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If we denote the number <math>z=ae+be^*</math> for real numbers {{mvar|a}} and {{mvar|b}} by {{math|(''a'', ''b'')}}, then split-complex multiplication is given by
<math display="block">\left( a_1, b_1 \right) \left( a_2, b_2 \right) = \left( a_1 a_2, b_1 b_2 \right) ~.</math>
In this basis, 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 split-complex conjugate in the diagonal basis is given by
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and the modulus by
<math display="block">\lVert (a, b) \rVert = ab.</math>
===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}}]]
▲
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}}.
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