Circular and hyperbolic quaternions, octonions, and sedenions

摘要

The complex number algebra can be extended to 4-, 8-, and 16-dimensional algebras, some of which are noncommutative and nonassociative. Various assumptions about the norm, including multiplicativity, motivate the arithmetic of these algebras. The 16-dimensional algebra contains nonreal square roots of +1, zero divisors, and nilpotents, and possesses a nonreal norm and a positive modulus. For some of these algebras, exponentials and logarithms are calculated, and striking multilevel exponential polar forms are developed.