title="Quaternion group">quaternion group</a> and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof: </p> <ul><li>Biquaternions when the coefficients are <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>.</li> <li><a href="/wiki/Split-biquaternion" title="Split-biquaternion">Split-biquaternions</a> when the coefficients are <a href="/wiki/Split-complex_number" title="Split-complex number">split-complex numbers</a>.</li> <li><a href="/wiki/Dual_quaternion" title="Dual quaternion">Dual quaternions</a> when the coefficients are <a href="/wiki/Dual_numbers" class="mw-redirect" title="Dual numbers">dual numbers</a>.</li></ul> <p>This article is about the <i>ordinary biquaternions</i> named by <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a> in 1844.<sup id="cite_ref-FOOTNOTEHamilton1850_1-0" class="reference"><a href="#cite_note-FOOTNOTEHamilton1850-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Some of the more prominent proponents of these biquaternions include <a href="/wiki/Alexander_Macfarlane" title="Alexander Macfarlane">Alexander Macfarlane</a>, <a href="/wiki/Arthur_W._Conway" title="Arthur W. Conway">Arthur W. Conway</a>, <a href="/wiki/Ludwik_Silberstein" title="Ludwik Silberstein">Ludwik Silberstein</a>, and <a href="/wiki/Cornelius_Lanczos" title="Cornelius Lanczos">Cornelius Lanczos</a>. As developed below, the unit <a href="/wiki/Quasi-sphere" title="Quasi-sphere">quasi-sphere</a> of the biquaternions provides a representation of the <a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a>, which is the foundation of <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>. </p><p>The algebra of biquaternions can be considered as a <a href="/wiki/Tensor_product_of_algebras" title="Tensor product of algebras">tensor product</a> <span class="texhtml"><b>C</b> ⊗<sub><b>R</b></sub> <b>H</b></span>, where <span class="texhtml"><b>C</b></span> is the <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> of complex numbers and <span class="texhtml"><b>H</b></span> is the <a href="/wiki/Division_algebra" title="Division algebra">division algebra</a> of (real) <a href="/wiki/Quaternions" class="mw-redirect" title="Quaternions">quaternions</a>. In other words, the biquaternions are just the <a href="/wiki/Complexification" title="Complexification">complexification</a> of the quaternions. Viewed as a complex algebra, the biquaternions are isomorphic to the algebra of <span class="texhtml">2 × 2</span> complex matrices <span class="texhtml">M<sub>2</sub>(<b>C</b>)</span>. They are also isomorphic to several <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebras</a> including <span class="texhtml"><span class="texhtml"><b>C</b> ⊗<sub><b>R</b></sub> <b>H</b></span> = Cl<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:0.9em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">[0]</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">3</sub></span></span>(<b>C</b>) = Cl<sub>2</sub>(<b>C</b>) = Cl<sub>1,2</sub>(<b>R</b>)</span>,<sup id="cite_ref-FOOTNOTEGarling2011112,_113_2-0" class="reference"><a href="#cite_note-FOOTNOTEGarling2011112,_113-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> the <a href="/wiki/Pauli_algebra" class="mw-redirect" title="Pauli algebra">Pauli algebra</a> <span class="texhtml">Cl<sub>3,0</sub>(<b>R</b>)</span>,<sup id="cite_ref-FOOTNOTEGarling2011112_3-0" class="reference"><a href="#cite_note-FOOTNOTEGarling2011112-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEFrancisKosowsky2005404_4-0" class="reference"><a href="#cite_note-FOOTNOTEFrancisKosowsky2005404-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and the even part <span class="texhtml">Cl<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:0.9em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">[0]</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">1,3</sub></span></span>(<b>R</b>) = Cl<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:0.9em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">[0]</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">3,1</sub></span></span>(<b>R</b>)</span> of the <a href="/wiki/Spacetime_algebra" title="Spacetime algebra">spacetime algebra</a>.<sup id="cite_ref-FOOTNOTEFrancisKosowsky2005386_5-0" class="reference"><a href="#cite_note-FOOTNOTEFrancisKosowsky2005386-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Biquaternion&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="texhtml">{<b>1</b>, <b>i</b>, <b>j</b>, <b>k</b>}</span> be the basis for the (real) <a href="/wiki/Quaternion" title="Quaternion">quaternions</a> <span class="texhtml"><b>H</b></span>, and let <span class="texhtml"><i>u</i>, <i>v</i>, <i>w</i>, <i>x</i></span> be complex numbers, then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=u\mathbf {1} +v\mathbf {i} +w\mathbf {j} +x\mathbf {k} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mo>+</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>+</mo> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>+</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=u\mathbf {1} +v\mathbf {i} +w\mathbf {j} +x\mathbf {k} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5147d89b35452d618683f2653f3dbf6df233e9fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.446ex; height:2.509ex;" alt="{\displaystyle q=u\mathbf {1} +v\mathbf {i} +w\mathbf {j} +x\mathbf {k} }"></span></dd></dl> <p>is a <i>biquaternion</i>.<sup id="cite_ref-FOOTNOTEHamilton1853639_6-0" class="reference"><a href="#cite_note-FOOTNOTEHamilton1853639-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> To distinguish square roots of minus one in the biquaternions, Hamilton<sup id="cite_ref-FOOTNOTEHamilton1853730_7-0" class="reference"><a href="#cite_note-FOOTNOTEHamilton1853730-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEHamilton1866289_8-0" class="reference"><a href="#cite_note-FOOTNOTEHamilton1866289-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Arthur_W._Conway" title="Arthur W. Conway">Arthur W. Conway</a> used the convention of representing the square root of minus one in the scalar field <span class="texhtml"><b>C</b></span> by <span class="texhtml"><i>h</i></span> to avoid confusion with the <span class="texhtml"><b>i</b></span> in the quaternion group. <a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">Commutativity</a> of the scalar field with the quaternion group is assumed: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\mathbf {i} =\mathbf {i} h,\ \ h\mathbf {j} =\mathbf {j} h,\ \ h\mathbf {k} =\mathbf {k} h.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mi>h</mi> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mi>h</mi> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mi>h</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\mathbf {i} =\mathbf {i} h,\ \ h\mathbf {j} =\mathbf {j} h,\ \ h\mathbf {k} =\mathbf {k} h.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68b75bbc05bea4f3b30702d8da8059608f641943" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.305ex; height:2.509ex;" alt="{\displaystyle h\mathbf {i} =\mathbf {i} h,\ \ h\mathbf {j} =\mathbf {j} h,\ \ h\mathbf {k} =\mathbf {k} h.}"></span></dd></dl> <p>Hamilton introduced the terms <i><a href="/wiki/Bivector_(complex)" title="Bivector (complex)">bivector</a></i>, <i>biconjugate</i>, <i>bitensor</i>, and <i>biversor</i> to extend notions used with real quaternions <span class="texhtml"><b>H</b></span>. </p><p>Hamilton's primary exposition on biquaternions came in 1853 in his <i>Lectures on Quaternions</i>. The editions of <i>Elements of Quaternions</i>, in 1866 by <a href="/wiki/William_Edwin_Hamilton" title="William Edwin Hamilton">William Edwin Hamilton</a> (son of Rowan), and in 1899, 1901 by <a href="/wiki/Charles_Jasper_Joly" title="Charles Jasper Joly">Charles Jasper Joly</a>, reduced the biquaternion coverage in favour of the real quaternions. </p><p>Considered with the operations of component-wise addition, and multiplication according to the quaternion group, this collection forms a <a href="/wiki/Four-dimensional_space" title="Four-dimensional space">4-dimensional</a> <a href="/wiki/Algebra_over_a_field" title="Algebra over a field">algebra</a> over the complex numbers <span class="texhtml"><b>C</b></span>. The algebra of biquaternions is <a href="/wiki/Associative" class="mw-redirect" title="Associative">associative</a>, but not <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a>. A biquaternion is either a <a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">unit</a> or a <a href="/wiki/Zero_divisor" title="Zero divisor">zero divisor</a>. The algebra of biquaternions forms a <a href="/wiki/Composition_algebra" title="Composition algebra">composition algebra</a> and can be constructed from <a href="/wiki/Bicomplex_number" title="Bicomplex number">bicomplex numbers</a>. See <i><a href="#As_a_composition_algebra">§ As a composition algebra</a></i> below. </p> <div class="mw-heading mw-heading2"><h2 id="Place_in_ring_theory">Place in ring theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Biquaternion&action=edit&section=2" title="Edit section: Place in ring theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Linear_representation">Linear representation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Biquaternion&action=edit&section=3" title="Edit section: Linear representation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Note that the <a href="/wiki/Matrix_product" class="mw-redirect" title="Matrix product">matrix product</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}h&0\\0&-h\end{pmatrix}}{\begin{pmatrix}0&1\\-1&0\end{pmatrix}}={\begin{pmatrix}0&h\\h&0\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>h</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>h</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>h</mi> </mtd> </mtr> <mtr> <mtd> <mi>h</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}h&0\\0&-h\end{pmatrix}}{\begin{pmatrix}0&1\\-1&0\end{pmatrix}}={\begin{pmatrix}0&h\\h&0\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88628533666111f73d77d8c8e9e97a3ecd0a5040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.881ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}h&0\\0&-h\end{pmatrix}}{\begin{pmatrix}0&1\\-1&0\end{pmatrix}}={\begin{pmatrix}0&h\\h&0\end{pmatrix}}}"></span>.</dd></dl> <p>Because <span class="texhtml"><i>h</i></span> is the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>, each of these three arrays has a square equal to the negative of the <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a>. When this matrix product is interpreted as <span class="texhtml"><b>i</b> <b>j</b> = <b>k</b></span>, then one obtains a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of matrices that is <a href="/wiki/Isomorphism" title="Isomorphism">isomorphic</a> to the <a href="/wiki/Quaternion_group" title="Quaternion group">quaternion group</a>. Consequently, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}u+hv&w+hx\\-w+hx&u-hv\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>u</mi> <mo>+</mo> <mi>h</mi> <mi>v</mi> </mtd> <mtd> <mi>w</mi> <mo>+</mo> <mi>h</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>w</mi> <mo>+</mo> <mi>h</mi> <mi>x</mi> </mtd> <mtd> <mi>u</mi> <mo>−</mo> <mi>h</mi> <mi>v</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}u+hv&w+hx\\-w+hx&u-hv\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88ef1c2d61a67b466430a8ccf757890e01188962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.65ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}u+hv&w+hx\\-w+hx&u-hv\end{pmatrix}}}"></span></dd></dl> <p>represents biquaternion <span class="texhtml"><i>q</i> = <i>u</i> <b>1</b> + <i>v</i> <b>i</b> + <i>w</i> <b>j</b> + <i>x</i> <b>k</b></span>. Given any <span class="texhtml">2 × 2</span> complex matrix, there are complex values <span class="texhtml"><i>u</i></span>, <span class="texhtml"><i>v</i></span>, <span class="texhtml"><i>w</i></span>, and <span class="texhtml"><i>x</i></span> to put it in this form so that the <a href="/wiki/Matrix_ring" title="Matrix ring">matrix ring</a> <span class="texhtml">M(2, <b>C</b>)</span> is isomorphic<sup id="cite_ref-FOOTNOTEDickson191413_9-0" class="reference"><a href="#cite_note-FOOTNOTEDickson191413-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> to the biquaternion <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Subalgebras">Subalgebras</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Biquaternion&action=edit&section=4" title="Edit section: Subalgebras"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Considering the biquaternion algebra over the scalar field of real numbers <span class="texhtml"><b>R</b></span>, the set </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\mathbf {1} ,h,\mathbf {i} ,h\mathbf {i} ,\mathbf {j} ,h\mathbf {j} ,\mathbf {k} ,h\mathbf {k} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mo>,</mo> <mi>h</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>,</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>,</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>,</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\mathbf {1} ,h,\mathbf {i} ,h\mathbf {i} ,\mathbf {j} ,h\mathbf {j} ,\mathbf {k} ,h\mathbf {k} \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55162bfe71a1ea40b223f1bfe583848b5c14aa6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.194ex; height:2.843ex;" alt="{\displaystyle \{\mathbf {1} ,h,\mathbf {i} ,h\mathbf {i} ,\mathbf {j} ,h\mathbf {j} ,\mathbf {k} ,h\mathbf {k} \}}"></span></dd></dl> <p>forms a <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> so the algebra has eight real <a href="/wiki/Dimension" title="Dimension">dimensions</a>. The squares of the elements <span class="texhtml"><i>h</i><b>i</b>, <i>h</i><b>j</b></span>, and <span class="texhtml"><i>h</i><b>k</b></span> are all positive one, for example, <span class="texhtml">(<i>h</i><b>i</b>)<sup>2</sup> = <i>h</i><sup>2</sup><b>i</b><sup>2</sup> = (−<b>1</b>)(−<b>1</b>) = +<b>1</b></span>. </p><p>The <a href="/wiki/Subalgebra" title="Subalgebra">subalgebra</a> given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x+y(h\mathbf {i} ):x,y\in \mathbb {R} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo stretchy="false">)</mo> <mo>:</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x+y(h\mathbf {i} ):x,y\in \mathbb {R} \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c257b60b4ed93139fc7a992ea13d5d7c3222d972" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.516ex; height:2.843ex;" alt="{\displaystyle \{x+y(h\mathbf {i} ):x,y\in \mathbb {R} \}}"></span></dd></dl> <p>is <a href="/wiki/Ring_isomorphism" class="mw-redirect" title="Ring isomorphism">ring isomorphic</a> to the plane of <a href="/wiki/Split-complex_number" title="Split-complex number">split-complex numbers</a>, which has an algebraic structure built upon the <a href="/wiki/Unit_hyperbola" title="Unit hyperbola">unit hyperbola</a>. The elements <span class="texhtml"><i>h</i><b>j</b></span> and <span class="texhtml"><i>h</i><b>k</b></span> also determine such subalgebras. </p><p>Furthermore, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x+y\mathbf {j} :x,y\in \mathbb {C} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>:</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x+y\mathbf {j} :x,y\in \mathbb {C} \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/478da1d51fa4b3e8f6d4d6d83e73bcf340b7cf1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.442ex; height:2.843ex;" alt="{\displaystyle \{x+y\mathbf {j} :x,y\in \mathbb {C} \}}"></span></dd></dl> <p>is a subalgebra isomorphic to the bicomplex numbers. </p><p>A third subalgebra called <a href="/wiki/Coquaternion" class="mw-redirect" title="Coquaternion">coquaternions</a> is generated by <span class="texhtml"><i>h</i><b>j</b></span> and <span class="texhtml"><i>h</i><b>k</b></span>. It is seen that <span class="texhtml">(<i>h</i><b>j</b>)(<i>h</i><b>k</b>) = (−<b>1</b>)<b>i</b></span>, and that the square of this element is <span class="texhtml">−<b>1</b></span>. These elements generate the <a href="/wiki/Dihedral_group" title="Dihedral group">dihedral group</a> of the square. The <a href="/wiki/Linear_subspace" title="Linear subspace">linear subspace</a> with basis <span class="texhtml">{<b>1</b>, <b>i</b>, <i>h</i><b>j</b>, <i>h</i><b>k</b>}</span> thus is closed under multiplication, and forms the coquaternion algebra. </p><p>In the context of <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> and <a href="/wiki/Spinor" title="Spinor">spinor</a> algebra, the biquaternions <span class="texhtml"><i>h</i><b>i</b>, <i>h</i><b>j</b></span>, and <span class="texhtml"><i>h</i><b>k</b></span> (or their negatives), viewed in the <span class="texhtml">M<sub>2</sub>(<b>C</b>)</span> representation, are called <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Algebraic_properties">Algebraic properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Biquaternion&action=edit&section=5" title="Edit section: Algebraic properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The biquaternions have two <i>conjugations</i>: </p> <ul><li>the <b>biconjugate</b> or biscalar minus <a href="/wiki/Bivector_(complex)" title="Bivector (complex)">bivector</a> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q^{*}=w-x\mathbf {i} -y\mathbf {j} -z\mathbf {k} \!