Split-complex number: Difference between revisions

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{{redirect|Double number|the computer number format|double-precision floating-point format}}

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

The collection {{mvar|D}} of all split-complex numbers <math>z=x+yj</math> for {{tmath|x,y \in \R}} forms an [[algebra over a field|algebra over the field of real numbers]]. Two split-complex numbers {{mvar|w}} and {{mvar|z}} have a product {{mvar|wz}} that satisfies <math>N(wz)=N(w)N(z).</math> This composition of {{mvar|N}} over the algebra product makes {{math|(''D'', +, ×, *)}} a [[composition algebra]].