## Quaternions and Numbers of n Dimensions

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