\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mi>w</mi> <mo>−</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>−</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>−</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mspace width="negativethinmathspace" /> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q^{*}=w-x\mathbf {i} -y\mathbf {j} -z\mathbf {k} \!\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/655525a74a3ce1a6599e0ff626e2a29a342c632a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.801ex; height:2.676ex;" alt="{\displaystyle q^{*}=w-x\mathbf {i} -y\mathbf {j} -z\mathbf {k} \!\ ,}"></span> and</li> <li>the <a href="/wiki/Complex_conjugation" class="mw-redirect" title="Complex conjugation">complex conjugation</a> of biquaternion coefficients <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {q}}={\bar {w}}+{\bar {x}}\mathbf {i} +{\bar {y}}\mathbf {j} +{\bar {z}}\mathbf {k} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {q}}={\bar {w}}+{\bar {x}}\mathbf {i} +{\bar {y}}\mathbf {j} +{\bar {z}}\mathbf {k} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed161325b4b12610025454145892202ee729d366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.558ex; height:2.509ex;" alt="{\displaystyle {\bar {q}}={\bar {w}}+{\bar {x}}\mathbf {i} +{\bar {y}}\mathbf {j} +{\bar {z}}\mathbf {k} }"></span></li></ul> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {z}}=a-bh}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {z}}=a-bh}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6cb2b17f2771f40fc1e53ad1dae10f8a51b29b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.801ex; height:2.343ex;" alt="{\displaystyle {\bar {z}}=a-bh}"></span> when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=a+bh,\quad a,b\in \mathbb {R} ,\quad h^{2}=-\mathbf {1} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>h</mi> <mo>,</mo> <mspace width="1em" /> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mspace width="1em" /> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=a+bh,\quad a,b\in \mathbb {R} ,\quad h^{2}=-\mathbf {1} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d5c053c24c4eb64a136bb00aafcdf34254442" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:34.37ex; height:3.009ex;" alt="{\displaystyle z=a+bh,\quad a,b\in \mathbb {R} ,\quad h^{2}=-\mathbf {1} .}"></span> </p><p>Note that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (pq)^{*}=q^{*}p^{*},\quad {\overline {pq}}={\bar {p}}{\bar {q}},\quad {\overline {q^{*}}}={\bar {q}}^{*}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>p</mi> <mi>q</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mover> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (pq)^{*}=q^{*}p^{*},\quad {\overline {pq}}={\bar {p}}{\bar {q}},\quad {\overline {q^{*}}}={\bar {q}}^{*}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce63a541eb4d7c23a1ae8647e981f928dee0de4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.907ex; height:3.509ex;" alt="{\displaystyle (pq)^{*}=q^{*}p^{*},\quad {\overline {pq}}={\bar {p}}{\bar {q}},\quad {\overline {q^{*}}}={\bar {q}}^{*}.}"></span> </p><p>Clearly, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle qq^{*}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle qq^{*}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc76cd6616fd2298aed25c5ac77a01766d7270e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.464ex; height:2.676ex;" alt="{\displaystyle qq^{*}=0}"></span> then <span class="texhtml"><i>q</i></span> is a zero divisor. Otherwise <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lbrace qq^{*}\rbrace ^{-\mathbf {1} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>q</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lbrace qq^{*}\rbrace ^{-\mathbf {1} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4092421ff68b45cd413be7b18b80bfd2ff92088" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.984ex; height:3.176ex;" alt="{\displaystyle \lbrace qq^{*}\rbrace ^{-\mathbf {1} }}"></span> is a complex number. Further, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle qq^{*}=q^{*}q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle qq^{*}=q^{*}q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c30afe84888031fa8845de169a3aa009280b5c09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.505ex; height:2.676ex;" alt="{\displaystyle qq^{*}=q^{*}q}"></span> is easily verified. This allows the inverse to be defined by </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q^{-1}=q^{*}\lbrace qq^{*}\rbrace ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo fence="false" stretchy="false">{</mo> <mi>q</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q^{-1}=q^{*}\lbrace qq^{*}\rbrace ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/081ef4fbac9dc1bf91dbab1dda6b2292c60cde21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.505ex; height:3.176ex;" alt="{\displaystyle q^{-1}=q^{*}\lbrace qq^{*}\rbrace ^{-1}}"></span>, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle qq^{*}\neq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>≠</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle qq^{*}\neq 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a362fd45268b4e9efb4caa807ba4db948227aa5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.111ex; height:2.843ex;" alt="{\displaystyle qq^{*}\neq 0.}"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Relation_to_Lorentz_transformations">Relation to Lorentz transformations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Biquaternion&action=edit&section=6" title="Edit section: Relation to Lorentz transformations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/History_of_Lorentz_transformations#Lorentz_transformation_via_quaternions_and_hyperbolic_numbers" title="History of Lorentz transformations">Lorentz transformation via quaternions and hyperbolic numbers</a> and <a href="/wiki/History_of_Lorentz_transformations#Noether" title="History of Lorentz transformations">Relativistic biquaternions by Noether (1910), Klein (1910), Conway (1911), Silberstein (1911)</a></div> <p>Consider now the linear subspace<sup id="cite_ref-FOOTNOTELanczos1949See_equation_94.16,_page_305._The_following_algebra_compares_to_Lanczos,_except_he_uses_~_to_signify_quaternion_conjugation_and_*_for_complex_conjugation_10-0" class="reference"><a href="#cite_note-FOOTNOTELanczos1949See_equation_94.16,_page_305._The_following_algebra_compares_to_Lanczos,_except_he_uses_~_to_signify_quaternion_conjugation_and_*_for_complex_conjugation-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=\lbrace q\colon q^{*}={\bar {q}}\rbrace =\lbrace t+x(h\mathbf {i} )+y(h\mathbf {j} )+z(h\mathbf {k} )\colon t,x,y,z\in \mathbb {R} \rbrace .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>q</mi> <mo>:</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>t</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>z</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo stretchy="false">)</mo> <mo>:</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=\lbrace q\colon q^{*}={\bar {q}}\rbrace =\lbrace t+x(h\mathbf {i} )+y(h\mathbf {j} )+z(h\mathbf {k} )\colon t,x,y,z\in \mathbb {R} \rbrace .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c039daeedb77da4e02bc8677a795b07940c6b469" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:61.064ex; height:2.843ex;" alt="{\displaystyle M=\lbrace q\colon q^{*}={\bar {q}}\rbrace =\lbrace t+x(h\mathbf {i} )+y(h\mathbf {j} )+z(h\mathbf {k} )\colon t,x,y,z\in \mathbb {R} \rbrace .}"></span></dd></dl> <p><span class="texhtml"><i>M</i></span> is not a subalgebra since it is not <a href="/wiki/Closure_(mathematics)" title="Closure (mathematics)">closed under products</a>; for example <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (h\mathbf {i} )(h\mathbf {j} )=h^{2}\mathbf {ij} =-\mathbf {k} \notin M.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> <mi mathvariant="bold">j</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>∉</mo> <mi>M</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (h\mathbf {i} )(h\mathbf {j} )=h^{2}\mathbf {ij} =-\mathbf {k} \notin M.