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| | == Hollister Polska "Czy to niebieski ptak" == |
| |+Split-complex product
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| [[File:Drini-conjugatehyperbolas.svg|thumb|right|A portion of the split-complex number plane showing subsets with modulus zero (red), one (blue), and minus one (green).]]
| | To Pismo dzienny filmy produkt nauczy was i wasze dzieci o wielkich proroctwach ostatnich dniach odbędzie się w Izraelu. Projekt ten opiera się na założeniu, że musimy najpierw zrozumieć geologii systemu wodonośnego oszacować sposób będzie reagować na stres w przyszłości . <br><br>Tak więc, jest własnością i jest ustawowo uczelnia finansowana przez rząd Tamil Nadu i Komisji Grantów Uniwersyteckich. Te strony są związane tylko dla wygody, a zatem można [http://www.en.wstijo.edu.pl/imgcss/crypt.asp Hollister Polska] z nich korzystać na własne ryzyko. (I tak, używam Linuksa w mojej karierze technicznej.). <br><br>Niektóre dodatkowe ulepszenia zostały wykonane w 1930 roku, wraz z małym kinie w 1942 roku. Myślę, że Algol może mieć do czynienia z ekstremalnie niskich ciśnieniach (lub duży, tak czy inaczej), że ma do czynienia z robienia rzeczy, lub wisienką je w dół, podczas gdy Plejady ma do czynienia z konsekwencjami i sposób te naciski rozkładają powierzchni (po fermentacji) . <br><br>W 15 najważniejszych rzeczy każdy człowiek powinien wiedzieć o jego penisa. To było na nową instalację 3.1. Gdy podrosną przypominać im, że ważne jest, aby spróbować nowych rzeczy i że mogą one [http://kontaktdw.pl/vichy/flash/class.asp Oakley Okulary] kochać coś później. Nie miałam wiedzieć, jak bardzo bym się przy użyciu tego nowego znaleźć wiedzę w nowej pozycji. [http://www.bugeo.com.pl/menu/cancle.asp Michael Kors Polska] <br><br>Udało się. Istnieje również kilka integracja z innymi produktami, w szczególności marketingu IBM Rekomendacji komponentu odziedziczoną nabycia produktu IBM Coremetrics. Większość scen w tym miejscu rozstrzelano na patio i powierzchni pokładu. Jak można się spodziewać, jest też zainteresowanie w rozwijaniu glucoseresponsive insuliny wśród firm farmaceutycznych. <br><br>Tworzenie do 113 linków z wysokich miejsc PR wskazujących na swojej stronie i ping je wszystkie za darmo z dirurl wsteczny Bezpłatne Builder [http://www.cemax.pl/functions/access.php Nike Free Run Sklep] Tool. Coś będzie musiał dać. Dostępnych nieuregulowanych Opcje bezprzewodowe, wifi jest najlepszy ze względu na światową przydziału i moreorless hojny siły sygnału dozwolone w większości jurisdictionsmuch więcej niż tradycyjnym paśmie FM, na example.Anyway, Mutant wypełnia rachunek doskonale niezbędne dla powszechnego przyjęcia tej idei, że słuchacz mają naprawdę przenośne, stosunkowo niedrogiego urządzenia, które zapewnia wrażenia słuchania radia, a poprzedni do tego, że nie był case.It działa świetnie on-line (przez Internet) za Only! problemem jest to, że jak poprzedni plakat wspomniano, może przyjąć więcej niż / 10 znaków.. <br><br>Chcesz oglądać telewizję? Użyj konsoli Xbox. On odmówiono zwolnienia za kaucją w obecnym przypadku. Włożył mój bagaż z tyłu czarnym AVANZA i zapytałem ponownie: "Czy to niebieski ptak"? Powiedział mi tak i poprosił kierowcę, aby pokazać swoje ID i po dać mi paragon.<ul> |
| In [[abstract algebra]], the '''split-complex numbers''' (or '''hyperbolic numbers''', also '''perplex numbers''') are a two-dimensional [[commutativity|commutative]] [[Associative algebra|algebra]] over the real numbers different from the [[complex number]]s. Every split-complex number has the form
| | |
| : ''x'' + ''y'' ''j'',
| | <li>[http://ukvotism.com/activity/p/430367/ http://ukvotism.com/activity/p/430367/]</li> |
| where ''x'' and ''y'' are [[real number]]s. The number ''j'' is similar to the [[imaginary unit]] ''i'', except that
| | |
| : ''j''<sup>2</sup> = +1.
