(04/11/2011, 07:32 PM)bo198214 Wrote: I never heard about 3 dimensional numbers, the possible finite dimensional division algebras must have dimension 1 (real), 2 (complex), 4 (quaternion) or 8 (octonion), see wikipedia.
I remember that Hamilton tried to find to 3 dimensional numbers but failed, and came up in the end with quaternions.
yes and no : we have zero-divisors in 3D.
(04/06/2011, 04:20 PM)tommy1729 Wrote: it should be noted that there are 2 types of 3D numbers.
a + b P + c P^2 + d P^3 where 1 + P + P^2 + P^3 = 0 and P^4 = 1 and a , b , c , d are positive.
Quote:Isnt that 4 dimensional, a,b,c,d?
no , since like i said : a , b , c , d are POSITIVE.
the units are not orthogonal.
Quote:and the classical " 3D complex "
a + b w + c w^2 where a , b , c are real and w^3 = 1
Quote:How do you define division here?
just as the multiplicative inverse.
1 = (a' + b' w + c' w^2)(a + b w + c w^2)
if (a + b w + c w^2) is not a zero-divisor.
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in abstract algebra notation the 2 kinds of 3D numbers are RxRxR and RxC.
matrix representation is a must , and they also satisfy the modified Cauchy-Riemann equations.
( they also extend the gaussian integers , which makes fun number theory but that is a bit off topic , also its possible to use them for rotations rather than euler angles and quaternions )
perhaps the following are illuminating :
http://en.wikipedia.org/wiki/Group_ring
( amateur mathematician ) http://bandtech.com/PolySigned/PolySigned.html
( also check the links )
many papers by beresford or Silviu Olariu ( cant find them right now )
http://en.wikipedia.org/wiki/Tricomplex_number
http://arxiv.org/PS_cache/math/pdf/0008/0008120v1.pdf
http://arxiv.org/PS_cache/math/pdf/0011/0011044v1.pdf
and many others.
feel free to give more free references !
regards
tommy1729