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f12def42e612941225c78f887471040822470daf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.152ex; height:3.176ex;" alt="{\displaystyle (h\mathbf {i} )(h\mathbf {j} )=h^{2}\mathbf {ij} =-\mathbf {k} \notin M.}"></span> Indeed, <span class="texhtml"><i>M</i></span> cannot form an algebra if it is not even a <a href="/wiki/Magma_(algebra)" title="Magma (algebra)">magma</a>. </p><p><b>Proposition:</b> If <span class="texhtml mvar" style="font-style:italic;">q</span> is in <span class="texhtml mvar" style="font-style:italic;">M</span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle qq^{*}=t^{2}-x^{2}-y^{2}-z^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle qq^{*}=t^{2}-x^{2}-y^{2}-z^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c5c26fa8261ed002b40dda2a777c65548f04a1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.107ex; height:3.009ex;" alt="{\displaystyle qq^{*}=t^{2}-x^{2}-y^{2}-z^{2}.}"></span> </p><p>Proof: From the definitions, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}qq^{*}&=(t+xh\mathbf {i} +yh\mathbf {j} +zh\mathbf {k} )(t-xh\mathbf {i} -yh\mathbf {j} -zh\mathbf {k} )\\&=t^{2}-x^{2}(h\mathbf {i} )^{2}-y^{2}(h\mathbf {j} )^{2}-z^{2}(h\mathbf {k} )^{2}\\&=t^{2}-x^{2}-y^{2}-z^{2}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>q</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>x</mi> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>+</mo> <mi>y</mi> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>+</mo> <mi>z</mi> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>x</mi> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>−</mo> <mi>y</mi> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>−</mo> <mi>z</mi> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}qq^{*}&=(t+xh\mathbf {i} +yh\mathbf {j} +zh\mathbf {k} )(t-xh\mathbf {i} -yh\mathbf {j} -zh\mathbf {k} )\\&=t^{2}-x^{2}(h\mathbf {i} )^{2}-y^{2}(h\mathbf {j} )^{2}-z^{2}(h\mathbf {k} )^{2}\\&=t^{2}-x^{2}-y^{2}-z^{2}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eaf309413966d9f6b2fa91605bb288d2d552a21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:50.513ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}qq^{*}&=(t+xh\mathbf {i} +yh\mathbf {j} +zh\mathbf {k} )(t-xh\mathbf {i} -yh\mathbf {j} -zh\mathbf {k} )\\&=t^{2}-x^{2}(h\mathbf {i} )^{2}-y^{2}(h\mathbf {j} )^{2}-z^{2}(h\mathbf {k} )^{2}\\&=t^{2}-x^{2}-y^{2}-z^{2}.\end{aligned}}}"></span></dd></dl> <p><b>Definition:</b> Let biquaternion <span class="texhtml mvar" style="font-style:italic;">g</span> satisfy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle gg^{*}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle gg^{*}=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35778deed0affd0317b053420340cbd2eea91045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.196ex; height:2.676ex;" alt="{\displaystyle gg^{*}=1.}"></span> Then the <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a> associated with <span class="texhtml mvar" style="font-style:italic;">g</span> is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(q)=g^{*}q{\bar {g}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(q)=g^{*}q{\bar {g}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8876300edb4a0f766d40b926106c31488f62a4d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.734ex; height:2.843ex;" alt="{\displaystyle T(q)=g^{*}q{\bar {g}}.}"></span></dd></dl> <p><b>Proposition:</b> If <span class="texhtml mvar" style="font-style:italic;">q</span> is in <span class="texhtml mvar" style="font-style:italic;">M</span>, then <span class="texhtml"><i>T</i>(<i>q</i>)</span> is also in <span class="texhtml"><i>M</i></span>. </p><p>Proof: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g^{*}q{\bar {g}})^{*}={\bar {g}}^{*}q^{*}g={\overline {g^{*}}}{\bar {q}}g={\overline {g^{*}q{\bar {g}})}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>g</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mi>g</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (g^{*}q{\bar {g}})^{*}={\bar {g}}^{*}q^{*}g={\overline {g^{*}}}{\bar {q}}g={\overline {g^{*}q{\bar {g}})}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/752c04fd0585729a331a780a4ae2a8cf5ef0c70a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.089ex; height:3.676ex;" alt="{\displaystyle (g^{*}q{\bar {g}})^{*}={\bar {g}}^{*}q^{*}g={\overline {g^{*}}}{\bar {q}}g={\overline {g^{*}q{\bar {g}})}}.}"></span> </p><p><b>Proposition:</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad T(q)(T(q))^{*}=qq^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>T</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mi>q</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad T(q)(T(q))^{*}=qq^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/614f60443d5e1472766ebdac02ae34e86aa8ab16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.518ex; height:2.843ex;" alt="{\displaystyle \quad T(q)(T(q))^{*}=qq^{*}}"></span> </p><p>Proof: Note first that <i>gg</i>* = 1 implies that the sum of the squares of its four complex components is one. Then the sum of the squares of the <i>complex conjugates</i> of these components is also one. Therefore, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {g}}({\bar {g}})^{*}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {g}}({\bar {g}})^{*}=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f56b1e1fd1a59b28a7140da80ffe6d817d129a6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.235ex; height:2.843ex;" alt="{\displaystyle {\bar {g}}({\bar {g}})^{*}=1.}"></span> Now </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g^{*}q{\bar {g}})(g^{*}q{\bar {g}})^{*}=g^{*}q({\bar {g}}{\bar {g}}^{*})q^{*}g=g^{*}qq^{*}g=qq^{*}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>q</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">)</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>g</mi> <mo>=</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>q</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>g</mi> <mo>=</mo> <mi>q</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (g^{*}q{\bar {g}})(g^{*}q{\bar {g}})^{*}=g^{*}q({\bar {g}}{\bar {g}}^{*})q^{*}g=g^{*}qq^{*}g=qq^{*}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c86b23d143c23ec32e2f28d535df64bb127560b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.076ex; height:2.843ex;" alt="{\displaystyle (g^{*}q{\bar {g}})(g^{*}q{\bar {g}})^{*}=g^{*}q({\bar {g}}{\bar {g}}^{*})q^{*}g=g^{*}qq^{*}g=qq^{*}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Associated_terminology">Associated terminology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Biquaternion&action=edit&section=7" title="Edit section: Associated terminology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As the biquaternions have been a fixture of <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a> since the beginnings of <a href="/wiki/Mathematical_physics" title="Mathematical physics">mathematical physics</a>, there is an array of concepts that are illustrated or represented by biquaternion algebra. The <a href="/wiki/Transformation_group" class="mw-redirect" title="Transformation group">transformation group</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=\lbrace g:gg^{*}=1\rbrace }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>g</mi> <mo>:</mo> <mi>g</mi> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=\lbrace g:gg^{*}=1\rbrace }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b46dca2607c70b3de9bfd34c8d6d6a8f95d195b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.853ex; height:2.843ex;" alt="{\displaystyle G=\lbrace g:gg^{*}=1\rbrace }"></span> has two parts, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\cap H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>∩</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\cap H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/927a14c2f8dbc5358a6617feb5ed35471906a36e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.473ex; height:2.176ex;" alt="{\displaystyle G\cap H}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\cap M.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>∩</mo> <mi>M</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\cap M.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96d6229be3e5ebda12196ae52538edf8c44fdc4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.498ex; height:2.176ex;" alt="{\displaystyle G\cap M.