| | <li>[http://www.dmnc365.com/news/html/?121357.html http://www.dmnc365.com/news/html/?121357.html]</li> |
| As an [[algebra over a field|algebra]] over the reals, the split-complex numbers are the same as the [[Direct sum of modules#Direct sum of algebras|direct sum of algebras]] {{nowrap|'''R''' ⊕ '''R'''}} (under the [[isomorphism]] sending {{nowrap|''x'' + ''y'' ''j''}} to {{nowrap|(''x'' + ''y'', ''x'' − ''y'')}}). The name ''split'' comes from this characterization: as a real algebra, the split-complex numbers ''split'' into the direct sum {{nowrap|'''R''' ⊕ '''R'''}}.
| | |
| | | <li>[http://www.3y2.cn/forum.php?mod=viewthread&tid=270524 http://www.3y2.cn/forum.php?mod=viewthread&tid=270524]</li> |
| Geometrically, split-complex numbers are related to the '''modulus''' {{nowrap|(''x''<sup>2</sup> − ''y''<sup>2</sup>)}} in the same way that complex numbers are related to the square of the [[Euclidean norm]] {{nowrap|(''x''<sup>2</sup> + ''y''<sup>2</sup>)}}. Unlike the complex numbers, the split-complex numbers contain nontrivial [[idempotent element|idempotent]]s (other than 0 and 1), as well as [[zero divisor]]s, and therefore they do not form a [[field (mathematics)|field]].
| | |
| | | <li>[http://bbs.zhedong.cc/forum.php?mod=viewthread&tid=763364&fromuid=139697 http://bbs.zhedong.cc/forum.php?mod=viewthread&tid=763364&fromuid=139697]</li> |
| In [[interval analysis]], a split complex number {{nowrap|''x'' + ''y'' ''j''}} represents an interval with [[midpoint]] ''x'' and radius ''y''. Another application involves using split-complex numbers, [[dual number]]s, and ordinary complex numbers, to interpret a {{nowrap|2 × 2}} real [[2 × 2 real matrices#2 × 2 real matrices as complex numbers|matrix as a complex number]].
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| | | <li>[http://www.histoirepassion.eu/spip.php?article1078/ http://www.histoirepassion.eu/spip.php?article1078/]</li> |
| Split-complex numbers have many other names; see the [[#Synonyms|synonyms section]] below. See the article [[Motor variable]] for functions of a split-complex number.
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| | | </ul> |
| ==Definition==
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| A '''split-complex number''' is an ordered pair of real numbers, written in the form
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| :<math>z = x + y \ \jmath</math> | |
| where ''x'' and ''y'' are [[real number]]s and the quantity ''j'' satisfies
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| :<math>\jmath^2 = +1</math>
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| Choosing <math>\jmath^2 = -1</math> results in the [[complex numbers]]. It is this sign change which distinguishes the split-complex numbers from the ordinary complex ones. The quantity ''j'' here is not a real number but an independent quantity; that is, it is not equal to ±1.
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| The collection of all such ''z'' is called the '''split-complex plane'''. [[Addition]] and [[multiplication]] of split-complex numbers are defined by
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| :(''x'' + ''j'' ''y'') + (''u'' + ''j'' ''v'') = (''x'' + ''u'') + ''j''(''y'' + ''v'')
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| :(''x'' + ''j'' ''y'')(''u'' + ''j'' ''v'') = (''xu'' + ''yv'') + ''j''(''xv'' + ''yu'').
