Editing Split-complex number (section)

{{Short description|The reals with an extra square root of +1 adjoined}}
{{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]].

A similar algebra based on {{tmath|\R^2}} and component-wise operations of addition and multiplication, {{tmath|(\R^2, +, \times, xy),}} where {{mvar|xy}} is the [[quadratic form]] on {{tmath|\R^2,}} also forms a [[quadratic space]]. The [[ring isomorphism]]
<math display=block>\begin{align}
       D &\to \mathbb{R}^2 \\
  x + yj &\mapsto (x - y, x + y)
\end{align}</math>
is an [[quadratic space#isometry|isometry]] of quadratic spaces.

Split-complex numbers have many other names; see ''{{section link||Synonyms}}'' below. See the article ''[[Motor variable]]'' for functions of a split-complex number.