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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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@TheChaz Look at that set: only 4 elements, not 8... –  MathematicalOrchid May 7 '12 at 19:43
    
Oh, sorry I misread. –  The Chaz 2.0 May 7 '12 at 19:43
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3 Answers 3

up vote 3 down vote accepted

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... –  MathematicalOrchid May 7 '12 at 21:05
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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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There's also a sign change somewhere along the line; the OP has $ij = k$ but Wikipedia has $ik = -j$. –  Rahul May 7 '12 at 22:59
    
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. :) –  Micah May 7 '12 at 23:17
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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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