Talk:Geometric algebra: Difference between revisions
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:::No, I combine them with the geometric product. Let me remind you that quaternions here are used as a special case of a more general structure, the geometric algebra.--[[User:Grondilu|Grondilu]] ([[User talk:Grondilu|talk]]) 22:43, 17 October 2015 (UTC) |
:::No, I combine them with the geometric product. Let me remind you that quaternions here are used as a special case of a more general structure, the geometric algebra.--[[User:Grondilu|Grondilu]] ([[User talk:Grondilu|talk]]) 22:43, 17 October 2015 (UTC) |
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== Task still needs something... == |
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Re-reading the task, and the implementation you have, I think the task needs some additional qualification. |
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Specifically, it's entirely possible to satisfy |
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:<math>\begin{array}{c} |
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(ab)c = a(bc)\\ |
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a(b+c) = ab+ac\\ |
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(a+b)c = ac+bc\\ |
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\forall \mathbf{x}\in\mathcal{V},\,\mathbf{x}^2\in\R |
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\end{array}</math> |
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in a fashion which does not support quaternion multiplication. For example, an algebra where <math>\forall \mathbf{x}\in\mathcal{V},\forall \mathbf{y}\in\mathcal{V},\,\mathbf{xy}\in\R</math> would not support quaternions but would completely satisfy the given constraints. |
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And, in fact, there are arbitrarily many complex rules one could apply to achieve quaternion multiplication and still satisfy those constraints. This means that the specification is incomplete. --[[User:Rdm|Rdm]] ([[User talk:Rdm|talk]]) 22:49, 17 October 2015 (UTC) |