Editing Split-complex number (section)

==Definition==
A '''split-complex number''' is an ordered pair of real numbers, written in the form

<math display=block>z = x + jy</math>

where {{mvar|x}} and {{mvar|y}} are [[real number]]s and the '''hyperbolic unit'''<ref>Vladimir V. Kisil (2012) ''Geometry of Mobius Transformations: Elliptic, Parabolic, and Hyperbolic actions of SL(2,R)'', pages 2, 161, Imperial College Press {{ISBN|978-1-84816-858-9}}</ref> {{mvar|j}} satisfies

<math display=block>j^2 = +1</math>

In the field of [[complex number]]s  the [[imaginary unit]] i satisfies <math>i^2 = -1 .</math> The change of sign distinguishes the split-complex numbers from the ordinary complex ones. The hyperbolic unit {{mvar|j}} is ''not'' a real number but an independent quantity.

The collection of all such {{mvar|z}} is called the '''split-complex plane'''. [[Addition]] and [[multiplication]] of split-complex numbers are defined by

<math display=block>\begin{align}
  (x + jy) + (u + jv) &= (x + u) + j(y + v) \\
     (x + jy)(u + jv) &= (xu + yv) + j(xv + yu).
\end{align}</math>

This multiplication is [[commutative]], [[associative]] and [[distributive property|distributes]] over addition.

===Conjugate, modulus, and bilinear form===
Just as for complex numbers, one can define the notion of a '''split-complex conjugate'''. If

<math display=block> z = x + jy ~,</math>

then the conjugate of {{mvar|z}} is defined as

<math display=block> z^* = x - jy ~.</math>

The conjugate is an [[involution (mathematics)|involution]] which satisfies similar properties to the [[complex conjugate]]. Namely,

<math display=block>\begin{align}
           (z + w)^* &= z^* + w^* \\
              (zw)^* &= z^* w^* \\
  \left(z^*\right)^* &= z.
\end{align}</math>

The squared '''modulus''' of a split-complex number <math>z=x+jy</math> is given by the [[isotropic quadratic form]]

<math display=block>\lVert z \rVert^2 = z z^* = z^* z = x^2 - y^2 ~.</math>

It has the [[composition algebra]] property:

<math display=block>\lVert z w \rVert = \lVert z \rVert \lVert w \rVert ~.</math>

However, this quadratic form is not [[definite bilinear form|positive-definite]] but rather has [[metric signature|signature]] {{math|(1, −1)}}, so the modulus is ''not'' a [[norm (mathematics)|norm]].

The associated [[bilinear form]] is given by

<math display=block>\langle z, w \rangle = \operatorname\mathrm{Re}\left(zw^*\right) = \operatorname\mathrm{Re} \left(z^* w\right) = xu - yv ~,</math>

where <math>z=x+jy</math> and <math>w=u+jv.</math> Here, the ''real part'' is defined by <math>\operatorname\mathrm{Re}(z) = \tfrac{1}{2}(z + z^*) = x</math>. Another expression for the squared modulus is then

<math display=block> \lVert z \rVert^2 = \langle z, z \rangle ~.</math>

Since it is not positive-definite, this bilinear form is not an [[inner product]]; nevertheless the bilinear form is frequently referred to as an ''indefinite inner product''. A similar abuse of language refers to the modulus as a norm.

A split-complex number is invertible [[if and only if]] its modulus is nonzero {{nowrap|(<math>\lVert z \rVert \ne 0</math>),}} thus numbers of the form {{math|''x'' ± ''j x''}} have no inverse. The [[multiplicative inverse]] of an invertible element is given by

<math display=block>z^{-1} = \frac{z^*}{ {\lVert z \rVert}^2} ~.</math>

Split-complex numbers which are not invertible are called [[null vector]]s. These are all of the form {{math|(''a'' ± ''j a'')}} for some real number {{mvar|a}}.

===The diagonal basis===
There are two nontrivial [[idempotent element (ring theory)|idempotent element]]s given by <math>e=\tfrac{1}{2}(1-j)</math> and <math>e^* = \tfrac{1}{2}(1+j).</math> Idempotency means that <math>ee=e</math> and <math>e^*e^*=e^*.</math> Both of these elements are null:

<math display=block>\lVert e \rVert = \lVert e^* \rVert = e^* e = 0 ~.</math>

It is often convenient to use {{mvar|e}} and {{mvar|e}}<sup>∗</sup> as an alternate [[basis (linear algebra)|basis]] for the split-complex plane. This basis is called the '''diagonal basis''' or '''null basis'''. The split-complex number {{mvar|z}} can be written in the null basis as

<math display=block> z = x + jy = (x - y)e + (x + y)e^* ~.</math>

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>

The split-complex conjugate in the diagonal basis is given by
<math display=block>(a, b)^* = (b, a)</math>
and the squared modulus by

<math display=block> \lVert (a, b) \rVert^2  = 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}}]]

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}}.