algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

, a split complex number (or hyperbolic number, also perplex number, double number) has two

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

components and , and is written

z=x+yj

, where

j^2=1.

The ''conjugate'' of is

z^*=x-yj.

Since

j^2=1,

the product of a number with its conjugate is

N(z) := zz^* = x^2 - y^2,

isotropic quadratic form In mathematics, a quadratic form over a field ''F'' is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely, if ''q'' is a quadratic form on a vector s ...

. The collection of all split complex numbers

z=x+yj

for forms an algebra over the field of real numbers. Two split-complex numbers and have a product that satisfies

N(wz)=N(w)N(z).

This composition of over the algebra product makes a

composition algebra In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies :N(xy) = N(x)N(y) for all and in . A composition algebra includes an involution ...

. A similar algebra based on and component-wise operations of addition and multiplication, where is the

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...

on also forms a

quadratic space In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...

. The

ring isomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preservi ...

\begin
       D &\to \mathbb^2 \\
  x + yj &\mapsto (x - y, x + y)
\end

relates proportional quadratic forms, but the mapping is an

isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...

since the multiplicative identity of is at a distance from 0, which is normalized in . Split-complex numbers have many other names; see ' below. See the article ''

Motor variable In mathematics, a function of a motor variable is a function with arguments and values in the split-complex number plane, much as functions of a complex variable involve ordinary complex numbers. William Kingdon Clifford coined the term motor fo ...

'' for functions of a split-complex number.

Definition

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

z = x + jy

where and are

s and the quantity satisfies

j^2 = +1

Choosing

j^2 = -1

results in the

complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...

. It is this sign change which distinguishes the split-complex numbers from the ordinary complex ones. The quantity here is not a real number but an independent quantity. The collection of all such is called the split-complex plane. Addition and multiplication of split-complex numbers are defined by

\begin
  (x + jy) + (u + jv) &= (x + u) + j(y + v) \\
     (x + jy)(u + jv) &= (xu + yv) + j(xv + yu).
\end

This multiplication is

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...

, associative and distributes over addition.

Conjugate, modulus, and bilinear form

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

z = x + jy ~,

then the conjugate of is defined as

z^* = x - jy ~.

The conjugate satisfies similar properties to usual complex conjugate. Namely,

\begin
           (z + w)^* &= z^* + w^* \\
              (zw)^* &= z^* w^* \\
  \left(z^*\right)^* &= z.
\end

These three properties imply that the split-complex conjugate is an automorphism of order 2. The squared modulus of a split-complex number

z=x+jy

is given by the

\lVert z \rVert^2 = z z^* = z^* z = x^2 - y^2 ~.

It has the

property:

\lVert z w \rVert = \lVert z \rVert \lVert w \rVert ~.

However, this quadratic form is not

positive-definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite fu ...

but rather has

signature A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...

, so the modulus is ''not'' a

norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...

. The associated

bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear i ...

is given by

\langle z, w \rangle = \operatorname\mathcal\left(zw^*\right) = \operatorname\mathcal \left(z^* w\right) = xu - yv ~,

where

z=x+jy

and

w=u+jv.

Another expression for the squared modulus is then

\lVert z \rVert^2 = \langle z, z \rangle ~.

Since it is not positive-definite, this bilinear form is not an

inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...

; 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 In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

its modulus is nonzero thus numbers of the form have no inverse. The

multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a rat ...

of an invertible element is given by

z^ = \frac ~.

Split-complex numbers which are not invertible are called

null vector In mathematics, given a vector space ''X'' with an associated quadratic form ''q'', written , a null vector or isotropic vector is a non-zero element ''x'' of ''X'' for which . In the theory of real bilinear forms, definite quadratic forms an ...

s. These are all of the form for some real number .

The diagonal basis

There are two nontrivial

idempotent element Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...

s given by

e=\tfrac(1-j)

and

e^* = \tfrac(1+j).

