Inalgebra, asplit-complex number (orhyperbolic number, alsoperplex number,double number) is based on ahyperbolic unitj satisfying${\displaystyle j^2=1}$, where${\displaystyle j\ne \pm 1}$. A split-complex number has tworeal number componentsx andy, and is written${\displaystyle z=x+yj.}$ Theconjugate ofz is${\displaystyle z^{\ast }=x-yj.}$ Since${\displaystyle j^2=1,}$ the product of a numberz with its conjugate is${\displaystyle N(z):=z{z}^{\ast }=x^2-y^2,}$ anisotropic quadratic form.

The collectionD of all split-complex numbers${\displaystyle z=x+yj}$ for⁠${\displaystyle x,y\in \mathbb{R}}$⁠ forms analgebra over the field of real numbers. Two split-complex numbersw andz have a productwz that satisfies${\displaystyle N(wz)=N(w)N(z).}$ This composition ofN over the algebra product makes(D, +, ×, *) acomposition algebra.

A similar algebra based on⁠${\displaystyle {\mathbb{R}}^2}$⁠ and component-wise operations of addition and multiplication,⁠${\displaystyle ({\mathbb{R}}^2,+,\times ,xy),}$⁠ wherexy is thequadratic form on⁠${\displaystyle {\mathbb{R}}^2,}$⁠ also forms aquadratic space. Thering isomorphismis anisometry of quadratic spaces.

Split-complex numbers have many other names; see§ Synonyms below. See the articleMotor variable for functions of a split-complex number.

Asplit-complex number is an ordered pair of real numbers, written in the form

$${\displaystyle z=x+jy}$$

wherex andy arereal numbers and thehyperbolic unit^[1]j satisfies

$${\displaystyle j^2=+1}$$

In the field ofcomplex numbers theimaginary unit i satisfies${\displaystyle i^2=-1.}$ The change of sign distinguishes the split-complex numbers from the ordinary complex ones. The hyperbolic unitj isnot a real number but an independent quantity.

The collection of all suchz is called thesplit-complex plane.Addition andmultiplication of split-complex numbers are defined by

This multiplication iscommutative,associative anddistributes over addition.

Just as for complex numbers, one can define the notion of asplit-complex conjugate. If

$${\displaystyle z=x+jy,}$$

then the conjugate ofz is defined as

$${\displaystyle z^{\ast }=x-jy.}$$

The conjugate is aninvolution which satisfies similar properties to thecomplex conjugate. Namely,

The squaredmodulus of a split-complex number${\displaystyle z=x+jy}$ is given by theisotropic quadratic form

$${\displaystyle \Vert z{\Vert }^2=z{z}^{\ast }=z^{\ast }z=x^2-y^2.}$$

It has thecomposition algebra property:

$${\displaystyle \Vert zw\Vert =\Vert z\Vert \Vert w\Vert .}$$

However, this quadratic form is notpositive-definite but rather hassignature(1, −1), so the modulus isnot anorm.

The associatedbilinear form is given by

$${\displaystyle ⟨z,w⟩=\mathrm{R}\mathrm{e}\left(z{w}^{\ast }\right)=\mathrm{R}\mathrm{e}\left(z^{\ast }w\right)=xu-yv,}$$

where${\displaystyle z=x+jy}$ and${\displaystyle w=u+jv.}$ Here, thereal part is defined by${\displaystyle \mathrm{R}\mathrm{e}(z)=\frac{1}{2}(z+z^{\ast })=x}$. Another expression for the squared modulus is then

$${\displaystyle \Vert z{\Vert }^2=⟨z,z⟩.}$$

Since it is not positive-definite, this bilinear form is not aninner product; nevertheless the bilinear form is frequently referred to as anindefinite inner product. A similar abuse of language refers to the modulus as a norm.

A split-complex number is invertibleif and only if its modulus is nonzero(${\displaystyle \Vert z\Vert \ne 0}$), thus numbers of the formx ±j x have no inverse. Themultiplicative inverse of an invertible element is given by

$${\displaystyle z^{-1}=\frac{z^{\ast }}{{\Vert z\Vert }^2}.}$$

Split-complex numbers which are not invertible are callednull vectors. These are all of the form(a ±j a) for some real numbera.

