Wed . Wed Jul 2018

Split-complex number


In 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 split-complex 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 component-wise 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,x-y 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 D

Split-complex numbers have many other names; see the synonyms section below See the article Motor variable for functions of a split-complex 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 split-complex number is an ordered pair of real numbers, written in the form

z = x + j y

where x and y are real numbers and the quantity j satisfies

j 2 = + 1 =+1

Choosing j 2 = − 1 =-1 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

The collection of all such z is called the split-complex plane Addition and multiplication of split-complex 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 u

This 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 split-complex conjugate If

z = x + j y

the conjugate of z is defined as

z ∗ = x − j y =x-jy

The conjugate satisfies similar properties to usual complex conjugate Namely,

z + w ∗ = z ∗ + w ∗ =z^+w^ z w ∗ = z ∗ w ∗ =z^w^ z ∗ ∗ = z ^=z

These three properties imply that the split-complex conjugate is an automorphism of order 2

The modulus of a split-complex 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 positive-definite 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=xu-yv,

where z = x + j y and w = u + j v Another expression for the modulus is then

∥ z ∥ = ⟨ z , z ⟩

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 ∥ 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\rVert

Split-complex 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=0

It is often convenient to use e and e∗ as an alternate 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 ∗

If we denote the number z = ae + be∗ for real numbers a and b by a, b, then split-complex 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 split-complex numbers are ring-isomorphic to the direct sum RR with addition and multiplication defined pairwise

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

a , b ∗ = b , a =b,a

and the modulus by

∥ a , b ∥ = a b

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 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 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 R ⊕ R \oplus \mathbf

Geometryedit

Unit hyperbola with ||z||=1 blue,
conjugate hyperbola with ||z||=−1 green,
and asymptotes ||z||=0 red

A two-dimensional real vector space with the Minkowski inner product is called 1 + 1-dimensional 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 split-complex 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

Split-complex numbers z and w are said to be hyperbolic-orthogonal 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 split-complex numbers is

exp ⁡ j θ = cosh ⁡ θ + j sinh ⁡ θ

This 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 split-complex 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 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

The set of all transformations of the split-complex 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 ^1,1

sending 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 split-complex number z does not lie on one of the diagonals, then z has a polar decomposition

Algebraic propertiesedit

In abstract algebra terms, the split-complex numbers can be described as the quotient of the polynomial ring Rx by the ideal generated by the polynomial x2 − 1,

Rx/x2 − 1

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 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 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 split-complex numbers form a topological ring

The algebra of split-complex numbers forms a composition algebra since

∥ z w ∥ = ∥ z ∥ ∥ w ∥  for any numbers z and w

From the definition it is apparent that the ring of split-complex numbers is isomorphic to the group ring RC2 of the cyclic group C2 over the real numbers R

Matrix representationsedit

One can easily represent split-complex numbers by matrices The split-complex number

z = x + j y

can be represented by the matrix

z ↦ x y y x x&y\\y&x\end

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

C = 1 0 0 − 1 1&0\\0&-1\end

For 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 R2

The diagonal basis for the split-complex 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,yS

Now 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 b

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 z = x, y given by the following matrix representation:

Z = x y y x x&y\\y&x\end

its Jordan canonical form is given by:

J z = x + y 0 0 x − y , =x+y&0\\0&x-y\end,

where Z = S J z S − 1   , S^\ , and

S = 1 − 1 1 1 1&-1\\1&1\end

Historyedit

The use of split-complex numbers dates back to 1848 when James Cockle revealed his tessarines1 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-biquaternions 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 split-complex 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 spacio-temporal plane, where x is measured in nanoseconds and y in Mermin’s feet The future corresponds to the quadrant of events , which has the split-complex 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 split-complex 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\rbrace

is the line of events simultaneous with the origin in the frame of reference with rapidity a Two events z and w are hyperbolic-orthogonal 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 split-octonions 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 split-octonions His innovation was perpetuated by Adrian Albert, Richard D Schafer, and others8 The gamma factor, with 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 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 split-complex 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 split-complex geometric arithmetic to establish the nine-point 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 split-complex 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
  • anormal-complex 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
  • semi-complex numbers, F Antonuccio 1994
  • split binarions, K McCrimmon 2004
  • split-complex numbers, B Rosenfeld 199713
  • spacetime numbers, N Borota 2000
  • Study numbers, P Lounesto 2001
  • twocomplex numbers, S Olariu 2002

Split-complex numbers and their higher-dimensional relatives split-quaternions / coquaternions and split-octonions 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
  • Split-quaternion
  • Hypercomplex numbers

Referencesedit

  1. ^ James Cockle 1849 On a New Imaginary in Algebra 34:37–47, London-Edinburgh-Dublin Philosophical Magazine 3 33:435–9, link from Biodiversity Heritage Library
  2. ^ Francesco Antonuccio 1994 Semi-complex analysis and mathematical physics
  3. ^ 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
  4. ^ Francesco Catoni; Dino Boccaletti; Roberto Cannata; Vincenzo Catoni; Paolo Zampetti 2011 "Chapter 2: Hyperbolic Numbers" Geometry of Minkowski Space-Time Springer Science & Business Media ISBN 978-3-642-17977-8 
  5. ^ Fjelstadt, P 1986 "Extending Special Relativity with Perplex Numbers", American Journal of Physics 54 :416
  6. ^ Louis Kauffman 1985 "Transformations in Special Relativity", International Journal of Theoretical Physics 24:223–36
  7. ^ Sobczyk, G1995 Hyperbolic Number Plane, also published in College Mathematics Journal 26:268–80
  8. ^ Robert B Brown 1967On Generalized Cayley-Dickenson Algebras, Pacific Journal of Mathematics 203:415–22, link from Project Euclid
  9. ^ NH McCoy 1942 Review of "Quadratic forms permitting composition" by AA Albert, Mathematical Reviews #0006140
  10. ^ 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
  11. ^ Allen, EF 1941 "On a Triangle Inscribed in a Rectangular Hyperbola", American Mathematical Monthly 4810: 675–681
  12. ^ M Warmus 1956 "Calculus of Approximations", Bulletin de l'Academie Polonaise de Sciences, Vol 4, No 5, pp 253–257, MR0081372
  13. ^ Rosenfeld, B 1997 Geometry of Lie Groups, page 30, Kluwer Academic Publishers ISBN 0-7923-4390-5
  • Bencivenga, Uldrico 1946 "Sulla rappresentazione geometrica delle algebre doppie dotate di modulo", Atti della Reale Accademia delle Scienze e Belle-Lettere 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: 159-168
  • N A Borota and T J Osler 2002 "Functions of a spacetime variable", Mathematics and Computer Education 36: 231-239
  • 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 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 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, North-Holland Mathematics Studies #190, Elsevier ISBN 0-444-51123-7
  • 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 CISM-IFToMM Symposium on Theory and Practice of Robots and Manipulators Springer ISBN 978-3-319-07058-2 doi:101007/978-3-319-07058-2_7