Recall that idempotent means that

ee=e

and

e^*e^*=e^*.

Both of these elements are null:

\lVert e \rVert = \lVert e^* \rVert = e^* e = 0 ~.

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

basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...

for the split-complex plane. This basis is called the diagonal basis or null basis. The split-complex number can be written in the null basis as

z = x + jy = (x - y)e + (x + y)e^* ~.

If we denote the number

z=ae+be^*

for real numbers and by , then split-complex multiplication is given by

\left( a_1, b_1 \right) \left( a_2, b_2 \right) = \left( a_1 a_2, b_1 b_2 \right) ~.

In this basis, it becomes clear that the split-complex numbers are ring-isomorphic to the direct sum with addition and multiplication defined pairwise. The split-complex conjugate in the diagonal basis is given by

(a, b)^* = (b, a)

and the modulus by

\lVert (a, b) \rVert = ab.

Though lying in the same isomorphism class in the

category of rings In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of ring ...

, the split-complex plane and the direct sum of two real lines differ in their layout in the

Cartesian plane A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...

. The isomorphism, as a planar mapping, consists of a counter-clockwise rotation by 45° and a

dilation Dilation (or dilatation) may refer to: Physiology or medicine * Cervical dilation, the widening of the cervix in childbirth, miscarriage etc. * Coronary dilation, or coronary reflex * Dilation and curettage, the opening of the cervix and surgic ...

by . The dilation in particular has sometimes caused confusion in connection with areas of a

hyperbolic sector A hyperbolic sector is a region of the Cartesian plane bounded by a hyperbola and two rays from the origin to it. For example, the two points and on the rectangular hyperbola , or the corresponding region when this hyperbola is re-scaled and ...

. Indeed,

hyperbolic angle In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of ''xy'' = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrises the unit hyperbola, which has hyperbolic function ...

corresponds to

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...

of a sector in the plane with its "unit circle" given by

\.

The contracted

unit hyperbola In geometry, the unit hyperbola is the set of points (''x'',''y'') in the Cartesian plane that satisfy the implicit equation x^2 - y^2 = 1 . In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an ''alternative radi ...

\

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 .

Geometry

A two-dimensional real

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...

with the Minkowski inner product is called -dimensional

Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...

, often denoted Just as much of the

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

of the Euclidean plane can be described with complex numbers, the geometry of the Minkowski plane can be described with split-complex numbers. The set of points

\left\

is a

hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...

for every nonzero in The hyperbola consists of a right and left branch passing through and . The case is called the

. The conjugate hyperbola is given by

\left\

with an upper and lower branch passing through and . The hyperbola and conjugate hyperbola are separated by two diagonal

asymptote In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related context ...

s which form the set of null elements:

\left\.

These two lines (sometimes called the null cone) are

perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...

in and have slopes ±1. Split-complex numbers and are said to be

hyperbolic-orthogonal In geometry, the relation of hyperbolic orthogonality between two lines separated by the asymptotes of a hyperbola is a concept used in special relativity to define simultaneous events. Two events will be simultaneous when they are on a line hyperb ...

if . 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 Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that fo ...

for the split-complex numbers is

\exp(j\theta) = \cosh(\theta) + j\sinh(\theta).

This formula can be derived from a

power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...

expansion using the fact that cosh has only even powers while that for sinh has odd powers. For all real values of the

the split-complex number has norm 1 and lies on the right branch of the unit hyperbola. Numbers such as have been called hyperbolic versors. Since has modulus 1, multiplying any split-complex number by preserves the modulus of and represents a ''hyperbolic rotation'' (also called a

Lorentz boost In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...

or a

squeeze mapping In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is ''not'' a rotation or shear mapping. For a fixed positive real number , th ...

). Multiplying by 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 A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

called the generalized orthogonal group . This group consists of the hyperbolic rotations, which form a

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

denoted , combined with four

discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a g ...

reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...

s given by :

z \mapsto \pm z

and

z \mapsto \pm z^*.