There are two nontrivialidempotent elements given by${\displaystyle e=\frac{1}{2}(1-j)}$ and${\displaystyle e^{\ast }=\frac{1}{2}(1+j).}$ Idempotency means that${\displaystyle ee=e}$ and${\displaystyle e^{\ast }{e}^{\ast }=e^{\ast }.}$ Both of these elements are null:

$${\displaystyle \Vert e\Vert =\Vert e^{\ast }\Vert =e^{\ast }e=0.}$$

It is often convenient to usee ande^∗ as an alternatebasis for the split-complex plane. This basis is called thediagonal basis ornull basis. The split-complex numberz can be written in the null basis as

$${\displaystyle z=x+jy=(x-y)e+(x+y)e^{\ast }.}$$

If we denote the number${\displaystyle z=ae+b{e}^{\ast }}$ for real numbersa andb by(a,b), then split-complex multiplication is given by

$${\displaystyle \left(a_1,b_1\right)\left(a_2,b_2\right)=\left(a_1{a}_2,b_1{b}_2\right).}$$

The split-complex conjugate in the diagonal basis is given by$${\displaystyle (a,b{)}^{\ast }=(b,a)}$$and the squared modulus by

$${\displaystyle \Vert (a,b){\Vert }^2=ab.}$$

On the basis {e, e*} it becomes clear that the split-complex numbers arering-isomorphic to the direct sum⁠${\displaystyle \mathbb{R}\oplus \mathbb{R}}$⁠ with addition and multiplication defined pairwise.

The diagonal basis for the split-complex number plane can be invoked by using an ordered pair(x,y) for${\displaystyle z=x+jy}$ and making the mapping

Now the quadratic form is${\displaystyle uv=(x+y)(x-y)=x^2-y^2.}$ Furthermore,

so the twoparametrized hyperbolas are brought into correspondence withS.

Theaction ofhyperbolic versor${\displaystyle e^{bj}}$ then corresponds under this linear transformation to asqueeze mapping

$${\displaystyle \sigma :(u,v)\mapsto \left(ru,\frac{v}{r}\right),r=e^b.}$$

Though lying in the same isomorphism class in thecategory of rings, the split-complex plane and the direct sum of two real lines differ in their layout in theCartesian plane. The isomorphism, as a planar mapping, consists of a counter-clockwise rotation by 45° and adilation by√2. The dilation in particular has sometimes caused confusion in connection with areas of ahyperbolic sector. Indeed,hyperbolic angle corresponds toarea of a sector in the⁠${\displaystyle \mathbb{R}\oplus \mathbb{R}}$⁠ plane with its "unit circle" given by${\displaystyle \{(a,b)\in \mathbb{R}\oplus \mathbb{R}:ab=1\}.}$ The contractedunit hyperbola${\displaystyle \{\cosh a+j\sinh a:a\in \mathbb{R}\}}$ of the split-complex plane has onlyhalf 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${\displaystyle \mathbb{R}\oplus \mathbb{R}}$⁠.

A two-dimensional realvector space with the Minkowski inner product is called(1 + 1)-dimensionalMinkowski space, often denoted⁠${\displaystyle {\mathbb{R}}^{1,1}.}$⁠ Just as much of thegeometry of the Euclidean plane⁠${\displaystyle {\mathbb{R}}^2}$⁠ can be described with complex numbers, the geometry of the Minkowski plane⁠${\displaystyle {\mathbb{R}}^{1,1}}$⁠ can be described with split-complex numbers.

The set of points

$${\displaystyle \left\{z:\Vert z{\Vert }^2=a^2\right\}}$$

is ahyperbola for every nonzeroa in⁠${\displaystyle \mathbb{R}.}$⁠ The hyperbola consists of a right and left branch passing through(a, 0) and(−a, 0). The casea = 1 is called theunit hyperbola. Theconjugate hyperbola is given by

$${\displaystyle \left\{z:\Vert z{\Vert }^2=-a^2\right\}}$$

with an upper and lower branch passing through(0,a) and(0, −a). The hyperbola and conjugate hyperbola are separated by two diagonalasymptotes which form the set of null elements:

$${\displaystyle \left\{z:\Vert z\Vert =0\right\}.}$$

These two lines (sometimes called thenull cone) areperpendicular in⁠${\displaystyle {\mathbb{R}}^2}$⁠ and have slopes ±1.

Split-complex numbersz andw are said to behyperbolic-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 thesimultaneous hyperplane concept in spacetime.

