It isn't nonsense at all. Subspaces, fields, and vector spaces are all interrelated. The complex numbers are a vector space and the quadratics are a subspace of them. I was hoping that instead of just complex numbers, you could use the special type for all sorts of vector math, tensors, quadratics, values with units attached. Why make complex numbers the special case? Why not have two tuples or n-tuples that know about co-efficients so you can do all kinds of interesting algebra in the same way you can use complex numbers.
All you have to do to make this really really flexible is allow users to change the value of that sqrt(-1) co-efficient(s) and change the length of the vector/sum/value. Tada.
UPDATE: Yes, well, I believe the quadratics are a 2-dimensional vector space and a field of reals. It happens to be a subspace of the complexes, so maybe it doesn't matter, but I fail to see why there couldn't be an interface to use the complex number system for more.
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*I personally believe* it is still a nonsense, wrt the **"special case"** (your literal words) bit: do not misunderstand me! I am too hoping that -to quote you verbatim- we *"could use the special type for all sorts of vector math, tensors, quadratics, values with units attached."* (And more!) Nevertheless, complex numbers are something **more:** i.e. a field, and that cannot be told of general n-dimensional vector spaces: given the confidence you talk about these topics, you certainly know that the only three real algebras (with division) are the reals themselves, the complex numbers and quaternions. (Then you have the more esoteric Cayley numbers, if you're willing to give up on associativity.) Or else, *are you taking, say, two n-dimensional vectors and multiply them together?!?* Certainly, you can have an inner product, but that's an entirely different beast, and *not* a generalization of complex multiplication. **Thus, definitely, not as a "special case."**
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