Re:How the modern world arose from imaginary numbers
by xyzzyz on Thursday February 10, 2022 @11:22PM
(#30296933)
You can with n=4, you get quaternions. These are not commutative, but that’s really not too bad.
Re:How the modern world arose from imaginary numbers
by hgomersall on Friday February 11, 2022 @03:04AM
(#30298470)
You can for any positive n with geometric algebra (which encompasses the complex and quaternion algebras, the latter being much easier to understand in GA).
Note that, while the complex numbers are isomorphic to two-dimensional vectors, they are not used the same way. In particular, there is no equivalent to complex multiplication or division that is normally used with two dimensional vectors (though you could define them of course, they are not normally used).
The difference is also reinforced by the fact that there is no equivalent of the complex numbers for 3-dimensional vectors, or really any other n-dimensional vectors. That is, you can't define a multiplication and addition operation for n-tuples of real numbers for n>2, with the usual semantics of multiplication and addition (associativity, distributivity, inverse, neutral element).