In algebra, a split complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit j satisfying ${\displaystyle j^{2}=1.}$ 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

Split-complex numbers have many other names; see § Synonyms below. See the article Motor variable for functions of a split-complex number.