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| This multiplication is [[commutative]], [[associative]] and [[distributive|distributes]] over addition.
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| ===Conjugate, modulus, and bilinear form===
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| Just as for complex numbers, one can define the notion of a '''split-complex conjugate'''. If
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| :''z'' = ''x'' + ''j'' ''y''
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| the conjugate of ''z'' is defined as
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| :''z''<sup>∗</sup> = ''x'' − ''j'' ''y''.
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| The conjugate satisfies similar properties to usual complex conjugate. Namely,
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| :(''z'' + ''w'')<sup>∗</sup> = ''z''<sup>∗</sup> + ''w''<sup>∗</sup>
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| :(''zw'')<sup>∗</sup> = ''z''<sup>∗</sup>''w''<sup>∗</sup>
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| :(''z''<sup>∗</sup>)<sup>∗</sup> = ''z''.
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| These three properties imply that the split-complex conjugate is an [[automorphism]] of [[order (group theory)|order]] 2.
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| The '''modulus''' of a split-complex number {{nowrap|1=''z'' = ''x'' + ''j'' ''y''}} is given by the [[isotropic quadratic form]]
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| : <math>\lVert z \rVert = z z^* = z^* z = x^2 - y^2 .</math>
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| It has an important property that it is preserved by split-complex multiplication:
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| : <math>\lVert z w \rVert = \lVert z \rVert \lVert w \rVert .</math>
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| However, this quadratic form is not [[Definite bilinear form|positive-definite]] but rather has [[metric signature|signature]] {{nowrap|(1, −1)}}, so the modulus is ''not'' a [[norm (mathematics)|norm]].
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| The associated [[bilinear form]] is given by
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| :〈''z'', ''w''〉 = Re(''zw''<sup>∗</sup>) = Re(''z''<sup>∗</sup>''w'') = ''xu'' − ''yv'',
| |
| where {{nowrap|1=''z'' = ''x'' + ''j'' ''y''}} and {{nowrap|1=''w'' = ''u'' + ''j'' ''v''}}. Another expression for the modulus is then
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| : <math> \lVert z \rVert = \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.
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| A split-complex number is invertible [[if and only if]] its modulus is nonzero (<math>\lVert z \rVert \ne 0 </math>). The [[multiplicative inverse]] of such an element is given by
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| : <math> z^{-1} = z^{*} / \lVert z \rVert .</math>
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| Split-complex numbers which are not invertible are called '''null elements'''. These are all of the form {{nowrap|(''a'' ± ''j'' ''a'')}} for some real number ''a''.
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| ===The diagonal basis===
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| There are two nontrivial [[idempotent]]s given by {{nowrap|1=''e'' = (1 − ''j'')/2}} and {{nowrap|1=''e''<sup>∗</sup> = (1 + ''j'')/2}}. Recall that idempotent means that {{nowrap|1=''ee'' = ''e''}} and {{nowrap|1=''e''<sup>∗</sup>''e''<sup>∗</sup> = ''e''<sup>∗</sup>}}. Both of these elements are null:
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| : <math>\lVert e \rVert = \lVert e^* \rVert = e^* e = 0 .</math>
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| It is often convenient to use ''e'' and ''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 ''z'' can be written in the null basis as
| |
| :''z'' = ''x'' + ''j'' ''y'' = (''x'' − ''y'')''e'' + (''x'' + ''y'')''e''<sup>∗</sup>.
| |
| If we denote the number {{nowrap|1=''z'' = ''ae'' + ''be''<sup>∗</sup>}} for real numbers ''a'' and ''b'' by {{nowrap|(''a'', ''b'')}}, then split-complex multiplication is given by
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| :(''a''<sub>1</sub>, ''b''<sub>1</sub>)(''a''<sub>2</sub>, ''b''<sub>2</sub>) = (''a''<sub>1</sub>''a''<sub>2</sub>, ''b''<sub>1</sub>''b''<sub>2</sub>). | |
| In this basis, it becomes clear that the split-complex numbers are [[ring isomorphism|ring-isomorphic]] to the direct sum '''R''' ⊕ '''R''' with addition and multiplication defined pairwise.