The exponential map

\exp\colon (\R, +) \to \mathrm^(1, 1)

sending to rotation by is a group isomorphism since the usual exponential formula applies:

e^ = e^e^.

If a split-complex number does not lie on one of the diagonals, then has a

polar decomposition In mathematics, the polar decomposition of a square real or complex matrix A is a factorization of the form A = U P, where U is an orthogonal matrix and P is a positive semi-definite symmetric matrix (U is a unitary matrix and P is a positive se ...

Algebraic properties

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...

terms, the split-complex numbers can be described as the

quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...

of the

polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...

by the

ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...

generated by the

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...

x^2-1,

\R (x^2-1 ).

The image of in the quotient is the "imaginary" unit . With this description, it is clear that the split-complex numbers form a

commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...

over the real numbers. The algebra is ''not'' a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

since the null elements are not invertible. All of the nonzero null elements are

zero divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zer ...

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

topological ring In mathematics, a topological ring is a ring R that is also a topological space such that both the addition and the multiplication are continuous as maps: R \times R \to R where R \times R carries the product topology. That means R is an additive ...

. The algebra of split-complex numbers forms a

since :

\lVert zw \rVert = \lVert z \rVert \lVert w \rVert ~

for any numbers and . From the definition it is apparent that the ring of split-complex numbers is isomorphic to the

group ring In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the give ...

of the

cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...

over the real numbers

Matrix representations

One can easily represent split-complex numbers by

matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

. The split-complex number

z = x + jy

can be represented by the matrix

z \mapsto \beginx & y \\ y & x\end.

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

determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...

of the corresponding matrix. In this representation, split-complex conjugation corresponds to multiplying on both sides by the matrix

C = \begin1 & 0 \\ 0 & -1\end.

For any real number , a hyperbolic rotation by a

corresponds to multiplication by the matrix

\begin
\cosh a & \sinh a \\ 
\sinh a & \cosh a
\end.

Commutative diagram split-complex number 2

The diagonal basis for the split-complex number plane can be invoked by using an ordered pair for

z = x + jy

and making the mapping

(u, v) = (x, y) \begin1 & 1 \\1 & -1\end = (x, y) S ~.

Now the quadratic form is

uv = (x + y)(x - y) = x^2 - y^2 ~.

Furthermore,

(\cosh a, \sinh a) \begin
1 & 1 \\
1 & -1
\end = \left(e^a, e^\right)

so the two parametrized hyperbolas are brought into correspondence with . The

action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...

hyperbolic versor In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Wil ...

e^ \!

then corresponds under this linear transformation to a

\sigma: (u, v) \mapsto \left(ru, \frac\right),\quad r = e^b ~.

There are many different representations of split-complex numbers in the 2×2 real matrices. In fact, every matrix whose square is the identity matrix gives such a representation. The above diagonal representation represents the

Jordan canonical form In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF), is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to so ...

of the matrix representation of the split-complex numbers. For a split-complex number given by the following matrix representation:

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

its Jordan canonical form is given by:

J_z = \beginx + y & 0 \\ 0 & x - y\end ~,

where

Z = SJ_z S^\, ,

and

S = \begin 1 & -1 \\ 1 & 1 \end ~.

History

The use of split-complex numbers dates back to 1848 when

James Cockle Sir James Cockle FRS FRAS FCPS (14 January 1819 – 27 January 1895) was an English lawyer and mathematician. Cockle was born on 14 January 1819. He was the second son of James Cockle, a surgeon, of Great Oakley, Essex. Educated at Charte ...

revealed his

tessarine In abstract algebra, a bicomplex number is a pair of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate (w,z)^* = (w, -z), and the product of two bicomplex numbers as :(u,v)(w,z) = (u w - v z, u z ...