}"></span> The first part is characterized by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g={\bar {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g={\bar {g}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d2bd861fcb5a883851e44db187b212c7a693fe4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.446ex; height:2.343ex;" alt="{\displaystyle g={\bar {g}}}"></span> ; then the Lorentz transformation corresponding to <span class="texhtml mvar" style="font-style:italic;">g</span> is given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(q)=g^{-1}qg}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>q</mi> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(q)=g^{-1}qg}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b95b96f583da71a5b89ac57b0f06013d3d7b2714" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.25ex; height:3.176ex;" alt="{\displaystyle T(q)=g^{-1}qg}"></span> since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{*}=g^{-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g^{*}=g^{-1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a46419068d3378ac2d05c742b09bd564e42878b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.369ex; height:3.009ex;" alt="{\displaystyle g^{*}=g^{-1}.}"></span> Such a transformation is a <a href="/wiki/Quaternions_and_spatial_rotation" title="Quaternions and spatial rotation">rotation by quaternion multiplication</a>, and the collection of them is <span class="texhtml"><a href="/wiki/SO(3)" class="mw-redirect" title="SO(3)">SO(3)</a></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cong G\cap H.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≅</mo> <mi>G</mi> <mo>∩</mo> <mi>H</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cong G\cap H.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ca1af911c05fdce9fb90f774bdfeb053acf543d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.573ex; height:2.176ex;" alt="{\displaystyle \cong G\cap H.}"></span> But this subgroup of <span class="texhtml mvar" style="font-style:italic;">G</span> is not a <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroup</a>, so no <a href="/wiki/Quotient_group" title="Quotient group">quotient group</a> can be formed. </p><p>To view <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\cap M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>∩</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\cap M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c859c09300bfe42f268295cb628f7a5d51cfd879" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.852ex; height:2.176ex;" alt="{\displaystyle G\cap M}"></span> it is necessary to show some subalgebra structure in the biquaternions. Let <span class="texhtml mvar" style="font-style:italic;">r</span> represent an element of the <a href="/wiki/Quaternion#Square_roots_of_−1" title="Quaternion">sphere of square roots of minus one</a> in the real quaternion subalgebra <span class="texhtml"><b>H</b></span>. Then <span class="texhtml">(<i>hr</i>)<sup>2</sup> = +1</span> and the plane of biquaternions given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{r}=\lbrace z=x+yhr:x,y\in \mathbb {R} \rbrace }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mi>h</mi> <mi>r</mi> <mo>:</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{r}=\lbrace z=x+yhr:x,y\in \mathbb {R} \rbrace }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d9354c920ed71716f0b4964121f07b9116d50fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.196ex; height:2.843ex;" alt="{\displaystyle D_{r}=\lbrace z=x+yhr:x,y\in \mathbb {R} \rbrace }"></span> is a commutative subalgebra isomorphic to the plane of split-complex numbers. Just as the ordinary complex plane has a unit circle, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8783bccae1e365d5e58cd502bea46ce4eee7fe34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.898ex; height:2.509ex;" alt="{\displaystyle D_{r}}"></span> has a <a href="/wiki/Unit_hyperbola" title="Unit hyperbola">unit hyperbola</a> given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(ahr)=\cosh(a)+hr\ \sinh(a),\quad a\in R.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>h</mi> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>h</mi> <mi>r</mi> <mtext> </mtext> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>a</mi> <mo>∈</mo> <mi>R</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp(ahr)=\cosh(a)+hr\ \sinh(a),\quad a\in R.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1a064687caed15a77ef68f718d92e69847117b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.741ex; height:2.843ex;" alt="{\displaystyle \exp(ahr)=\cosh(a)+hr\ \sinh(a),\quad a\in R.}"></span></dd></dl> <p>Just as the unit circle turns by multiplication through one of its elements, so the hyperbola turns because <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(ahr)\exp(bhr)=\exp((a+b)hr).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>h</mi> <mi>r</mi> <mo stretchy="false">)</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mi>h</mi> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mi>h</mi> <mi>r</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp(ahr)\exp(bhr)=\exp((a+b)hr).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/805e19e5ec523e34a88389c9505148fd1986939a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.485ex; height:2.843ex;" alt="{\displaystyle \exp(ahr)\exp(bhr)=\exp((a+b)hr).}"></span> Hence these algebraic operators on the hyperbola are called <a href="/wiki/Versor#Hyperbolic_versor" title="Versor">hyperbolic versors</a>. The unit circle in <span class="texhtml"><b>C</b></span> and unit hyperbola in <span class="texhtml"><i>D</i><sub><i>r</i></sub></span> are examples of <a href="/wiki/One-parameter_group" title="One-parameter group">one-parameter groups</a>. For every square root <span class="texhtml"><i>r</i></span> of minus one in <span class="texhtml"><b>H</b></span>, there is a one-parameter group in the biquaternions given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\cap D_{r}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>∩</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\cap D_{r}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa8fe593065636d39922a4a2453f98abd601e2c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.954ex; height:2.509ex;" alt="{\displaystyle G\cap D_{r}.}"></span> </p><p>The space of biquaternions has a natural <a href="/wiki/Topology" title="Topology">topology</a> through the <a href="/wiki/Euclidean_metric" class="mw-redirect" title="Euclidean metric">Euclidean metric</a> on <span class="texhtml">8</span>-space. With respect to this topology, <span class="texhtml mvar" style="font-style:italic;">G</span> is a <a href="/wiki/Topological_group" title="Topological group">topological group</a>. Moreover, it has analytic structure making it a six-parameter <a href="/wiki/Lie_group" title="Lie group">Lie group</a>. Consider the subspace of bivectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\lbrace q:q^{*}=-q\rbrace }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>q</mi> <mo>:</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi>q</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\lbrace q:q^{*}=-q\rbrace }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0385035eb234168099d3424c5d93bc3ec0642cb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.283ex; height:2.843ex;" alt="{\displaystyle A=\lbrace q:q^{*}=-q\rbrace }"></span>. Then the <a href="/wiki/Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">exponential map</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp :A\to G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp :A\to G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/231d3ed5ec4f6756e44e213b3fb4687eb15014be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.673ex; height:2.509ex;" alt="{\displaystyle \exp :A\to G}"></span> takes the real vectors to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\cap H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>∩</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\cap H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/927a14c2f8dbc5358a6617feb5ed35471906a36e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.473ex; height:2.176ex;" alt="{\displaystyle G\cap H}"></span> and the <span class="texhtml mvar" style="font-style:italic;">h</span>-vectors to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\cap M.