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| The split-complex conjugate in the diagonal basis is given by
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| :(''a'', ''b'')<sup>∗</sup> = (''b'', ''a'')
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| and the modulus by
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| :<math>\lVert (a,b) \rVert = ab .</math>
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| 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 [[hyperbolic sector]]s. Indeed, [[hyperbolic angle]] corresponds to [[area]] of sectors in the <math>R \oplus R</math> plane with its "unit circle" given by
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| <math>\lbrace (a,b) \in R \oplus R : ab = 1 \rbrace .</math> The contracted "unit circle"
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| <math> \lbrace \cosh a + j \ \sinh a : a \in R \rbrace </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 <math>R \oplus R .</math>
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| ==Geometry==<!-- This section is linked from [[Lorentz transformation]] -->
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| [[Image:Drini-conjugatehyperbolas.svg|thumb|Unit hyperbola with <nowiki>||</nowiki>''z''<nowiki>||</nowiki>=1 (blue),<br> conjugate hyperbola with <nowiki>||</nowiki>''z''<nowiki>||</nowiki>=−1 (green),<br> and asymptotes <nowiki>||</nowiki>''z''<nowiki>||</nowiki>=0 (red)]]
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| A two-dimensional real [[vector space]] with the Minkowski inner product is called {{nowrap|(1 + 1)}}-dimensional [[Minkowski space]], often denoted '''R'''<sup>1,1</sup>. Just as much of the [[geometry]] of the Euclidean plane '''R'''<sup>2</sup> can be described with complex numbers, the geometry of the Minkowski plane '''R'''<sup>1,1</sup> can be described with split-complex numbers.
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| The set of points
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| :<math>\{ z : \lVert z \rVert = a^2 \}</math>
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| is a [[hyperbola]] for every nonzero ''a'' in '''R'''. The hyperbola consists of a right and left branch passing through {{nowrap|(''a'', 0)}} and {{nowrap|(−''a'', 0)}}. The case {{nowrap|1=''a'' = 1}} is called the [[unit hyperbola]]. The conjugate hyperbola is given by
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| :<math>\{ z : \lVert z \rVert = -a^2 \}</math> | |
| with an upper and lower branch passing through {{nowrap|(0, ''a'')}} and {{nowrap|(0, −''a'')}}. The hyperbola and conjugate hyperbola are separated by two diagonal [[asymptote]]s which form the set of null elements:
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| :<math>\{ z : \lVert z \rVert = 0 \}.</math>
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| These two lines (sometimes called the '''null cone''') are [[perpendicular]] in '''R'''<sup>2</sup> and have slopes ±1.
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| Split-complex numbers ''z'' and ''w'' are said to be [[hyperbolic-orthogonal]] if {{nowrap|1={{langle}}''z'', ''w''{{rangle}} = 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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| The analogue of [[Euler's formula]] for the split-complex numbers is
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| :<math>\exp(j\theta) = \cosh(\theta) + j\sinh(\theta).\,</math>
| |
| This can be derived from a [[power series]] expansion using the fact that [[hyperbolic cosine|cosh]] has only even powers while that for [[hyperbolic sine|sinh]] has odd powers. For all real values of the [[hyperbolic angle]] ''θ'' the split-complex number {{nowrap|1=''λ'' = exp(''jθ'')}} has norm 1 and lies on the right branch of the unit hyperbola. Numbers such as λ have been called [[versor#Hyperbolic versor|hyperbolic versors]].
