A spherical-number is a point on a sphere in a complex volume that is defined by orthogonal orientated imaginary unit numbers . The group is a set with four operations; these are the two binary operators for spherical-multiplication and spherical-addition (not to be confused with cartesian- or vector-addition), and two unitary operators for spherical-exponentiation and spherical-negation. The group is characterised by:
- is an abelian group with respect to spherical-multiplication with identity one,
- is an abelian group with respect to spherical-addition with identity zero,
- the distributive laws hold.
If the term spherical is dropped, (i) and (ii) remain true and the distributive laws over cartesian-addition is lost.
The innovative step to define these numbers, was not in defining a Hamilton like operation , but using the same rule that defines the spherical numbers to eliminate the product terms . The complex numbers are a subset of .