The treatment of "imaginary" numbers and quaternions really caught my fancy. In particular, I feel that it begs an extension to n dimensions. If we have found a huge number of applications of imaginary numbers, and a good number of applications for quaternions, it seems that there ought to be similarly promising applications for numbers of any dimensionality. Even if there aren't a whole lot of practical applications, it should at least be interesting math.
Did the chapter only mention 2-numbers and 4-numbers because they are actually the only ones that have been explored, or because they are the ones with the most practical applications, or for some other reason? It seems so obvious to talk about numbers of other dimensionality that it seems odd to me that there wouldn't already be extensive investigations of n-numbers.
The formulae for multiplication grabbed my fancy somewhat as an example of the territory an exploration of n-numbers could explore. The multiplication formula should ideally preserve the property of 1-number multiplication, wherein iterations of addition give the same result as multiplication. With the use of the dimensional vector analogy, it seems that one could offer a synthetic proof of a formula for the multiplication of an n-number.
Prof. Banchoff's response.Keith_Adams@brown.edu