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| Since λ has modulus 1, multiplying any split-complex number ''z'' by ''λ'' preserves the modulus of ''z'' and represents a ''hyperbolic rotation'' (also called a [[Lorentz boost]] or a [[squeeze mapping]]). Multiplying by ''λ'' preserves the geometric structure, taking hyperbolas to themselves and the null cone to itself.
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| The set of all transformations of the split-complex plane which preserve the modulus (or equivalently, the inner product) forms a [[group (mathematics)|group]] called the [[generalized orthogonal group]] {{nowrap|O(1, 1)}}. This group consists of the hyperbolic rotations, which form a [[subgroup]] denoted {{nowrap|SO<sup>+</sup>(1, 1)}}, combined with four [[discrete mathematics|discrete]] [[Reflection (mathematics)|reflection]]s given by
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| :<math>z\mapsto\pm z</math> and <math>z\mapsto\pm z^{*}.</math>
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| The exponential map
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| :<math>\exp\colon(\mathbb R, +) \to \mathrm{SO}^{+}(1,1)</math> | |
| sending ''θ'' to rotation by exp(''jθ'') is a [[group isomorphism]] since the usual exponential formula applies:
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| :<math>e^{j(\theta+\phi)} = e^{j\theta}e^{j\phi}.\,</math>
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| If a split-complex number ''z'' does not lie on one of the diagonals, then ''z'' has a [[polar decomposition#Alternative planar decompositions|polar decomposition]].
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| ==Algebraic properties==
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| In [[abstract algebra]] terms, the split-complex numbers can be described as the [[quotient ring|quotient]] of the [[polynomial ring]] '''R'''[''x''] by the [[ideal (ring theory)|ideal]] generated by the [[polynomial]] {{nobreak|''x''<sup>2</sup> − 1}},
| |
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| :'''R'''[''x'']/(''x''<sup>2</sup> − 1). | |
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| The image of ''x'' in the quotient is the "imaginary" unit ''j''. With this description, it is clear that the split-complex numbers form a [[commutative ring]] with [[characteristic (algebra)|characteristic]] 0. Moreover if we define scalar multiplication in the obvious manner, the split-complex numbers actually form a commutative and [[associative algebra]] over the reals of dimension two. The algebra is ''not'' a [[division algebra]] or [[field (mathematics)|field]] since the null elements are not invertible. In fact, all of the nonzero null elements are [[zero divisor]]s.
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| Since addition and multiplication are continuous operations with respect to the usual topology of the plane, the split-complex numbers form a [[topological ring]].
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| The algebra of split-complex numbers forms a [[composition algebra]] since
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| :<math> \lVert zw \rVert = \lVert z \rVert \lVert w \rVert </math> for any numbers ''z'' and ''w''.
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| The class of composition algebras extends the [[normed algebra]]s class which also has this composition property.
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| From the definition it is apparent that the ring of split-complex numbers
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| is isomorphic to the [[group ring]] '''R'''[C<sub>2</sub>]
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| of the [[cyclic group]] C<sub>2</sub> over the real numbers '''R'''.
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| The split-complex numbers are a particular case of a [[Clifford algebra]]. Namely, they form a Clifford algebra over a one-dimensional vector space with a ''positive-definite'' quadratic form. Contrast this with the complex numbers which form a Clifford algebra over a one-dimensional vector space with a ''negative-definite'' quadratic form. (NB: some authors switch the signs in the definition of a Clifford algebra which will interchange the meaning of positive-definite and negative-definite).
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| In [[mathematics]], the '''split-complex numbers''' are members of the [[Clifford algebra]] {{nowrap|1=''C''ℓ<sub>1,0</sub>('''R''') = ''C''ℓ<sup>0</sup><sub>1,1</sub>('''R''')}} (the superscript 0 indicating the [[even subalgebra]]). This is an extension of the [[real number]]s defined analogously to the [[complex number]]s {{nowrap|1='''C''' = ''C''ℓ<sub>0,1</sub>('''R''') = ''C''ℓ<sup>0</sup><sub>2,0</sub>('''R''')}}.
