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{{Short description|
{{redirect|Double number|the computer number format|double-precision floating-point format}}
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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]].
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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===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>
<math display=block>\lVert e \rVert = \lVert e^* \rVert = e^* e = 0 ~.</math>
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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==
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The number <math>z = x + jy</math> can be represented by the matrix <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 |pages=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 seconds and {{mvar|y}} in [[light-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
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<sup>''e''</sup> over ''F'' generalizing Cayley–Dickson algebras
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 {{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"] {{
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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