Splitcomplex number
split complex numbers, split complex numberIn abstract algebra, a split complex number or hyperbolic number, also perplex number, double number has two real number components x and y, and is written z = x + y j , j 2 = + 1 =+1 The conjugate of z is z = x  y j Since j2 = +1, z z ∗ = x 2 − y 2 , =x^y^, , an isotropic quadratic form, N z = x 2 − y 2 y^
The collection D of all split complex numbers j2 = +1 , forms an algebra over the field of real numbers Two splitcomplex numbers w and z have a product wz that satisfies N w z = N w N z This composition of N over the algebra product makes D, +, ×, a composition algebra
A similar algebra based on R2 and componentwise operations of addition and multiplication, R2, +, ×, xy, where xy is the quadratic form on R2, also forms a quadratic space The ring isomorphism
D → R 2 by x + y j ↦ x + y , x − y \ \ x+yj\mapsto x+y,xy relates proportional quadratic forms, but the mapping is not an isometry since the multiplicative identity 1,1 of R2 is at a distance √2 from 0, which is normalized in DSplitcomplex numbers have many other names; see the synonyms section below See the article Motor variable for functions of a splitcomplex number
Contents
 1 Definition
 11 Conjugate, modulus, and bilinear form
 12 The diagonal basis
 2 Geometry
 3 Algebraic properties
 4 Matrix representations
 5 History
 6 Synonyms
 7 See also
 8 References
Definitionedit
A splitcomplex number is an ordered pair of real numbers, written in the form
z = x + j ywhere x and y are real numbers and the quantity j satisfies
j 2 = + 1 =+1Choosing j 2 = − 1 =1 results in the complex numbers It is this sign change which distinguishes the splitcomplex 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
The collection of all such z is called the splitcomplex plane Addition and multiplication of splitcomplex numbers are defined by
x + j y + u + j v = x + u + j y + v x + j y u + j v = x u + y v + j x v + y uThis multiplication is commutative, associative and distributes over addition
Conjugate, modulus, and bilinear formedit
Just as for complex numbers, one can define the notion of a splitcomplex conjugate If
z = x + j ythe conjugate of z is defined as
z ∗ = x − j y =xjyThe conjugate satisfies similar properties to usual complex conjugate Namely,
z + w ∗ = z ∗ + w ∗ =z^+w^ z w ∗ = z ∗ w ∗ =z^w^ z ∗ ∗ = z ^=zThese three properties imply that the splitcomplex conjugate is an automorphism of order 2
The modulus of a splitcomplex number z = x + j y is given by the isotropic quadratic form
∥ z ∥ = z z ∗ = z ∗ z = x 2 − y 2 =z^z=x^y^It has the composition algebra property:
∥ z w ∥ = ∥ z ∥ ∥ w ∥However, this quadratic form is not positivedefinite but rather has signature 1, −1, so the modulus is not a norm
The associated bilinear form is given by
⟨ z , w ⟩ = Re z w ∗ = Re z ∗ w = x u − y v , zw^=\operatorname z^w=xuyv,where z = x + j y and w = u + j v Another expression for the modulus is then
∥ z ∥ = ⟨ z , z ⟩Since it is not positivedefinite, 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 splitcomplex number is invertible if and only if its modulus is nonzero ∥ z ∥ ≠ 0 , thus x ± j x have no inverse The multiplicative inverse of an invertible element is given by
z − 1 = z ∗ / ∥ z ∥ =z^/\lVert z\rVertSplitcomplex numbers which are not invertible are called null vectors These are all of the form a ± j a for some real number a
The diagonal basisedit
There are two nontrivial idempotent elements given by e = 1 − j/2 and e∗ = 1 + j/2 Recall that idempotent means that ee = e and e∗e∗ = e∗ Both of these elements are null:
∥ e ∥ = ∥ e ∗ ∥ = e ∗ e = 0 \rVert =e^e=0It is often convenient to use e and e∗ as an alternate basis for the splitcomplex plane This basis is called the diagonal basis or null basis The splitcomplex number z can be written in the null basis as
z = x + j y = x − y e + x + y e ∗If we denote the number z = ae + be∗ for real numbers a and b by a, b, then splitcomplex multiplication is given by
a 1 , b 1 a 2 , b 2 = a 1 a 2 , b 1 b 2 ,b_a_,b_=a_a_,b_b_In this basis, it becomes clear that the splitcomplex numbers are ringisomorphic to the direct sum R ⊕ R with addition and multiplication defined pairwise
The splitcomplex conjugate in the diagonal basis is given by
a , b ∗ = b , a =b,aand the modulus by