The analogue ofEuler's formula for the split-complex numbers is

$${\displaystyle \exp (j\theta )=\cosh (\theta )+j\sinh (\theta ).}$$

This formula can be derived from apower series expansion using the fact thatcosh has only even powers while that forsinh has odd powers.^[2] For all real values of thehyperbolic angleθ the split-complex numberλ = exp(jθ) has norm 1 and lies on the right branch of the unit hyperbola. Numbers such asλ have been calledhyperbolic versors.

Sinceλ has modulus 1, multiplying any split-complex numberz byλ preserves the modulus ofz and represents ahyperbolic rotation (also called aLorentz boost or asqueeze mapping). Multiplying byλ 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 agroup called thegeneralized orthogonal groupO(1, 1). This group consists of the hyperbolic rotations, which form asubgroup denotedSO⁺(1, 1), combined with fourdiscretereflections given by

$${\displaystyle z\mapsto \pm z}$$ and${\displaystyle z\mapsto \pm z^{\ast }.}$

The exponential map

$${\displaystyle \exp :(\mathbb{R},+)\to {\mathrm{S}\mathrm{O}}^{+}(1,1)}$$

sendingθ to rotation byexp(jθ) is agroup isomorphism since the usual exponential formula applies:

$${\displaystyle e^{j(\theta +\varphi )}=e^{j\theta }{e}^{j\varphi }.}$$

If a split-complex numberz does not lie on one of the diagonals, thenz has apolar decomposition.

Inabstract algebra terms, the split-complex numbers can be described as thequotient of thepolynomial ring⁠${\displaystyle \mathbb{R}[x]}$⁠ by theideal generated by thepolynomial${\displaystyle x^2-1,}$

$${\displaystyle \mathbb{R}[x]/(x^2-1).}$$

The image ofx in the quotient is the "imaginary" unitj. With this description, it is clear that the split-complex numbers form acommutative algebra over the real numbers. The algebra isnot afield since the null elements are not invertible. All of the nonzero null elements arezero divisors.

Since addition and multiplication are continuous operations with respect to the usual topology of the plane, the split-complex numbers form atopological ring.

The algebra of split-complex numbers forms acomposition algebra since

$${\displaystyle \Vert zw\Vert =\Vert z\Vert \Vert w\Vert }$$

for any numbersz andw.

From the definition it is apparent that the ring of split-complex numbers is isomorphic to thegroup ring⁠${\displaystyle \mathbb{R}[C_2]}$⁠ of thecyclic groupC₂ over the real numbers⁠${\displaystyle \mathbb{R}.}$⁠

Elements of theidentity component in thegroup of units inD have four square roots.: say${\displaystyle p=\exp (q),q\in D.\text{then}\pm \exp (\frac{q}{2})}$ are square roots ofp. Further,${\displaystyle \pm j\exp (\frac{q}{2})}$ are also square roots ofp.

Theidempotents${\displaystyle \frac{1\pm j}{2}}$ are their own square roots, and the square root of${\displaystyle s\frac{1\pm j}{2},s>0,\text{is}\sqrt{s}\frac{1\pm j}{2}}$

One can easily represent split-complex numbers bymatrices. The split-complex number${\displaystyle z=x+jy}$ can be represented by the matrix

Addition and multiplication of split-complex numbers are then given by matrix addition and multiplication. The squared modulus ofz is given by thedeterminant of the corresponding matrix.

In fact there are many representations of the split-complex plane in the four-dimensionalring of 2x2 real matrices. The real multiples of theidentity matrix form areal line in the matrix ring M(2,R). Any hyperbolic unitm provides abasis element with which to extend the real line to the split-complex plane. The matrices

which square to the identity matrix satisfy${\displaystyle a^2+bc=1.}$For example, whena = 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 asubring of M(2,R).^{[3][better source needed]}

The number${\displaystyle z=x+jy}$ can be represented by the matrix  ${\displaystyle xI+ym.}$

The use of split-complex numbers dates back to 1848 whenJames Cockle revealed histessarines.^[4]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 calledsplit-biquaternions. He called its elements "motors", a term in parallel with the "rotor" action of an ordinary complex number taken from thecircle group. Extending the analogy, functions of amotor variable contrast to functions of an ordinarycomplex variable.

