Split-complex number: Difference between revisions

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

The collection {{mvar|D}} of all split complex numbers {{<math|1=''>z'' = ''x'' + ''y{{tsp}}j''}}yj</math> for {{mathtmath|''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 {{mathmvar|''wz''}} that satisfies {{<math|1=''>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 {{mathtmath|'''\R'''^{^2}}} and component-wise operations of addition and multiplication, {{mathtmath|('''\R'''^{^2}, +, ×\times, ''xy''),}}, where {{mvar|xy}} is the [[quadratic form]] on {{mathtmath|'''\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>

relates proportional quadratic forms, but the mapping is {{em|not}} an [[isometry]] since the multiplicative identity {{math|(1, 1)}} of {{mathtmath|'''\R'''^{^2}}} is at a distance {{radictmath|\sqrt 2}} from 0, which is normalized in {{mvar|D}}.

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

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These three properties imply that the split-complex conjugate is an [[automorphism]] of [[order (group theory)|order]] 2.

The squared '''modulus''' of a split-complex number {{<math|1=''>z'' = ''x'' + ''j'' ''y''}}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>

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The associated [[bilinear form]] is given by

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

where {{<math|1=''>z'' = ''x'' + ''j'' ''y''}}jy</math> and {{<math|1=''>w'' = ''u'' + ''j'' ''v''}}jv.</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|1=''>e'' = \tfrac{1}{1/2}}(1 − ''-j'') }}</math> and {{<math|1=''>e''^∗^* = \tfrac{1}{1/2}}(1 + ''j'') }}.</math> Recall that idempotent means that {{<math|1=''>ee'' = ''e''}}</math> and {{<math|1=''e''^∗''e''^∗ ^*e^*= ''e''<sup>∗^*.</supmath>}}. Both of these elements are null:

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

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

If we denote the number {{<math|1=''>z'' = ''ae'' + ''be''<sup>∗^*</supmath>}} 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 {{mathtmath|'''\R''' ⊕\oplus '''\R'''}} with addition and multiplication defined pairwise.

The split-complex conjugate in the diagonal basis is given by

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<math display="block">\lVert (a, b) \rVert = ab.</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 {{mathtmath|'''\R''' ⊕\oplus '''\R'''}} plane with its "unit circle" given by {{<math|1=>\{(''a'', ''b'') ∈\in '''\R''' ⊕\oplus '''\R''' : ''ab'' = 1\}.<nowiki/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 {{mathtmath|'''\R''' ⊕\oplus '''\R'''}}.

==Geometry==

<!-- This section is linked from [[Lorentz transformation]] -->

[[Image:Drini-conjugatehyperbolas.svg|thumb|
{{colorboxlegend-line|solid blue}} |Unit hyperbola: {{math|1=‖''z''‖ = 1}}, <br>}}
{{colorboxlegend-line|green}}solid green|Conjugate hyperbola: {{math|1=‖''z''‖ = −1}} <br>}}
{{colorboxlegend-line|red}}solid red|Asymptotes: {{math|1=‖''z''‖ = 0}}.}}]]

A two-dimensional real [[vector space]] with the Minkowski inner product is called {{nobrmath|(1 + 1)}}-dimensional [[Minkowski space]], often denoted {{mathtmath|'''\R'''^{{sup|1,1}.}}}. Just as much of the [[geometry]] of the Euclidean plane {{mathtmath|'''\R'''{{sup|^2}}}} can be described with complex numbers, the geometry of the Minkowski plane {{mathtmath|'''\R'''^{{sup|1,1}} }} can be described with split-complex numbers.

The set of points

<math display="block">\left\{ z : \lVert z \rVert^2 = a^2 \right\}</math>

is a [[hyperbola]] for every nonzero {{mvar|a}} in {{mathtmath|'''\R'''.}}. The hyperbola consists of a right and left branch passing through {{math|(''a'', 0)}} and {{math|(−''a'', 0)}}. The case {{math|1=''a'' = 1}} is called the [[unit hyperbola]]. The conjugate hyperbola is given by

<math display="block">\left\{ z : \lVert z \rVert^2 = -a^2 \right\}</math>

with an upper and lower branch passing through {{math|(0, ''a'')}} and {{math|(0, −''a'')}}. The hyperbola and conjugate hyperbola are separated by two diagonal [[asymptote]]s which form the set of null elements:

<math display="block">\left\{ z : \lVert z \rVert = 0 \right\}.</math>

These two lines (sometimes called the '''null cone''') are [[perpendicular]] in {{mathtmath|'''\R'''{{sup|^2}}}} and have slopes ±1.

