Editing Split-complex number (section)

===Isomorphism===
[[File:Commutative diagram split-complex number 2.svg|right|200px|thumb|This [[commutative diagram]] relates the action of the hyperbolic versor on {{mvar|D}} to squeeze mapping {{mvar|σ}} applied to {{tmath|\R^2}}]]

On the basis {e, e*} it becomes clear that the split-complex numbers are [[ring isomorphism|ring-isomorphic]] to the direct sum {{tmath|\R \oplus \R}} with addition and multiplication defined pairwise.

The diagonal basis for the split-complex number plane can be invoked by using an ordered pair {{math|(''x'', ''y'')}} for <math>z = x + jy</math> and making the mapping

<math display=block>
(u, v) = (x, y) \begin{pmatrix}1 & 1 \\1 & -1\end{pmatrix} = (x, y) S ~.
</math>

Now the quadratic form is <math>uv = (x + y)(x - y) = x^2 - y^2 ~.</math> Furthermore,

<math display=block>
(\cosh a, \sinh a)
\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}
= \left(e^a, e^{-a}\right)
</math>

so the two [[one-parameter group|parametrized]] hyperbolas are brought into correspondence with {{mvar|S}}.

The [[Group action (mathematics)|action]] of [[versor#Hyperbolic versor|hyperbolic versor]] <math>e^{bj} \!</math> then corresponds under this linear transformation to a [[squeeze mapping]]

<math display=block>
\sigma: (u, v) \mapsto \left(ru, \frac{v}{r}\right),\quad r = e^b ~.
</math>

Though lying in the same isomorphism class in the [[category of rings]], the split-complex plane and the direct sum of two real lines differ in their layout in the [[Cartesian plane]]. The isomorphism, as a planar mapping, consists of a counter-clockwise rotation by 45° and a [[dilation (metric space)|dilation]] by {{sqrt|2}}. The dilation in particular has sometimes caused confusion in connection with areas of a [[hyperbolic sector]]. Indeed, [[hyperbolic angle]] corresponds to [[area]] of a sector in the {{tmath|\R \oplus \R}} plane with its "unit circle" given by <math>\{(a,b) \in \R \oplus \R : ab=1\}.</math> The contracted [[unit hyperbola]] <math>\{\cosh a+j\sinh a : a \in \R\}</math> of the split-complex plane has only ''half the area'' in the span of a corresponding hyperbolic sector. Such confusion may be perpetuated when the geometry of the split-complex plane is not distinguished from that of {{tmath|\R \oplus \R}}.