Since the late twentieth century, the split-complex multiplication has commonly been seen as aLorentz boost of aspacetime plane.^{[5][6][7][8][9][10]} In that model, the numberz =x +yj represents an event in a spatio-temporal plane, wherex is measured in seconds andy inlight-seconds. The future corresponds to the quadrant of events {z : |y| <x}, which has the split-complex polar decomposition${\displaystyle z=\rho e^{aj}}$. The model says thatz can be reached from the origin by entering aframe of reference ofrapiditya and waitingρ nanoseconds. The split-complex equation

$${\displaystyle e^{aj}{e}^{bj}=e^{(a+b)j}}$$

expressing products on the unit hyperbola illustrates the additivity of rapidities for collinear velocities. Simultaneity of events depends on rapiditya;

$${\displaystyle \{z=\sigma j{e}^{aj}:\sigma \in \mathbb{R}\}}$$

is the line of events simultaneous with the origin in the frame of reference with rapiditya.

Two eventsz andw arehyperbolic-orthogonal when${\displaystyle z^{\ast }w+z{w}^{\ast }=0.}$ Canonical events exp(aj) andj 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 toj exp(aj).

In 1933Max Zorn was using thesplit-octonions and noted thecomposition algebra property. He realized that theCayley–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 byAdrian Albert, Richard D. Schafer, and others.^[11] The gamma factor, withR as base field, builds split-complex numbers as a composition algebra. Reviewing Albert forMathematical Reviews, N. H. McCoy wrote that there was an "introduction of some new algebras of order 2^e overF generalizing Cayley–Dickson algebras."^[12] TakingF =R ande = 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 inContribución 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 appreciation.^[13]

In 1941 E.F. Allen used the split-complex geometric arithmetic to establish thenine-point hyperbola of a triangle inscribed in zz^∗ = 1.^[14]

In 1956 Mieczyslaw Warmus published "Calculus of Approximations" inBulletin 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.^[15]D. H. Lehmer reviewed the article inMathematical 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.

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, W.K. Clifford (1882)
• hyperbolic complex numbers, J.C. Vignaux (1935), G. Cree (1949)^[16]
• bireal numbers, U. Bencivenga (1946)
• real hyperbolic numbers, N. Smith (1949)^[17]
• approximate numbers, Warmus (1956), for use ininterval analysis
• double numbers,I.M. Yaglom (1968), Kantor and Solodovnikov (1989),Hazewinkel (1990), Rooney (2014)
• hyperbolic numbers, W. Miller & R. Boehning (1968),^[18] G. Sobczyk (1995)
• anormal-complex numbers, W. Benz (1973)
• perplex numbers, P. Fjelstad (1986) and Poodiack & LeClair (2009)
• countercomplex orhyperbolic, Carmody (1988)
• Lorentz numbers, F.R. Harvey (1990)
• semi-complex numbers, F. Antonuccio (1994)
• paracomplex numbers, Cruceanu, Fortuny & Gadea (1996)
• split-complex numbers, B. Rosenfeld (1997)^[19]
• spacetime numbers, N. Borota (2000)
• Study numbers, P. Lounesto (2001)
• twocomplex numbers, S. Olariu (2002)
• split binarions, K. McCrimmon (2004)

The WikibookAssociative Composition Algebra has a page on the topic of:Split binarions

• Minkowski space
• Split-quaternion
• Hypercomplex number

• 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) v.2 No7.MR0021123.
• Walter Benz (1973)Vorlesungen 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 (1882)Mathematical Works, A. W. Tucker editor, page 392, "Further Notes on Biquaternions"
• V.Cruceanu, P. Fortuny & P.M. Gadea (1996)A Survey on Paracomplex Geometry,Rocky Mountain Journal of Mathematics 26(1): 83–115, link fromProject Euclid.
• De Boer, R. (1987) "An also known as list for perplex numbers",American Journal of Physics 55(4):296.
• Anthony A. Harkin & Joseph B. Harkin (2004)Geometry of Generalized Complex Numbers,Mathematics Magazine 77(2):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, SpringerISBN 0-387-95447-3MR2014924
• 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,ElsevierISBN 0-444-51123-7.
• Poodiack, Robert D. & Kevin J. LeClair (2009) "Fundamental theorems of algebra for the perplexes",The College Mathematics Journal 40(5):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 (ed.).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. Mechanisms and Machine Science. Vol. 22. Springer. pp. 55–62.doi:10.1007/978-3-319-07058-2_7.ISBN 978-3-319-07058-2.

• Composition algebras
• Linear algebra
• Hypercomplex numbers

• Webarchive template wayback links
• Articles with short description
• Short description is different from Wikidata
• All articles lacking reliable references
• Articles lacking reliable references from May 2023