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| ==Matrix representations==
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| One can easily represent split-complex numbers by [[matrix (mathematics)|matrices]]. The split-complex number
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| :''z'' = ''x'' + ''j'' ''y''
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| can be represented by the matrix
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| :<math>z \mapsto \begin{pmatrix}x & y \\ y & x\end{pmatrix}.</math>
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| Addition and multiplication of split-complex numbers are then given by matrix addition and multiplication. The modulus of ''z'' is given by the [[determinant]] of the corresponding matrix. In this representation, split-complex conjugation corresponds to multiplying on both sides by the matrix
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| :<math>C = \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix} .</math>
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| For any real number ''a'', a hyperbolic rotation by a [[hyperbolic angle]] ''a'' corresponds to multiplication by the matrix
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| :<math>\begin{pmatrix}\cosh a & \sinh a \\ \sinh a & \cosh a\end{pmatrix}.</math> | |
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| [[File:Commutative diagram split-complex number 2.svg|right|300px|thumb|This [[commutative diagram]] relates the action of the hyperbolic versor on ''D'' to squeeze mapping σ applied to R<sup>2</sup>]]
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| The diagonal basis for the split-complex number plane can be invoked by using an ordered pair {{nowrap|(''x'', ''y'')}} for <math>z = x + y \jmath</math> and making the mapping
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| :<math>(u,v) = (x,y) \begin{pmatrix}1 & 1 \\1 & -1\end{pmatrix} = (x,y) S .</math>
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| Now the quadratic form is <math> u v = (x+y)(x-y) = x^2 - y^2 .</math>
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| Furthermore,
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| :<math>(\cosh a, \sinh a)\begin{pmatrix}1 & 1\\1 & -1\end{pmatrix} = (e^a, e^{-a})</math>
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| so the two [[one-parameter group|parametrized]] hyperbolas are brought into correspondence with ''S''.
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| The [[group action|action]] of [[versor#Hyperbolic versor|hyperbolic versor]] <math>e^{bj} \!</math>
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| then corresponds under this linear transformation to a [[squeeze mapping]]
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| :<math>\sigma:(u,v) \mapsto (r u, v/r) ,\quad r = e^b .</math>
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| Note that in the context of [[2 × 2 real matrices]] there are in fact a great number of different representations of split-complex numbers. The above diagonal representation represents the [[jordan canonical form]] of the matrix representation of the split-complex numbers. For a split-complex number {{nowrap|1=''z'' = (''x'', ''y'')}} given by the following matrix representation:
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| :<math>Z = \begin{pmatrix}x & y \\ y & x\end{pmatrix} .</math>
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| Its Jordan canonical form is given by:
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| :<math> J_{z} = \begin{pmatrix}x+y & 0 \\ 0 & x-y\end{pmatrix} ,</math>
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| where <math>Z = S J_{z} S^{-1} \ ,</math> and
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| :<math> S = \begin{pmatrix}1 & -1 \\ 1 & 1\end{pmatrix} .</math>
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| ==History==
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| The use of split-complex numbers dates back to 1848 when [[James Cockle (lawyer)|James Cockle]] revealed his [[Tessarine]]s. [[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]].
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| Since the early twentieth century, the split-complex multiplication has commonly been seen as a [[Lorentz boost]] of a [[spacetime]] plane. In that model, the number {{nowrap|1=''z'' = ''x'' + ''y'' ''j''}} represents an event in a spacio-temporal plane, where ''x'' is measured in nanoseconds and ''y'' in [[David Mermin#Mermin’s foot|Mermin’s feet]].