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>∩</mo> <mi>M</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\cap M.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96d6229be3e5ebda12196ae52538edf8c44fdc4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.498ex; height:2.176ex;" alt="{\displaystyle G\cap M.}"></span> When equipped with the <a href="/wiki/Commutator" title="Commutator">commutator</a>, <span class="texhtml mvar" style="font-style:italic;">A</span> forms the <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> of <span class="texhtml mvar" style="font-style:italic;">G</span>. Thus this study of a <a href="/wiki/Six-dimensional_space" title="Six-dimensional space">six-dimensional space</a> serves to introduce the general concepts of <a href="/wiki/Lie_theory" title="Lie theory">Lie theory</a>. When viewed in the matrix representation, <span class="texhtml mvar" style="font-style:italic;">G</span> is called the <a href="/wiki/Special_linear_group" title="Special linear group">special linear group</a> <span class="texhtml"><a href="/wiki/SL(2,C)" class="mw-redirect" title="SL(2,C)">SL(2,C)</a></span> in <span class="texhtml">M(2, <b>C</b>)</span>. </p><p>Many of the concepts of <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> are illustrated through the biquaternion structures laid out. The subspace <span class="texhtml mvar" style="font-style:italic;">M</span> corresponds to <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a>, with the four coordinates giving the time and space locations of events in a resting <a href="/wiki/Frame_of_reference" title="Frame of reference">frame of reference</a>. Any hyperbolic versor <span class="texhtml">exp(<i>ahr</i>)</span> corresponds to a <a href="/wiki/Velocity" title="Velocity">velocity</a> in direction <span class="texhtml mvar" style="font-style:italic;">r</span> of speed <span class="texhtml"><i>c</i> tanh <i>a</i></span> where <span class="texhtml mvar" style="font-style:italic;">c</span> is the <a href="/wiki/Velocity_of_light" class="mw-redirect" title="Velocity of light">velocity of light</a>. The inertial frame of reference of this velocity can be made the resting frame by applying the <a href="/wiki/Lorentz_boost" class="mw-redirect" title="Lorentz boost">Lorentz boost</a> <span class="texhtml mvar" style="font-style:italic;">T</span> given by <span class="texhtml"><i>g</i> = exp(0.5<i>ahr</i>)</span> since then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{\star }=\exp(-0.5ahr)=g^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msup> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>0.5</mn> <mi>a</mi> <mi>h</mi> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g^{\star }=\exp(-0.5ahr)=g^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/775be315baa35903ecad6fb68f914b7799c3d1d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.301ex; height:2.843ex;" alt="{\displaystyle g^{\star }=\exp(-0.5ahr)=g^{*}}"></span> so that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(\exp(ahr))=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>h</mi> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(\exp(ahr))=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1f117e8a7b65d191d11386ef38f0f725f0ed7cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.332ex; height:2.843ex;" alt="{\displaystyle T(\exp(ahr))=1.}"></span> Naturally the <a href="/wiki/Hyperboloid" title="Hyperboloid">hyperboloid</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\cap M,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>∩</mo> <mi>M</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\cap M,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e280eadc8eb742cd4def84de121574ba1f8a552" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.498ex; height:2.509ex;" alt="{\displaystyle G\cap M,}"></span> which represents the range of velocities for sub-luminal motion, is of physical interest. There has been considerable work associating this "velocity space" with the <a href="/wiki/Hyperboloid_model" title="Hyperboloid model">hyperboloid model</a> of <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a>. In special relativity, the <a href="/wiki/Hyperbolic_angle" title="Hyperbolic angle">hyperbolic angle</a> parameter of a hyperbolic versor is called <a href="/wiki/Rapidity" title="Rapidity">rapidity</a>. Thus we see the biquaternion group <span class="texhtml mvar" style="font-style:italic;">G</span> provides a <a href="/wiki/Group_representation" title="Group representation">group representation</a> for the <a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a>.<sup id="cite_ref-FOOTNOTEHermann1974chapter_6.4_Complex_Quaternions_and_Maxwell's_Equations_11-0" class="reference"><a href="#cite_note-FOOTNOTEHermann1974chapter_6.4_Complex_Quaternions_and_Maxwell's_Equations-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>After the introduction of <a href="/wiki/Spinor" title="Spinor">spinor</a> theory, particularly in the hands of <a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Wolfgang Pauli</a> and <a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a>, the biquaternion representation of the Lorentz group was superseded. The new methods were founded on <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis vectors</a> in the set </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{q\ :\ qq^{*}=0\}=\left\{w+x\mathbf {i} +y\mathbf {j} +z\mathbf {k} \ :\ w^{2}+x^{2}+y^{2}+z^{2}=0\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>q</mi> <mtext> </mtext> <mo>:</mo> <mtext> </mtext> <mi>q</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mi>w</mi> <mo>+</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>+</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>+</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mtext> </mtext> <mo>:</mo> <mtext> </mtext> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{q\ :\ qq^{*}=0\}=\left\{w+x\mathbf {i} +y\mathbf {j} +z\mathbf {k} \ :\ w^{2}+x^{2}+y^{2}+z^{2}=0\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49b8b7f16b20af280a26920d404f6b32100b30f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:61.836ex; height:3.343ex;" alt="{\displaystyle \{q\ :\ qq^{*}=0\}=\left\{w+x\mathbf {i} +y\mathbf {j} +z\mathbf {k} \ :\ w^{2}+x^{2}+y^{2}+z^{2}=0\right\}}"></span></dd></dl> <p>which is called the <i>complex light cone</i>. The above <a href="/wiki/Representation_theory_of_the_Lorentz_group" title="Representation theory of the Lorentz group">representation of the Lorentz group</a> coincides with what physicists refer to as <a href="/wiki/Four-vector" title="Four-vector">four-vectors</a>. Beyond four-vectors, the <a href="/wiki/Standard_model" class="mw-redirect" title="Standard model">standard model</a> of particle physics also includes other Lorentz representations, known as <a href="/wiki/Lorentz_scalar" title="Lorentz scalar">scalars</a>, and the <span class="texhtml">(1, 0) ⊕ (0, 1)</span>-representation associated with e.g. the <a href="/wiki/Electromagnetic_field_tensor" class="mw-redirect" title="Electromagnetic field tensor">electromagnetic field tensor</a>. Furthermore, particle physics makes use of the <span class="texhtml">SL(2, <b>C</b>)</span> representations (or <a href="/wiki/Projective_representation" title="Projective representation">projective representations</a> of the Lorentz group) known as left- and right-handed <a href="/wiki/Weyl_spinor" class="mw-redirect" title="Weyl spinor">Weyl spinors</a>, <a href="/wiki/Majorana_spinor" class="mw-redirect" title="Majorana spinor">Majorana spinors</a>, and <a href="/wiki/Dirac_spinor" title="Dirac spinor">Dirac spinors</a>. It is known that each of these seven representations can be constructed as invariant subspaces within the biquaternions.