Split-complex number

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2024-04-03 22:30:07

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

The collection D of all split complex numbers z = x + y j {\displaystyle z=x+yj} for x , y ∈ R {\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 N ( w z ) = N ( w ) N ( z ) . {\displaystyle N(wz)=N(w)N(z).} This composition of N over the algebra product makes (D, +, ×, *) a composition algebra.