∥ a , b ∥ = a bThough lying in the same isomorphism class in the category of rings, the splitcomplex 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 counterclockwise rotation by 45° and a dilation by √2 The dilation in particular has sometimes caused confusion in connection with areas of hyperbolic sectors Indeed, hyperbolic angle corresponds to area of sectors in the R ⊕ R \oplus \mathbf plane with its "unit circle" given by \oplus \mathbf :ab=1\rbrace The contracted "unit circle" \rbrace of the splitcomplex plane has only half the area in the span of a corresponding hyperbolic sector Such confusion may be perpetuated when the geometry of the splitcomplex plane is not distinguished from that of R ⊕ R \oplus \mathbf
Geometryedit
Unit hyperbola with z=1 blue,conjugate hyperbola with z=−1 green,
and asymptotes z=0 red
A twodimensional real vector space with the Minkowski inner product is called 1 + 1dimensional Minkowski space, often denoted R1,1 Just as much of the geometry of the Euclidean plane R2 can be described with complex numbers, the geometry of the Minkowski plane R1,1 can be described with splitcomplex numbers
The set of points
\is a hyperbola for every nonzero a in R The hyperbola consists of a right and left branch passing through a, 0 and −a, 0 The case a = 1 is called the unit hyperbola The conjugate hyperbola is given by
\with an upper and lower branch passing through 0, a and 0, −a The hyperbola and conjugate hyperbola are separated by two diagonal asymptotes which form the set of null elements:
These two lines sometimes called the null cone are perpendicular in R2 and have slopes ±1
Splitcomplex numbers z and w are said to be hyperbolicorthogonal if ⟨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 simultaneous hyperplane concept in spacetime
The analogue of Euler's formula for the splitcomplex numbers is
exp j θ = cosh θ + j sinh θThis can be derived from a power series expansion using the fact that cosh has only even powers while that for sinh has odd powers For all real values of the hyperbolic angle θ the splitcomplex number λ = expjθ has norm 1 and lies on the right branch of the unit hyperbola Numbers such as λ have been called hyperbolic versors
Since λ has modulus 1, multiplying any splitcomplex 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
The set of all transformations of the splitcomplex plane which preserve the modulus or equivalently, the inner product forms a group called the generalized orthogonal group O1, 1 This group consists of the hyperbolic rotations, which form a subgroup denoted SO+1, 1, combined with four discrete reflections given by
z ↦ ± z and z ↦ ± z ∗The exponential map
exp : R , + → S O + 1 , 1 ,+\to \mathrm ^1,1sending θ to rotation by expjθ is a group isomorphism since the usual exponential formula applies:
e j θ + ϕ = e j θ e j ϕ =e^e^\,If a splitcomplex number z does not lie on one of the diagonals, then z has a polar decomposition
Algebraic propertiesedit
In abstract algebra terms, the splitcomplex numbers can be described as the quotient of the polynomial ring Rx by the ideal generated by the polynomial x2 − 1,
Rx/x2 − 1The image of x in the quotient is the "imaginary" unit j With this description, it is clear that the splitcomplex numbers form a commutative ring with characteristic 0 Moreover, if we define scalar multiplication in the obvious manner, the splitcomplex numbers actually form a commutative and associative algebra over the reals of dimension two The algebra is not a division algebra or field since the null elements are not invertible In fact, all of the nonzero null elements are zero divisors Since addition and multiplication are continuous operations with respect to the usual topology of the plane, the splitcomplex numbers form a topological ring
The algebra of splitcomplex numbers forms a composition algebra since
∥ z w ∥ = ∥ z ∥ ∥ w ∥ for any numbers z and wFrom the definition it is apparent that the ring of splitcomplex numbers is isomorphic to the group ring RC2 of the cyclic group C2 over the real numbers R
Matrix representationsedit
One can easily represent splitcomplex numbers by matrices The splitcomplex number
z = x + j ycan be represented by the matrix
z ↦ x y y x x&y\\y&x\endAddition and multiplication of splitcomplex 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, splitcomplex conjugation corresponds to multiplying on both sides by the matrix
C = 1 0 0 − 1 1&0\\0&1\endFor any real number a, a hyperbolic rotation by a hyperbolic angle a corresponds to multiplication by the matrix
