Properties lost when going from real number system to quaternions and octonions What properties do we lose as we go from real numbers to quaternions, then to octonions? Do any new properties arise, or do calculations just become more "path dependant"?

The most important property (IMHO) that $\mathbb{R}$ has that its extensions don't have is that it is a totally ordered field with least upper bound property. You don't have an order (that works nicely with the field) in these extensions.