<sup id="cite_ref-FOOTNOTEFurey2012_12-0" class="reference"><a href="#cite_note-FOOTNOTEFurey2012-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="As_a_composition_algebra">As a composition algebra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Biquaternion&action=edit&section=8" title="Edit section: As a composition algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although W. R. Hamilton introduced biquaternions in the 19th century, its delineation of its <a href="/wiki/Mathematical_structure" title="Mathematical structure">mathematical structure</a> as a special type of <a href="/wiki/Algebra_over_a_field" title="Algebra over a field">algebra over a field</a> was accomplished in the 20th century: the biquaternions may be generated out of the <a href="/wiki/Bicomplex_number" title="Bicomplex number">bicomplex numbers</a> in the same way that <a href="/wiki/Adrian_Albert" class="mw-redirect" title="Adrian Albert">Adrian Albert</a> generated the real quaternions out of complex numbers in the so-called <a href="/wiki/Cayley%E2%80%93Dickson_construction" title="Cayley–Dickson construction">Cayley–Dickson construction</a>. In this construction, a bicomplex number <span class="texhtml">(<i>w</i>, <i>z</i>)</span> has conjugate <span class="texhtml">(<i>w</i>, <i>z</i>)* = (<i>w</i>, – <i>z</i>)</span>. </p><p>The biquaternion is then a pair of bicomplex numbers <span class="texhtml">(<i>a</i>, <i>b</i>)</span>, where the product with a second biquaternion <span class="texhtml">(<i>c</i>, <i>d</i>)</span> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)(c,d)=(ac-d^{*}b,da+bc^{*}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>c</mi> <mo>−</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>b</mi> <mo>,</mo> <mi>d</mi> <mi>a</mi> <mo>+</mo> <mi>b</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)(c,d)=(ac-d^{*}b,da+bc^{*}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7877c1398f55be9a974a7d7ee97afd0a046029e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.417ex; height:2.843ex;" alt="{\displaystyle (a,b)(c,d)=(ac-d^{*}b,da+bc^{*}).}"></span></dd></dl> <p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=(u,v),b=(w,z),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>w</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=(u,v),b=(w,z),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e27f4c83a21db5a86166144dc6fcd6bf853a98b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.001ex; height:2.843ex;" alt="{\displaystyle a=(u,v),b=(w,z),}"></span> then the <i>biconjugate</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)^{*}=(a^{*},-b).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>,</mo> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)^{*}=(a^{*},-b).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bc2ee60c3dc6f28b711cb3419eb68fbb557ee7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.803ex; height:2.843ex;" alt="{\displaystyle (a,b)^{*}=(a^{*},-b).}"></span> </p><p>When <span class="texhtml">(<i>a</i>, <i>b</i>)*</span> is written as a 4-vector of ordinary complex numbers, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u,v,w,z)^{*}=(u,-v,-w,-z).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo>,</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mo>−</mo> <mi>v</mi> <mo>,</mo> <mo>−</mo> <mi>w</mi> <mo>,</mo> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (u,v,w,z)^{*}=(u,-v,-w,-z).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26a27e2e34da8496dc57e349a3cf84879d9fef49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.465ex; height:2.843ex;" alt="{\displaystyle (u,v,w,z)^{*}=(u,-v,-w,-z).}"></span></dd></dl> <p>The biquaternions form an example of a <a href="/wiki/Quaternion_algebra" title="Quaternion algebra">quaternion algebra</a>, and it has norm </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(u,v,w,z)=u^{2}+v^{2}+w^{2}+z^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(u,v,w,z)=u^{2}+v^{2}+w^{2}+z^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86604132256b5c43661ffe32cc5d8214a02a720e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.88ex; height:3.176ex;" alt="{\displaystyle N(u,v,w,z)=u^{2}+v^{2}+w^{2}+z^{2}.}"></span></dd></dl> <p>Two biquaternions <span class="texhtml"><i>p</i></span> and <span class="texhtml"><i>q</i></span> satisfy <span class="texhtml"><i>N</i>(<i>pq</i>) = <i>N</i>(<i>p</i>) <i>N</i>(<i>q</i>)</span>, indicating that <span class="texhtml"><i>N</i></span> is a quadratic form admitting composition, so that the biquaternions form a <a href="/wiki/Composition_algebra" title="Composition algebra">composition algebra</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Biquaternion&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Biquaternion_algebra" title="Biquaternion algebra">Biquaternion algebra</a></li> <li><a href="/wiki/Hypercomplex_number" title="Hypercomplex number">Hypercomplex number</a></li> <li><a href="/wiki/Hypercomplex_analysis" title="Hypercomplex analysis">Hypercomplex analysis</a></li> <li><a href="/wiki/Joachim_Lambek" title="Joachim Lambek">Joachim Lambek</a></li> <li><a href="/wiki/Hyperbolic_quaternion#MacFarlane's_hyperbolic_quaternion_paper_of_1900" title="Hyperbolic quaternion">MacFarlane's use</a></li> <li><a href="/wiki/Quotient_ring#Quaternions_and_alternatives" title="Quotient ring">Quotient ring</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Biquaternion&action=edit&section=10" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width reflist-columns-2"> <ol class="references"> <li id="cite_note-FOOTNOTEHamilton1850-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHamilton1850_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHamilton1850">Hamilton 1850</a>.</span> </li> <li id="cite_note-FOOTNOTEGarling2011112,_113-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGarling2011112,_113_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGarling2011">Garling 2011</a>, pp. 112, 113.</span> </li> <li id="cite_note-FOOTNOTEGarling2011112-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGarling2011112_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGarling2011">Garling 2011</a>, p. 112.</span> </li> <li id="cite_note-FOOTNOTEFrancisKosowsky2005404-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFrancisKosowsky2005404_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFrancisKosowsky2005">Francis & Kosowsky 2005</a>, p. 404.</span> </li> <li id="cite_note-FOOTNOTEFrancisKosowsky2005386-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFrancisKosowsky2005386_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFrancisKosowsky2005">Francis & Kosowsky 2005</a>, p. 386.</span> </li> <li id="cite_note-FOOTNOTEHamilton1853639-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHamilton1853639_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHamilton1853">Hamilton 1853</a>, p. 639.</span> </li> <li id="cite_note-FOOTNOTEHamilton1853730-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHamilton1853730_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHamilton1853">Hamilton 1853</a>, p. 730.</span> </li> <li id="cite_note-FOOTNOTEHamilton1866289-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHamilton1866289_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHamilton1866">Hamilton 1866</a>, p. 289.</span> </li> <li id="cite_note-FOOTNOTEDickson191413-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDickson191413_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDickson1914">Dickson 1914</a>, p. 13.</span> </li> <li id="cite_note-FOOTNOTELanczos1949See_equation_94.16,_page_305._The_following_algebra_compares_to_Lanczos,_except_he_uses_~_to_signify_quaternion_conjugation_and_*_for_complex_conjugation-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELanczos1949See_equation_94.16,_page_305._The_following_algebra_compares_to_Lanczos,_except_he_uses_~_to_signify_quaternion_conjugation_and_*_for_complex_conjugation_10-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLanczos1949">Lanczos 1949</a>, See equation 94.16, page 305. The following algebra compares to Lanczos, except he uses ~ to signify quaternion conjugation and * for complex conjugation.</span> </li> <li id="cite_note-FOOTNOTEHermann1974chapter_6.4_Complex_Quaternions_and_Maxwell's_Equations-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHermann1974chapter_6.4_Complex_Quaternions_and_Maxwell's_Equations_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHermann1974">Hermann 1974</a>, chapter 6.4 Complex Quaternions and Maxwell's Equations.</span> </li> <li id="cite_note-FOOTNOTEFurey2012-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFurey2012_12-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFurey2012">Furey 2012</a>.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Biquaternion&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wikibooks-logo-en-noslogan.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/60px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/80px-Wikibooks-logo-en-noslogan.