Circular and hyperbolic quaternions, octonions, and sedenions—Further results

摘要

A sedenion, a 16-dimensional hypercomplex number z = a + Σbnin + Σcnεn + di0, where in are the classical octonions, εn are corresponding proper square roots of +1, and i0 = inεn, has three conjugates and three norms, which when composed yield one real modulus. The numbers 12(1 ± ε) generate a unique arithmetic containing zero divisors, idempotents, and multivalued identities. All three quaternionic subalgebras of the sedenions can be factored into polar forms involving multilevel exponents, and the geometry of these polar forms is explored. The conic quaternionic subalgebra is shown to have only four square roots of +1 and four square roots of −1.