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| {{mergeto|non-associative algebra|date=February 2013|discuss=Talk:Non-associative algebra#Merge example}}
| | 54 year-old Jeweller Rave from Manotick, has lots of passions which include beatboxing, property developers in new launch ec singapore ([http://www.isoevent.fr/en/node/2912492 similar webpage]) and cave diving. Gets enormous encouragement from life by touring locales such as Tino and Tinetto). |
| {{Unreferenced|date=December 2009}}
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| This page presents and discusses an example of a non-associative [[division algebra]] over the [[real number]]s.
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| The multiplication is defined by taking the [[complex conjugate]] of the usual multiplication: <math>a*b=\overline{ab}</math>. This is a commutative, non-associative division algebra of dimension 2 over the reals, and has no unit element.
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| ==Proof that <math>(\mathbb{C},*)</math> is a division algebra==
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| For a proof that <math>\mathbb{R}</math> is a [[field (mathematics)|field]], see [[real number]].
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| Then, the complex numbers themselves clearly form a [[vector space]].
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| It remains to prove that the [[binary operation]] given above satisfies the requirements of a division algebra
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| * ('''x''' + '''y''')'''z''' = '''x''' '''z''' + '''y''' '''z''';
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| * '''x'''('''y''' + '''z''') = '''x''' '''y''' + '''x''' '''z''';
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| * (''a'' '''x''')'''y''' = ''a''('''x''' '''y'''); and
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| * '''x'''(''b'' '''y''') = ''b''('''x''' '''y''');
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| [[for all]] scalars ''a'' and ''b'' in <math>\mathbb{R}</math> and all vectors '''x''', '''y''', and '''z''' (also in <math>\mathbb{C}</math>).
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| For [[distributivity]]:
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| :<math>x*(y+z)=\overline{x(y+z)}=\overline{xy+xz}=\overline{xy}+\overline{xz}=x*y+x*z,</math>
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| (similarly for right distributivity); and for the third and fourth requirements
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| :<math> (ax)*y=\overline{(ax)y}=\overline{a(xy)}=\overline{a}\cdot\overline{xy}=\overline{a}(x*y).</math>
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| ==Non associativity of <math>(\mathbb{C},*)</math>==
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| *:<math>a * (b * c) = a * \overline{b c} = \overline{a \overline{b c}} = \overline{a} b c </math>
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| *:<math>(a * b) * c = \overline{a b} * c = \overline{\overline{a b} c} = a b \overline{c} </math>
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| So, if ''a'', ''b'', and ''c'' are all non-zero, and if ''a'' and ''c'' do not differ by a real multiple, <math>a * (b * c) \neq (a * b) * c</math>.
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| {{DEFAULTSORT:Example Of A Non-Associative Algebra}}
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| [[Category:Algebras]]
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| [[Category:Non-associative algebras]]
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| [[Category:Articles containing proofs]]
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54 year-old Jeweller Rave from Manotick, has lots of passions which include beatboxing, property developers in new launch ec singapore (similar webpage) and cave diving. Gets enormous encouragement from life by touring locales such as Tino and Tinetto).
