TITLE: What's the Difference Between a Vector and an Hypercomplex Number?
QUESTION [6 upvotes]: What's the difference between a vector and an hypercomplex number? For instance a 4-vector and a quaternion. They seem to share many properties.
Perhaps this question could be put more generally as: what's the difference between a vector space and a field?

REPLY [3 votes]: A hypercomplex number is an element of a certain kind of distributive unital algebra over $\mathbb{R}$. An algebra over $\mathbb{R}$ is, by definition, a vector space over $\mathbb{R}$ with additional structure so that you can multiply elements together. It's not very hard to turn a given finite-dimensional vector space into an algebra of hypercomplex numbers, but the question is whether you can do so in a natural, useful way. 
Of course, there are vector spaces over any field at all, so it's also easy to come up with a vector space which cannot be turned into an algebra of hypercomplex numbers. For instance, a finite dimensional vector space over a finite field has only finitely many elements, so obviously cannot be turned into an algebra over $\mathbb{R}$.
The interesting thing is that there are objects which are simultaneously all of these things, and in a recursive way: for example, $\mathbb{C}$ is both a field and an algebra over $\mathbb{R}$, and a fortiori a vector space over $\mathbb{R}$. But $\mathbb{R}$ is in turn both a field and an algebra over $\mathbb{Q}$... The study of fields which are algebras over other fields is called Galois theory, which also finds applications in algebraic geometry (though probably not in the way you expect!).