I'm curious as to why we don't have any useful numbers for the non-power-of-2 dimensions. E.g. 3-dimensional numbers.
Yes, this is exactly why the 16 dimensional sedenions are not in the same category as the others. For it to be a division algebra, you must be able to reverse any multiplication that isn't by zero. This is just the normal concept of division in the real numbers. It also allows us to define division in the complex, quaternion and octonion numbers. However, it is possible to multiply two non-zero sedenions together and get zero. This breaks this property completely.
> Is there a possibility of things that are distinct from our concept of numbers but are none-the-less useful being discovered in the future?
Absolutely! These four numbers systems are a tiny portion of spaces studied by mathematicians. There are a lot of interesting things that can be said just by generalising a few properties of numbers (e.g. without requiring that we can divide, or without requiring that we have both addition and multiplication). There are groups, fields, rings, algebras, topological spaces, and all sorts of interesting things that aren't 'numbers' as such.
I wonder if there's anything of numbers in infinite dimensions and, more interestingly, numbers in different degrees of infinite dimensions (yup, infinity comes in different degrees too - http://sites.google.com/site/degreesofinfinity/). It seems that math is only limited by the imagination.
(Infinite is often confused for "really big", but a better way of understanding it is often to use the term "unbounded". When we deal with n-dimensional vectors, we're not saying that we are always literally dealing with vectors with a millionybillionytertrillion dimensions, such that we can't even represent one in a real computer, we're saying that there isn't a firm upper limit on the number of dimensions we may encounter. It is often more the "unbounded" aspect of infinity that we are concerned with, rather than the "larger than anything else" aspect.)
Ah, so it's unbounded like how Haskell's lazy evaluation treats infinity.
Interestingness. Usefulness, perhaps, if you have a specific definition of 'useful' in mind.
> Is our commonly used system basically an arbitrary set of conditions?
It's an arbitrary set of conditions with interesting properties, and it's not always obvious which arbitrary conditions will have interesting properties.
> Do you think that some systems correlate better to how nature does things than others, and if so, which ones would they be?
Well, that is an interesting question, which means anyone who claims to know an absolute answer to it is a moron. We do know, for example, that vectors are a very useful tool to model a lot of what happens in physics, and that complex numbers are a compact way to talk about rotation, especially complex exponentiation (taking a real number, such as e, to complex powers).
> Ah, so it's unbounded like how Haskell's lazy evaluation treats infinity.
Yes. For example, a definition (the Peano axioms) of the set of the natural numbers (either the positive or non-negative integers; the set may or may not include zero, as conventions vary) states that the set contains 0 (or 1) and that for every number x it contains, it also contains x+1.
5i + 2j + 3k.
Since you can assume that 2 is getting multiplied by the unit real vector (equal to 1), you can think of 2+3i as
2r + 3i
but why write the 'r' if we know 2*1 = 2?