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By the Frobenius theorem, there are only three possible structures for a real finite-dimensional associative division algebra. Those structures correspond to the real numbers, the complex numbers, and what are called the quaternions. So essentially the above definition is not arbitrary because it's the only other possible way (besides R and C) to get that sort of algebraic system. Of course, this is not obvious at all. C famously is algebraically closed as a field, which makes it a ripe playground for much of topology, algebraic geometry, and analysis. There are some nonobvious generalizations of algebraic closure for the quaternions. (Naively, the quaternions are not algebraically closed in the classic sense because, evidently, ix + xi - j has no root.) As for why one might want to consider such a noncommutative division algebra in the first place, the answer I suppose is just that it manages to pop up in a variety of areas in mathematics. We've already seen the connection with rotations in 3-space (the topic of this post). Here's another. The 3-sphere (that is, a sphere in 4-dimensional space whose surface is itself 3-dimensional) can be realized as the multiplicative group of unit quaternions spanned by {1,i,j,k}. Consider the circle H = {cos(theta) + i * sin(theta)} for real values of theta; H is a subset of the 3-sphere. If r is any unit quaternion, then the coset rH is another circle. But given a subgroup H of any group G, the left cosets of H in G form a partition of G. Therefore, these circles just described form a partition of all of the 3-sphere (the Hopf fibration). Speaking of rotations, the involvement of quaternions should not be surprising. Indeed, complex numbers are intimately involved in rotations in 2-space (multiplication by a unit complex number e^(i*theta) corresponds to rotation about the origin by theta). Quaternions can similarly express rotations in 3-space, but one cannot just left- or right-multiply but must instead use conjugation. In general, one can generalize this using the techniques of geometric algebra. |