Split-complex numbers {{mvar|z}} and {{mvar|w}} are said to be [[hyperbolic-orthogonal]] if {{math|1=⟨''z'', ''w''⟩ = 0 }}. While analogous to ordinary orthogonality, particularly as it is known with ordinary complex number arithmetic, this condition is more subtle. It forms the basis for the [[Minkowski space#Causal structure|simultaneous hyperplane]] concept in spacetime.

The analogue of [[Euler's formula]] for the split-complex numbers is

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The exponential map

<math display="block">\exp\colon (\mathbb{R}, +) \to \mathrm{SO}^{+}(1, 1)</math>

sending {{mvar|θ}} to rotation by {{math|exp(''jθ'')}} is a [[group isomorphism]] since the usual exponential formula applies:

<math display="block">e^{j(\theta + \phi)} = e^{j\theta}e^{j\phi}.</math>

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==Algebraic properties==

In [[abstract algebra]] terms, the split-complex numbers can be described as the [[quotient ring|quotient]] of the [[polynomial ring]] {{mathtmath|'''\R'''[''x'']}} by the [[ideal (ring theory)|ideal]] generated by the [[polynomial]] {{<math|''>x''<sup>^2-1,</supmath> − 1}},

<math display="block">\R[x]/(x^2-1 ).</math>

{{block indent | em = 1.5 | text = {{math|'''R'''[''x'']/(''x''{{sup|2}} − 1 ).}}}}

The image of {{mvar|x}} in the quotient is the "imaginary" unit {{mvar|j}}. With this description, it is clear that the split-complex numbers form a [[commutative algebra (structure)|commutative algebra]] over the real numbers. The algebra is ''not'' a [[field (mathematics)|field]] since the null elements are not invertible. All of the nonzero null elements are [[zero divisor]]s.

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:<math>\lVert zw \rVert = \lVert z \rVert \lVert w \rVert ~</math> for any numbers {{mvar|z}} and {{mvar|w}}.

From the definition it is apparent that the ring of split-complex numbers is isomorphic to the [[group ring]] {{mathtmath|'''\R'''[C{{sub|^2}}]}} of the [[cyclic group]] {{math|C{{sub|2}}}} over the real numbers {{mathtmath|'''\R'''.}}.

==Matrix representations==

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For any real number {{mvar|a}}, a hyperbolic rotation by a [[hyperbolic angle]] {{mvar|a}} corresponds to multiplication by the matrix

<math display="block">\begin{pmatrix}
\cosh a & \sinh a \\
\sinh a & \cosh a
\end{pmatrix}.</math>

[[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 {{mathtmath|'''\R'''{{sup|^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

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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}}.

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is the line of events simultaneous with the origin in the frame of reference with rapidity ''a''.

Two events {{mvar|z}} and {{mvar|w}} are [[hyperbolic-orthogonal]] when {{<math|1=''>z''{{sup|∗}}''^*w'' + ''zw''{{sup|∗}}^* = 0}}.</math> Canonical events {{math| exp(''aj'')}} and {{math|''j'' exp(''aj'')}} are hyperbolic orthogonal and lie on the axes of a frame of reference in which the events simultaneous with the origin are proportional to {{math|''j'' exp(''aj'')}}.

In 1933 [[Max Zorn]] was using the [[split-octonion]]s and noted the [[composition algebra]] property. He realized that the [[Cayley–Dickson construction]], used to generate division algebras, could be modified (with a factor gamma, {{mvar|γ}}) to construct other composition algebras including the split-octonions. His innovation was perpetuated by [[Adrian Albert]], Richard D. Schafer, and others.<ref>Robert B. Brown (1967)[http://projecteuclid.org/euclid.pjm/1102992693 On Generalized Cayley-Dickson Algebras], [[Pacific Journal of Mathematics]] 20(3):415–22, link from [[Project Euclid]].</ref> The gamma factor, with {{math|'''R'''}} as base field, builds split-complex numbers as a composition algebra. Reviewing Albert for [[Mathematical Reviews]], N. H. McCoy wrote that there was an "introduction of some new algebras of order 2^e over ''F'' generalizing Cayley–Dickson algebras."<ref>N.H. McCoy (1942) Review of "Quadratic forms permitting composition" by A.A. Albert, [[Mathematical Reviews]] #0006140</ref> Taking {{math|1=''F'' = '''R'''}} and {{math|1=''e'' = 1 }} corresponds to the algebra of this article.