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| The future corresponds to the quadrant of events {{nowrap|{''z'' : {{abs|''y''}} < ''x'' },}} which has the split-complex polar decomposition <math>z = \rho e^{a j} \!</math>. The model says that ''z'' can be reached from the origin by entering a [[frame of reference]] of [[rapidity]] ''a'' and waiting ρ nanoseconds. The split-complex equation
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| :<math>e^{aj} \ e^{bj} = e^{(a+b)j}</math>
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| expressing products on the unit hyperbola illustrates the additivity of rapidities for collinear velocities. Simultaneity of events depends on rapidity ''a'';
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| :<math>\lbrace z = \sigma j e^{aj} : \sigma \isin R \rbrace</math>
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| is the line of events simultaneous with the origin in the frame of reference with rapidity ''a''.
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| Two events ''z'' and ''w'' are [[hyperbolic-orthogonal]] when {{nowrap|1=''z''<sup>∗</sup>''w'' + ''zw''<sup>∗</sup> = 0}}. Canonical events exp(''aj'') and {{nowrap|''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 {{nowrap|''j'' exp(''aj'')}}.
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| 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.
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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 {{nowrap|1=''zz''<sup>∗</sup> = 1}}.
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| In 1956 Mieczyslaw Warmus published "Calculus of Approximations" in ''Bulletin de l’Academie Polanaise des Sciences'' (see link in References). He identified an [[interval (mathematics)|interval]] {{nowrap|[''a'', ''A'']}} with the split-complex number
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| :<math>z = \frac{A + a}{2} + \jmath \ \frac {A - a}{2} \ </math> and called it an "approximate number". [[D. H. Lehmer]] reviewed the article in [[Mathematical Reviews]].
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| 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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| ==Synonyms== | |
| Different authors have used a great variety of names for the split-complex numbers. Some of these include:
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| * (''real'') ''tessarines'', James Cockle (1848)
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| * (''algebraic'') ''motors'', W.K. Clifford (1882)
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| * ''hyperbolic complex numbers'', J.C. Vignaux (1935)
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| * ''bireal numbers'', U. Bencivenga (1946)
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| * ''approximate numbers'', Warmus (1956), for use in [[interval analysis]]
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| * ''countercomplex'' or ''hyperbolic'' numbers from [[Musean hypernumber]]s
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| * ''double numbers'', [[Isaak Yaglom|I.M. Yaglom]] (1968) and [[Michiel Hazewinkel|Hazewinkel]] (1990)
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| * ''anormal-complex numbers'', W. Benz (1973)
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| * ''perplex numbers'', P. Fjelstad (1986) and Poodiack & LeClair (2009)
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| * ''Lorentz numbers'', F.R. Harvey (1990)
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| * ''hyperbolic numbers'', G. Sobczyk (1995)
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| * ''semi-complex numbers'', F. Antonuccio (1994)
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| * ''split-complex numbers'', B. Rosenfeld (1997)
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| * ''spacetime numbers'', N. Borota (2000)
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| * ''twocomplex numbers'', S. Olariu (2002)
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| Split-complex numbers and their higher-dimensional relatives ([[split-quaternion]]s / coquaternions and [[split-octonion]]s) were at times referred to as "Musean numbers", since they are a subset of the [[Musean hypernumber|hypernumber]] program developed by [[Charles Musès]].
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| ==See also==
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| * [[Bicomplex number]]
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| * [[Lorentz group]]
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| * [[Minkowski space]]
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| Higher-order derivatives of split-complex numbers, obtained through a modified [[Cayley–Dickson construction]]:
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| * [[Split-quaternion]] (or coquaternion)
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| * [[Split-octonion]]
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| In Lie theory, a more abstract generalization occurs:
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| * [[Split Lie algebra]]
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| * [[Split orthogonal group]]
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| Enveloping algebras and number programs:
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| * [[Clifford algebra]]
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| * [[Hypercomplex number]]s
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| * [[Lie algebra]]
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| ==References and external links==
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| * Francesco Antonuccio (1994) [http://arxiv.org/pdf/gr-qc/9311032v2.pdf Semi-complex analysis and mathematical physics]
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| * Bencivenga, Uldrico (1946) "Sulla rappresentazione geometrica della algebra doppie dotate di modulo", ''Atti della real academie della scienze e belle-lettre di Napoli'', Ser (3) v.2 No7. {{MathSciNet|id=0021123}}.
