Split-complex number

In algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit j satisfying ${\displaystyle j^{2}=1}$, where ${\displaystyle j\neq \pm 1}$. A split-complex number has two real number components x and y, and is written ${\displaystyle z=x+yj.}$ The conjugate of z is ${\displaystyle z^{*}=x-yj.}$ Since ${\displaystyle j^{2}=1,}$ the product of a number z with its conjugate is ${\displaystyle N(z):=zz^{*}=x^{2}-y^{2},}$ an isotropic quadratic form.

The collection D of all split-complex numbers ${\displaystyle z=x+yj}$ for ⁠${\displaystyle x,y\in \mathbb {R} }$⁠ forms an algebra over the field of real numbers. Two split-complex numbers w and z have a product wz that satisfies ${\displaystyle N(wz)=N(w)N(z).}$ This composition of N over the algebra product makes (D, +, ×, *) a composition algebra.

A similar algebra based on ⁠${\displaystyle \mathbb {R} ^{2}}$⁠ and component-wise operations of addition and multiplication, ⁠${\displaystyle (\mathbb {R} ^{2},+,\times ,xy),}$⁠ where xy is the quadratic form on ⁠${\displaystyle \mathbb {R} ^{2},}$⁠ also forms a quadratic space. The ring isomorphism

$${\displaystyle {\begin{aligned}D&\to \mathbb {R} ^{2}\\x+yj&\mapsto (x-y,x+y)\end{aligned}}}$$

relates proportional quadratic forms, but the mapping is not an isometry since the multiplicative identity (1, 1) of ⁠${\displaystyle \mathbb {R} ^{2}}$⁠ is at a distance ⁠${\displaystyle {\sqrt {2}}}$⁠ 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.