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span></div> <div class="side-box-text plainlist">The Wikibook <i><a href="https://en.wikibooks.org/wiki/Associative_Composition_Algebra" class="extiw" title="wikibooks:Associative Composition Algebra">Associative Composition Algebra</a></i> has a page on the topic of: <i><b><a href="https://en.wikibooks.org/wiki/Associative_Composition_Algebra/Quaternions" class="extiw" title="wikibooks:Associative Composition Algebra/Quaternions">Biquaternions</a></b></i></div></div> </div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><a href="/wiki/Arthur_Buchheim" title="Arthur Buchheim">Arthur Buchheim</a> (1885) <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2369176">"A Memoir on biquaternions"</a>, <a href="/wiki/American_Journal_of_Mathematics" title="American Journal of Mathematics">American Journal of Mathematics</a> 7(4):293 to 326 from <a href="/wiki/Jstor" class="mw-redirect" title="Jstor">Jstor</a> early content.</li> <li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFConway1911" class="citation cs2"><a href="/wiki/Arthur_W._Conway" title="Arthur W. 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alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span>)</li> <li><a href="/wiki/Integer" title="Integer">Integers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>)</li> <li><a href="/wiki/Rational_number" title="Rational number">Rational numbers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>)</li> <li><a href="/wiki/Constructible_number" title="Constructible number">Constructible numbers</a></li> <li><a href="/wiki/Algebraic_number" title="Algebraic number">Algebraic numbers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb423c16a5f403edbaf66438b75e7a36e725af6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {A} }"></span>)</li> <li><a href="/wiki/Closed-form_expression#Closed-form_number" title="Closed-form expression">Closed-form numbers</a></li> <li><a href="/wiki/Period_(algebraic_geometry)" title="Period (algebraic geometry)">Periods</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>)</li> <li><a href="/wiki/Computable_number" title="Computable number">Computable numbers</a></li> <li><a href="/wiki/Definable_real_number#Definability_in_arithmetic" title="Definable real number">Arithmetical numbers</a></li> <li><a href="/wiki/Definable_real_number#Definability_in_models_of_ZFC" title="Definable real number">Set-theoretically definable numbers</a></li> <li><a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integers</a> <ul><li><a href="/wiki/Gaussian_rational" title="Gaussian rational">Gaussian rationals</a></li></ul></li> <li><a href="/wiki/Eisenstein_integer" title="Eisenstein integer">Eisenstein integers</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Composition_algebra" title="Composition algebra">Composition algebras</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Division_algebra" title="Division algebra">Division algebras</a>: <a href="/wiki/Real_number" title="Real number">Real numbers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>)</li> <li><a href="/wiki/Complex_number" title="Complex number">Complex numbers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>)</li> <li><a href="/wiki/Quaternion" title="Quaternion">Quaternions</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e050965453c42bcc6bd544546703c836bdafeac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} }"></span>)</li> <li><a href="/wiki/Octonion" title="Octonion">Octonions</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {O} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">O</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {O} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ed2664a4fe515e6fbed25a7193ce663b82920c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {O} }"></span>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Split<br />types</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>:</li> <li><a href="/wiki/Split-complex_number" title="Split-complex number">Split-complex numbers</a></li> <li><a href="/wiki/Split-quaternion" title="Split-quaternion">Split-quaternions</a></li> <li><a href="/wiki/Split-octonion" title="Split-octonion">Split-octonions</a><br /> Over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>:</li> <li><a href="/wiki/Bicomplex_number" title="Bicomplex number">Bicomplex numbers</a></li> <li><a class="mw-selflink selflink">Biquaternions</a></li> <li><a href="/wiki/Bioctonion" title="Bioctonion">Bioctonions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other <a href="/wiki/Hypercomplex_number" title="Hypercomplex number">hypercomplex</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dual_number" title="Dual number">Dual numbers</a></li> <li><a href="/wiki/Dual_quaternion" title="Dual quaternion">Dual quaternions</a></li> <li><a href="/wiki/Dual-complex_number" class="mw-redirect" title="Dual-complex number">Dual-complex numbers</a></li> <li><a href="/wiki/Hyperbolic_quaternion" title="Hyperbolic quaternion">Hyperbolic quaternions</a></li> <li><a href="/wiki/Sedenion" title="Sedenion">Sedenions</a>  (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }"></span>)</li> <li><a href="/wiki/Trigintaduonion" title="Trigintaduonion">Trigintaduonions</a>  (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {T} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c039979935c00b3b216cbb065999207872677f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {T} }"></span>)</li> <li><a href="/wiki/Split-biquaternion" title="Split-biquaternion">Split-biquaternions</a></li> <li><a href="/wiki/Multicomplex_number" title="Multicomplex number">Multicomplex numbers</a></li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a>/<a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a> <ul><li><a href="/wiki/Algebra_of_physical_space" title="Algebra of physical space">Algebra of physical space</a></li> <li><a href="/wiki/Spacetime_algebra" title="Spacetime algebra">Spacetime algebra</a></li> <li><a href="/wiki/Plane-based_geometric_algebra" title="Plane-based geometric algebra">Plane-based geometric algebra</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Infinity" title="Infinity">Infinities</a> and <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimals</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cardinal_number" title="Cardinal number">Cardinal numbers</a></li> <li><a href="/wiki/Extended_natural_numbers" title="Extended natural numbers">Extended natural numbers</a></li> <li><a href="/wiki/Extended_real_number_line" title="Extended real number line">Extended real numbers</a> <ul><li><a href="/wiki/Projectively_extended_real_line" title="Projectively extended real line">Projective</a></li></ul></li> <li><a href="/wiki/Riemann_sphere" title="Riemann sphere">Extended complex numbers</a></li> <li><a href="/wiki/Hyperreal_number" title="Hyperreal number">Hyperreal numbers</a></li> <li><a href="/wiki/Levi-Civita_field" title="Levi-Civita field">Levi-Civita field</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal numbers</a></li> <li><a href="/wiki/Supernatural_number" title="Supernatural number">Supernatural numbers</a></li> <li><a href="/wiki/Surreal_number" title="Surreal number">Surreal numbers</a></li> <li><a href="/wiki/Superreal_number" title="Superreal number">Superreal numbers</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other types</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Irrational_number" title="Irrational number">Irrational numbers</a></li> <li><a href="/wiki/Fuzzy_number" title="Fuzzy number">Fuzzy numbers</a></li> <li><a href="/wiki/Transcendental_number" title="Transcendental number">Transcendental numbers</a></li> <li><a href="/wiki/P-adic_number" title="P-adic number"><span class="nowrap"><i>p</i>-adic</span> numbers</a> (<a href="/wiki/Solenoid_(mathematics)#p-adic_solenoids" title="Solenoid (mathematics)"><span class="nowrap"><i>p</i>-adic</span> solenoids</a>)</li> <li><a href="/wiki/Profinite_integer" title="Profinite integer">Profinite integers</a></li> <li><a href="/wiki/Normal_number" title="Normal number">Normal numbers</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> <ul><li><a href="/wiki/Number#Main_classification" title="Number">Classification</a></li> <li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/List_of_types_of_numbers" title="List of types of numbers">List</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" 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