cosh a sinh a sinh a cosh a \cosh a&\sinh a\\\sinh a&\cosh a\end This commutative diagram relates the action of the hyperbolic versor on D to squeeze mapping σ applied to R2The diagonal basis for the splitcomplex number plane can be invoked by using an ordered pair x, y for z = x + j y and making the mapping
u , v = x , y 1 1 1 − 1 = x , y S 1&1\\1&1\end=x,ySNow the quadratic form is u v = x + y x − y = x 2 − y 2 y^ Furthermore,
cosh a , sinh a 1 1 1 − 1 = e a , e − a 1&1\\1&1\end=e^,e^so the two parametrized hyperbolas are brought into correspondence with S The action of hyperbolic versor e b j \! then corresponds under this linear transformation to a squeeze mapping
σ : u , v ↦ r u , v / r , r = e bNote that in the context of 2 × 2 real matrices there are in fact a great number of different representations of splitcomplex numbers The above diagonal representation represents the Jordan canonical form of the matrix representation of the splitcomplex numbers For a splitcomplex number z = x, y given by the following matrix representation:
Z = x y y x x&y\\y&x\endits Jordan canonical form is given by:
J z = x + y 0 0 x − y , =x+y&0\\0&xy\end,where Z = S J z S − 1 , S^\ , and
S = 1 − 1 1 1 1&1\\1&1\endHistoryedit
The use of splitcomplex numbers dates back to 1848 when James Cockle revealed his tessarines1 William Kingdon Clifford used splitcomplex numbers to represent sums of spins Clifford introduced the use of splitcomplex numbers as coefficients in a quaternion algebra now called splitbiquaternions 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 early twentieth century, the splitcomplex multiplication has commonly been seen as a Lorentz boost of a spacetime plane234567 In that model, the number z = x + y j represents an event in a spaciotemporal plane, where x is measured in nanoseconds and y in Mermin’s feet The future corresponds to the quadrant of events , which has the splitcomplex polar decomposition z = ρ e a j \! 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 splitcomplex equation
e a j e b j = e a + b j \ e^=e^expressing products on the unit hyperbola illustrates the additivity of rapidities for collinear velocities Simultaneity of events depends on rapidity a;
:\sigma \in R\rbraceis the line of events simultaneous with the origin in the frame of reference with rapidity a Two events z and w are hyperbolicorthogonal when z∗w + zw∗ = 0 Canonical events expaj and j expaj are hyperbolic orthogonal and lie on the axes of a frame of reference in which the events simultaneous with the origin are proportional to j expaj
In 1933 Max Zorn was using the splitoctonions 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 γ to construct other composition algebras including the splitoctonions His innovation was perpetuated by Adrian Albert, Richard D Schafer, and others8 The gamma factor, with ℝ as base field, builds splitcomplex 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 2e over F generalizing Cayley–Dickson algebras"9 Taking F = ℝ and e = 1 corresponds to the algebra of this article
In 1935 JC Vignaux and A Durañona y Vedia developed the splitcomplex geometric algebra and function theory in four articles in Contribución a las Ciencias Físicas y Matemáticas, National University of La Plata, República Argentina in Spanish These expository and pedagogical essays presented the subject for broad appreciation10
In 1941 EF Allen used the splitcomplex geometric arithmetic to establish the ninepoint hyperbola of a triangle inscribed in zz∗ = 111
In 1956 Mieczyslaw Warmus published "Calculus of Approximations" in Bulletin de l’Academie Polanaise 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 algebra12 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
Synonymsedit
Different authors have used a great variety of names for the splitcomplex numbers Some of these include:
 real tessarines, James Cockle 1848
 algebraic motors, WK Clifford 1882
 hyperbolic complex numbers, JC Vignaux 1935
 bireal numbers, U Bencivenga 1946
 approximate numbers, Warmus 1956, for use in interval analysis
 countercomplex or hyperbolic numbers from Musean hypernumbers
 double numbers, IM Yaglom 1968, Kantor and Solodovnikov 1989, Hazewinkel 1990, Rooney 2014
 anormalcomplex numbers, W Benz 1973
 perplex numbers, P Fjelstad 1986 and Poodiack & LeClair 2009
 Lorentz numbers, FR Harvey 1990
 hyperbolic numbers, G Sobczyk 1995
 paracomplex numbers, Cruceanu, Fortuny & Gadea 1996
 semicomplex numbers, F Antonuccio 1994
