There are no three dimensional numbers because it's impossible to construct such a system that behaves like 'numbers'. The real, complex, quaternion and octonion numbers are the only 'normed division algebras'. This basically means they are the only spaces with a notion of length in which mutliplication is invertible (division). These properties are needed for anything that we want to call 'numbers'. The fact that there are only 4 such spaces is quite a profound result and leads to a lot of other important results. See http://en.wikipedia.org/wiki/Hurwitz%27s_theorem_%28normed_d... for more information.
Why are there no 16, 32, 64, and so on, dimensions of numbers? Would multiplication not be invertible with them? 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?
> Why are there no 16, 32, 64, and so on, dimensions of numbers? Would multiplication not be invertible with them?
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.
This is pretty fascinating, do you know of any good sites or books which will get me up to scratch on different number systems? I haven't studied math in a long time, so do you think it would be worth going over the basics, like calculus and geometry, again to build a foundation before venturing so far out?
