  Extensions of real numbers in more than two dimensions, in particular
quaternions and octonions are finding applications in physics due to the fact
that they naturally capture certain symmetries of physical systems. Here it is
shown that the property of closure of coset groups can be used to generate the
basis and general multiplication rules for extensions of real numbers in a
systematic way. The coset approach has the advantage that multiplication rules
follow directly from group closure instead of being postulated. In this
approach, constraints on multiplication parameters can be formulated in ways
that capture the symmetry features of the coset group. A complete
classification of numbers systems is therefore obtained based on possible group
structures of a given order. General matrix representations are also obtained
through the coset procedure and by construction, the form of these matrices is
invariant under matrix addition and multiplication. Since group symmetries are
captured naturally into each number system, the coset group approach can add
insight into the utility of multidimensional number systems in describing
symmetries in nature.
