# What is the proper name for this number system?

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

To distinguish them from what others (including myself) might call the 'standard hypercomplex numbers,' I would call these the Davenport hypercomplex numbers.

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Davenport's page looks damned interesting, and quite readable... –  MathematicalOrchid May 7 '12 at 21:05
It appears to be the algebra of bicomplex numbers. Note that wikipedia's $j$ is your $k$ and vice versa.
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