Hypercomplex numbers | Composition algebras | Linear algebra
Split-complex number
In algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components x and y, and is written z = x + y j, where j2 = 1. The conjugate of z is z∗ = x − y j. Since j2 = 1, the product of a number z with its conjugate is N(z) := zz∗ = x2 − y2, an isotropic quadratic form. The collection D of all split complex numbers z = x + y j for x, y ∈ R forms an algebra over the field of real numbers. Two split-complex numbers w and z have a product wz that satisfies N(wz) = N(w)N(z). This composition of N over the algebra product makes (D, +, ×, *) a composition algebra. A similar algebra based on R2 and component-wise operations of addition and multiplication, (R2, +, ×, xy), where xy is the quadratic form on R2, 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 R2 is at a distance √2 from 0, which is normalized in D. Split-complex numbers have many other names; see below. See the article Motor variable for functions of a split-complex number. (Wikipedia).
