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Why is quaternion algebra 4D and not 3D? Complex algebra is 2D and what is known as quaternion algebra jumps to 4D.

$ i^2 = j^2 = k^2 = ijk = -1 $

Using $1, i, j,$ and $k$ as the base (where complex uses $1$ and $i$ (or $j$ if you are an EE)) which results in a 4-axis space. Why is there no 3D algebra?

  • 0
    possible duplicate of [Is there a third dimension of numbers?](http://math.stackexchange.com/questions/32100/is-there-a-third-dimension-of-numbers)2011-10-12
  • 0
    [This](http://math.stackexchange.com/questions/39522) is related...2011-10-12

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