 split binarions, K McCrimmon 2004
 splitcomplex numbers, B Rosenfeld 199713
 spacetime numbers, N Borota 2000
 Study numbers, P Lounesto 2001
 twocomplex numbers, S Olariu 2002
Splitcomplex numbers and their higherdimensional relatives splitquaternions / coquaternions and splitoctonions were at times referred to as "Musean numbers", since they are a subset of the hypernumber program developed by Charles Musès
See alsoedit
 Minkowski space
 Splitquaternion
 Hypercomplex numbers
Referencesedit
The Wikibook Associative Composition Algebra has a page on the topic of: Split binarions 
 ^ James Cockle 1849 On a New Imaginary in Algebra 34:37–47, LondonEdinburghDublin Philosophical Magazine 3 33:435–9, link from Biodiversity Heritage Library
 ^ Francesco Antonuccio 1994 Semicomplex analysis and mathematical physics
 ^ F Catoni, D Boccaletti, R Cannata, V Catoni, E Nichelatti, P Zampetti 2008 The Mathematics of Minkowski SpaceTime, Birkhäuser Verlag, Basel Chapter 4: Trigonometry in the Minkowski plane ISBN 9783764386139
 ^ Francesco Catoni; Dino Boccaletti; Roberto Cannata; Vincenzo Catoni; Paolo Zampetti 2011 "Chapter 2: Hyperbolic Numbers" Geometry of Minkowski SpaceTime Springer Science & Business Media ISBN 9783642179778
 ^ Fjelstadt, P 1986 "Extending Special Relativity with Perplex Numbers", American Journal of Physics 54 :416
 ^ Louis Kauffman 1985 "Transformations in Special Relativity", International Journal of Theoretical Physics 24:223–36
 ^ Sobczyk, G1995 Hyperbolic Number Plane, also published in College Mathematics Journal 26:268–80
 ^ Robert B Brown 1967On Generalized CayleyDickenson Algebras, Pacific Journal of Mathematics 203:415–22, link from Project Euclid
 ^ NH McCoy 1942 Review of "Quadratic forms permitting composition" by AA Albert, Mathematical Reviews #0006140
 ^ Vignaux, J1935 "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
 ^ Allen, EF 1941 "On a Triangle Inscribed in a Rectangular Hyperbola", American Mathematical Monthly 4810: 675–681
 ^ M Warmus 1956 "Calculus of Approximations", Bulletin de l'Academie Polonaise de Sciences, Vol 4, No 5, pp 253–257, MR0081372
 ^ Rosenfeld, B 1997 Geometry of Lie Groups, page 30, Kluwer Academic Publishers ISBN 0792343905
 Bencivenga, Uldrico 1946 "Sulla rappresentazione geometrica delle algebre doppie dotate di modulo", Atti della Reale Accademia delle Scienze e BelleLettere di Napoli, Ser 3 v2 No7 MR0021123
 Benz, W 1973Vorlesungen uber Geometrie der Algebren, Springer
 N A Borota, E Flores, and T J Osler 2000 "Spacetime numbers the easy way", Mathematics and Computer Education 34: 159168
 N A Borota and T J Osler 2002 "Functions of a spacetime variable", Mathematics and Computer Education 36: 231239
 K Carmody, 1988 "Circular and hyperbolic quaternions, octonions, and sedenions", Appl Math Comput 28:47–72
 K Carmody, 1997 "Circular and hyperbolic quaternions, octonions, and sedenions – further results", Appl Math Comput 84:27–48
 William Kingdon Clifford,Mathematical Works 1882 edited by AWTucker,pp 392,"Further Notes on Biquaternions"
 VCruceanu, P Fortuny & PM Gadea 1996 A Survey on Paracomplex Geometry, Rocky Mountain Journal of Mathematics 261: 83–115, link from Project Euclid
 De Boer, R 1987 "An also known as list for perplex numbers", American Journal of Physics 554:296
 Anthony A Harkin & Joseph B Harkin 2004 Geometry of Generalized Complex Numbers, Mathematics Magazine 772:118–29
 F Reese Harvey Spinors and calibrations Academic Press, San Diego 1990 ISBN 0123296501 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 0387954473 MR2014924
 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
 Olariu, Silviu 2002 Complex Numbers in N Dimensions, Chapter 1: Hyperbolic Complex Numbers in Two Dimensions, pages 1–16, NorthHolland Mathematics Studies #190, Elsevier ISBN 0444511237
 Poodiack, Robert D & Kevin J LeClair 2009 "Fundamental theorems of algebra for the perplexes", The College Mathematics Journal 405:322–35
 Isaak Yaglom 1968 Complex Numbers in Geometry, translated by E Primrose from 1963 Russian original, Academic Press, pp 18–20
 J Rooney 2014 "Generalised Complex Numbers in Mechanics" In Marco Ceccarelli and Victor A Glazunov Advances on Theory and Practice of Robots and Manipulators: Proceedings of Romansy 2014 XX CISMIFToMM Symposium on Theory and Practice of Robots and Manipulators Springer ISBN 9783319070582 doi:101007/9783319070582_7



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