Split-complex number
The reals with an extra square root of +1 adjoined / From Wikipedia, the free encyclopedia
In algebra, a split complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit j satisfying A split-complex number has two real number components x and y, and is written
The conjugate of z is
Since
the product of a number z with its conjugate is
an isotropic quadratic form.
The collection D of all split complex numbers for
forms an algebra over the field of real numbers. Two split-complex numbers w and z have a product wz that satisfies
This composition of N over the algebra product makes (D, +, ×, *) a composition algebra.
A similar algebra based on and component-wise operations of addition and multiplication,
where xy is the quadratic form on
also forms a quadratic space. The ring isomorphism
relates proportional quadratic forms, but the mapping is not an isometry since the multiplicative identity (1, 1) of is at a distance
from 0, which is normalized in D.
Split-complex numbers have many other names; see § Synonyms below. See the article Motor variable for functions of a split-complex number.