Talk:Geometric algebra: Difference between revisions
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--[[User:Grondilu|Grondilu]] ([[User talk:Grondilu|talk]]) 14:54, 21 October 2015 (UTC)
I'm not sure what your issues are, but it seems to me that quaternions are an algebra containing a vector space \mathcal{V} and obeying these axioms:
:<math>\begin{array}{c}
(ab)c = a(bc)\\
a(b+c) = ab+ac\\
(a+b)c = ac+bc\\
\forall \mathbf{x}\in\mathcal{V},\,\mathbf{x}^2\in\R
\end{array}
</math>
You've added other constraints onto the task (such as the calculation that generates a 19, the orthonormal basis for \mathcal{V} and the requirement for two different embedded quaternion algebras), and I guess I can agree that that aspect is something of a mess. Mathematically speaking, I suppose that this hints at the need for further axioms... but computationally speaking, usually all we can ever provide is an implementation which approximates the math.
For example, no real computer implementation can satisfy the Peano postulates, so there will be cases where simple addition fails. And multiplication is even worse. Roughly half of the result domain for addition is typically missing, but for multiplication the size of the valid argument domain approximates the square root of the size of the result domain. So that sort of thing is going to be completely valid, mathematically...
Which I think has to do with why we get into concrete examples and/or concrete requirements - those might be "unnecessary" from a mathematical point of view, but they can be critically important from an implementation point of view. I think this also has something to do with why it's often a good idea to avoid an overly-general implementation...
So, anyways, from a mathematical purist point of view, I can't imagine many tasks can be anything but a "mess".
Though, I guess we could use Boolean addition and multiplication (greatest common divisor and least common multiple) or maybe modulo arithmetic - that would work around the overflow problems with regular addition and multiplication. This probably is not what you intend - might even violate the task purpose (whatever that is) - but these sorts of implementations would be a better fit if axiomatic correctness is what you are asking for.
Anyways... if you can figure out what it is that you want, I guess go for it... --[[User:Rdm|Rdm]] ([[User talk:Rdm|talk]]) 15:35, 21 October 2015 (UTC)
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