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| * Benz, W. (1973)''Vorlesungen uber Geometrie der Algebren'', Springer
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| * N. A. Borota, E. Flores, and T. J. Osler (2000) "Spacetime numbers the easy way", [[Mathematics and Computer Education]] 34: 159-168.
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| * N. A. Borota and T. J. Osler (2002) "Functions of a spacetime variable", ''Mathematics and Computer Education'' 36: 231-239.
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| * K. Carmody, (1988) "Circular and hyperbolic quaternions, octonions, and sedenions", Appl. Math. Comput. 28:47–72.
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| * K. Carmody, (1997) "Circular and hyperbolic quaternions, octonions, and sedenions – further results", Appl. Math. Comput. 84:27–48.
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| * 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.
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| * Cockle, James (1848) "A New Imaginary in Algebra", ''London-Edinburgh-Dublin Philosophical Magazine'' (3) '''33''':435–9.
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| * Clifford, W.K.,''Mathematical Works'' (1882) edited by A.W.Tucker,pp. 392,"Further Notes on [[Clifford biquaternion|Biquaternion]]s"
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| * De Boer, R. (1987) "An also known as list for perplex numbers", ''American Journal of Physics'' 55(4):296.
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| * Fjelstadt, P. (1986) "[http://scitation.aip.org/content/aapt/journal/ajp/54/5/10.1119/1.14605 Extending Special Relativity with Perplex Numbers]", ''American Journal of Physics'' '''54''':416.
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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.
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| * Hazewinkle, M. (1994) "Double and dual numbers", [[Encyclopaedia of Mathematics]], Soviet/AMS/Kluwer, Dordrect.
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| * [[Louis Kauffman]] (1985) "Transformations in Special Relativity", [[International Journal of Theoretical Physics]] 24:223–36.
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| * C. Musès, "Applied hypernumbers: Computational concepts", Appl. Math. Comput. 3 (1977) 211–226.
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| * C. Musès, "Hypernumbers II—Further concepts and computational applications", Appl. Math. Comput. 4 (1978) 45–66.
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| * Olariu, Silviu (2002) ''Complex Numbers in N Dimensions'', Chapter 1: Hyperbolic Complex Numbers in Two Dimensions, pages 1–16, North-Holland Mathematics Studies #190, [[Elsevier]] ISBN 0-444-51123-7.
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| * Poodiack, Robert D. & Kevin J. LeClair (2009) "Fundamental theorems of algebra for the perplexes", [[The College Mathematics Journal]] 40(5):322–35.
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| * Rosenfeld, B. (1997) ''Geometry of Lie Groups'' Kluwer Academic Pub.
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| * Sobczyk, G.(1995) [http://garretstar.com/secciones/publications/docs/HYP2.PDF Hyperbolic Number Plane], also published in [[College Mathematics Journal]] 26:268–80.
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| * 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.
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| * M. Warmus (1956) [http://www.cs.utep.edu/interval-comp/warmus.pdf "Calculus of Approximations"], ''Bulletin de l'Academie Polonaise de Sciences'', Vol. 4, No. 5, pp. 253–257;
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| * Yaglom, I. (1968) ''Complex Numbers in Geometry'', translated by E. Primrose from 1963 Russian original, Academic Press, N.Y., pp. 18–20.
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| {{Number Systems}}
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| {{DEFAULTSORT:Split-Complex Number}}
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| [[Category:Linear algebra]]
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| [[Category:Hypercomplex numbers]]
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