Split-complex number: Difference between revisions

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{{Short description|The realsReals 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_form#Quadratic_space|quadratic spaces]].

relates proportional quadratic forms, but the mapping is {{em|not}} an [[isometry]] since the multiplicative identity {{math|(1, 1)}} of {{tmath|\R^2}} is at a distance {{tmath|\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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<math display=block> z^* = x - jy ~.</math>

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

<math display=block>\begin{align}

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\left(z^*\right)^* &= z.

\end{align}</math>

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>z=x+jy</math> is given by the [[isotropic quadratic form]]

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The associated [[bilinear form]] is given by

<math display=block>\langle z, w \rangle = \operatorname\mathcalmathrm{R_eRe}\left(zw^*\right) = \operatorname\mathcalmathrm{R_eRe} \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>

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===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> Recall that idempotentIdempotency 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>

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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===

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

is a [[hyperbola]] for every nonzero {{mvar|a}} in {{tmath|\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>

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

These two lines (sometimes called the '''[[null cone''']]) are [[perpendicular]] in {{tmath|\R^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.

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From the definition it is apparent that the ring of split-complex numbers is isomorphic to the [[group ring]] {{tmath|\R[C_2]}} of the [[cyclic group]] {{math|C{{sub|2}}}} over the real numbers {{tmath|\R.}}

Elements of the [[identity component]] in the [[group of units]] in '''D''' have four square roots.: say <math>p = \exp (q), \ \ q \in D. \text{then} \pm \exp(\frac{q}{2}) </math> are square roots of ''p''. Further, <math>\pm j \exp(\frac{q}{2})</math> are also square roots of ''p''.

The [[idempotent element (ring theory)|idempotents]] <math>\frac{1 \pm j}{2}</math> are their own square roots, and the square root of <math>s \frac{1 \pm j}{2}, \ \ s > 0, \ \text{is} \ \sqrt{s} \frac{1 \pm j}{2}</math>

==Matrix representations==

One can easily represent split-complex numbers by [[matrix (mathematics)|matrices]]. The split-complex number <math>z = x + jy</math> can be represented by the matrix <math>z \mapsto \begin{pmatrix}x & y \\ y & x\end{pmatrix}.</math>

Addition and multiplication of split-complex numbers are then given by matrix addition and multiplication. The squared modulus of {{mvar|z}} is given by the [[determinant]] of the corresponding matrix.

In fact there are many representations of the split-complex plane in the four-dimensional [[ring (mathematics)|ring]] of 2x2 real matrices. The real multiples of the [[identity matrix]] form a [[real line]] in the matrix ring M(2,R). Any hyperbolic unit ''m'' provides a [[basis (linear algebra)|basis]] element with which to extend the real line to the split-complex plane. The matrices

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which square to the identity matrix satisfy <math>a^2 + bc = 1 .</math>

For example, when ''a'' = 0, then (''b,c'') is a point on the standard hyperbola. More generally, there is a hypersurface in M(2,R) of hyperbolic units, any one of which serves in a basis to represent the split-complex numbers as a [[subring]] of M(2,R).<ref>{{wikibooks-inline|Abstract Algebra/2x2 real matrices}}</ref>{{Better source needed|reason=Wikibooks not a reliable source ([[WP:USERGENERATED]]) |date=May 2023}}

The number <math>z = x + jy</math> can be represented by the matrix &nbsp;<math>x\ I + y\ m .</math>

== History ==

The use of split-complex numbers dates back to 1848 when [[James Cockle (lawyer)|James Cockle]] revealed his [[tessarine]]s.<ref name=JC>[[James Cockle]] (1849) [https://www.biodiversitylibrary.org/item/20121#page/51/mode/1up On a New Imaginary in Algebra] 34:37–47, ''London-Edinburgh-Dublin Philosophical Magazine'' (3) '''33''':435–9, link from [[Biodiversity Heritage Library]].</ref> [[William Kingdon Clifford]] used split-complex numbers to represent sums of spins. Clifford introduced the use of split-complex numbers as coefficients in a quaternion algebra now called [[split-biquaternion]]s. He called its elements "motors", a term in parallel with the "rotor" action of an ordinary complex number taken from the [[circle group]]. Extending the analogy, functions of a [[motor variable]] contrast to functions of an ordinary [[complex variable]].

