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The FractInt documentation makes mention of two number systems which extend the complex numbers: the "quaternions" and the "hypercomplex numbers".

However, Wikipedia claims that "hypercomplex number" is not a number system but a type of number system. So, does anybody know specifically which system FractInt is referring to?

http://www.nahee.com/spanky/www/fractint/append_a_misc.html#hcpx_math_anchor

Relevant excerpt:

  • $\{1, i, j, k\}$ are the key elements of the set.

  • This is a field, but for the lack of inverses for all elements. (In particular, addition and multiplication are both associative and commutative.)

  • $i^2 = j^2 = -k^2 = -1$.

  • $ij = ji = k$.

  • $jk = kj = -i$.

  • $ki = ik = -j$.

PS. POV-Ray also refers to these same two number systems:

http://www.povray.org/documentation/view/3.6.1/280/

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    Oh, sorry I misread.2012-05-07

3 Answers 3

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These numbers are called the "Circular" hypercomplex numbers in Olariu, S. (2000), “Commutative Complex Numbers in Four Dimensions”, Institute of Physics and Nuclear Engineering, Bucharest, Romania

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It appears to be the algebra of bicomplex numbers. Note that wikipedia's $j$ is your $k$ and vice versa.

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    You're right, I missed that. They're definitely isomorphic (by some results on Davenport's page); finding an isomorphism would be a fun exercise. :)2012-05-07
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To distinguish them from what others (including myself) might call the 'standard hypercomplex numbers,' I would call these the Davenport hypercomplex numbers.

More information about them can be found at Mathworld or at Davenport's Page.

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    Davenport's page looks damned interesting, and quite readable...2012-05-07