Since the late twentieth century, the split-complex multiplication has commonly been seen as a [[Lorentz boost]] of a [[spacetime]] plane.<ref>Francesco Antonuccio (1994) [https://arxiv.org/abs/gr-qc/9311032 Semi-complex analysis and mathematical physics]</ref><ref>F. Catoni, D. Boccaletti, R. Cannata, V. Catoni, E. Nichelatti, P. Zampetti. (2008) ''The Mathematics of Minkowski Space-Time'', [[Birkhäuser Verlag]], Basel. Chapter 4: Trigonometry in the Minkowski plane. {{isbn|978-3-7643-8613-9}}.</ref><ref>{{cite book |author1=Francesco Catoni|author2=Dino Boccaletti |author3=Roberto Cannata |author4=Vincenzo Catoni |author5=Paolo Zampetti|title=Geometry of Minkowski Space-Time |year=2011 |publisher=Springer Science & Business Media |isbn=978-3-642-17977-8 |chapter=Chapter 2: Hyperbolic Numbers}}</ref><ref>{{cite journal |mode=cs2 |last=Fjelstad |first=Paul |year=1986 |title=Extending special relativity via the perplex numbers |journal=American Journal of Physics |volume=54 |issue=5 |pagepages=416–422 |doi=10.1119/1.14605 |bibcode=1986AmJPh..54..416F }}</ref><ref>[[Louis Kauffman]] (1985) "Transformations in Special Relativity", [[International Journal of Theoretical Physics]] 24:223–36.</ref><ref>Sobczyk, G.(1995) [http://garretstar.com/secciones/publications/docs/HYP2.PDF Hyperbolic Number Plane], also published in ''College Mathematics Journal'' 26:268–80.</ref> In that model, the number {{math|1=''z'' = ''x'' + ''y'' ''j''}} represents an event in a spatio-temporal plane, where ''x'' is measured in nanosecondsseconds and {{mvar|y}} in [[David Mermin#Word and phrase coinages|Mermin's feetlight-second]]s. The future corresponds to the quadrant of events {{math| {''z'' : {{abs|''y''}} < ''x''}<nowiki/>}}, which has the split-complex polar decomposition <math>z = \rho e^{aj} \!</math>. The model says that {{mvar|z}} can be reached from the origin by entering a [[frame of reference]] of [[rapidity]] {{mvar|a}} and waiting {{mvar|ρ}} nanoseconds. The split-complex equation

<math display=block>e^{aj} \ e^{bj} = e^{(a + b)j}</math>

expressing products on the unit hyperbola illustrates the additivity of rapidities for collinear velocities. Simultaneity of events depends on rapidity {{mvar|a}};

<math display=block>\{ z = \sigma j e^{aj} : \sigma \isin \R \}</math>

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>{{tmath|1= z^*w+zw^* = 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.

In 1935 J.C. Vignaux and A. Durañona y Vedia developed the split-complex geometric algebra and function theory in four articles in ''Contribución a las Ciencias Físicas y Matemáticas'', [[National University of La Plata]], [[Argentina|República Argentina]] (in Spanish). These expository and pedagogical essays presented the subject for broad appreciation.<ref>Vignaux, J.(1935) "Sobre el numero complejo hiperbolico y su relacion con la geometria de Borel", ''Contribucion al Estudio de las Ciencias Fisicas y Matematicas'', Universidad Nacional de la Plata, Republica Argentina</ref>

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In 1941 E.F. Allen used the split-complex geometric arithmetic to establish the [[nine-point hyperbola]] of a triangle inscribed in&nbsp;{{math|1=''zz''{{sup|∗}} = 1}}.<ref>Allen, E.F. (1941) "On a Triangle Inscribed in a Rectangular Hyperbola", [[American Mathematical Monthly]] 48(10): 675–681</ref>

In 1956 Mieczyslaw Warmus published "Calculus of Approximations" in ''Bulletin de l’Académie polonaise des sciences'' (see link in References). He developed two algebraic systems, each of which he called "approximate numbers", the second of which forms a real algebra.<ref>M. Warmus (1956) [http://www.cs.utep.edu/interval-comp/warmus.pdf "Calculus of Approximations"] {{webarchive |url=https://web.archive.org/web/20120309164421/http://www.cs.utep.edu/interval-comp/warmus.pdf |date=2012-03-09 }}, ''Bulletin de l'Académie polonaise des sciences'', Vol. 4, No. 5, pp.&nbsp;253–257, {{MR|id=0081372}}</ref> [[D. H. Lehmer]] reviewed the article in [[Mathematical Reviews]] and observed that this second system was isomorphic to the "hyperbolic complex" numbers, the subject of this article.

In 1961 Warmus continued his exposition, referring to the components of an approximate number as midpoint and radius of the interval denoted.

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* ''approximate numbers'', Warmus (1956), for use in [[interval analysis]]

* ''double numbers'', [[Isaak Yaglom|I.M. Yaglom]] (1968), Kantor and Solodovnikov (1989), [[Michiel Hazewinkel|Hazewinkel]] (1990), Rooney (2014)

* ''hyperbolic numbers'', W. Miller & R. Boehning (1968),<ref>{{cite journal |last1=Miller |first1=William |last2=Boehning |first2=Rochelle |title=Gaussian, parabolic, and hyperbolic numbers |journal=The Mathematics Teacher |volume=61 |number=4 |year=1968 |pages=377-382377–382 |doi=10.5951/MT.61.4.0377 |jstor=27957849 }}</ref> G. Sobczyk (1995)

* ''anormal-complex numbers'', W. Benz (1973)

* ''perplex numbers'', P. Fjelstad (1986) and Poodiack & LeClair (2009)

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* F. Reese Harvey. ''Spinors and calibrations.'' Academic Press, San Diego. 1990. {{isbn|0-12-329650-1}}. Contains a description of normed algebras in indefinite signature, including the Lorentz numbers.

* Hazewinkle, M. (1994) "Double and dual numbers", [[Encyclopaedia of Mathematics]], Soviet/AMS/Kluwer, Dordrect.

* [[Kevin McCrimmon]] (2004) ''A Taste of Jordan Algebras'', pp 66, 157, Universitext, Springer {{isbn|0-387-95447-3}} {{mr|id=2014924}}

* C. Musès, "Applied hypernumbers: Computational concepts", Appl. Math. Comput. 3 (1977) 211–226.

* C. Musès, "Hypernumbers II—Further concepts and computational applications", Appl. Math. Comput. 4 (1978) 45–66.