(1849
On a New Imaginary in Algebra
34:37–47, ''London-Edinburgh-Dublin Philosophical Magazine'' (3) 33:435–9, link from

Biodiversity Heritage Library The Biodiversity Heritage Library (BHL) is the world’s largest open access digital library for biodiversity literature and archives. BHL operates as worldwide consortiumof natural history, botanical, research, and national libraries working toge ...

. 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 In mathematics, a split-biquaternion is a hypercomplex number of the form :q = w + xi + yj + zk where ''w'', ''x'', ''y'', and ''z'' are split-complex numbers and i, j, and k multiply as in the quaternion group. Since each coefficient ''w'', ''x' ...

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 In mathematics, a function of a motor variable is a function with arguments and values in the split-complex number plane, much as functions of a complex variable involve ordinary complex numbers. William Kingdon Clifford coined the term motor fo ...

contrast to functions of an ordinary

complex variable Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...

. Since the late twentieth century, the split-complex multiplication has commonly been seen as a

of a

spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...

plane. In that model, the number represents an event in a spatio-temporal plane, where ''x'' is measured in nanoseconds and in Mermin's feet. The future corresponds to the quadrant of events , which has the split-complex polar decomposition

z = \rho e^ \!

. The model says that can be reached from the origin by entering a

frame of reference In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both mathema ...

rapidity In relativity, rapidity is commonly used as a measure for relativistic velocity. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being associated with di ...

and waiting nanoseconds. The split-complex equation

e^ \ e^ = e^

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

\

is the line of events simultaneous with the origin in the frame of reference with rapidity ''a''. Two events and are

when

z^*w+zw^* = 0.

Canonical events and are hyperbolic orthogonal and lie on the axes of a frame of reference in which the events simultaneous with the origin are proportional to . In 1933

Max Zorn Max August Zorn (; June 6, 1906 – March 9, 1993) was a German mathematician. He was an algebraist, group theorist, and numerical analyst. He is best known for Zorn's lemma, a method used in set theory that is applicable to a wide range of ...

was using the

split-octonion In mathematics, the split-octonions are an 8-dimensional nonassociative algebra over the real numbers. Unlike the standard octonions, they contain non-zero elements which are non-invertible. Also the signatures of their quadratic forms differ: t ...

s and noted the

property. He realized that the

Cayley–Dickson construction In mathematics, the Cayley–Dickson construction, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. The algebras produced b ...

, used to generate division algebras, could be modified (with a factor gamma, ) to construct other composition algebras including the split-octonions. His innovation was perpetuated by Adrian Albert, Richard D. Schafer, and others. The gamma factor, with as base field, builds split-complex numbers as a composition algebra. Reviewing Albert for

Mathematical Reviews ''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also pu ...

, N. H. McCoy wrote that there was an "introduction of some new algebras of order 2^e over ''F'' generalizing Cayley–Dickson algebras." Taking and 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 in ''Contribución a las Ciencias Físicas y Matemáticas'',

National University of La Plata The La Plata National University ( es, Universidad Nacional de La Plata, UNLP) is one of the most important Argentine national universities and the biggest one situated in the city of La Plata, capital of Buenos Aires Province. It has over 90 ...

, República Argentina (in Spanish). These expository and pedagogical essays presented the subject for broad appreciation. In 1941 E.F. Allen used the split-complex geometric arithmetic to establish the

nine-point hyperbola In Euclidean geometry with triangle , the nine-point hyperbola is an instance of the nine-point conic described by American mathematician Maxime Bôcher in 1892. The celebrated nine-point circle is a separate instance of Bôcher's conic: :Given ...

of a triangle inscribed in . In 1956 Mieczyslaw Warmus published "Calculus of Approximations" in ''Bulletin de l’Académie polonaise des sciences'' (see link in References). He developed two algebraic systems, each of which he called "approximate numbers", the second of which forms a real algebra. D. H. Lehmer reviewed the article in

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.

Synonyms

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) * ''bireal numbers'', U. Bencivenga (1946) * ''approximate numbers'', Warmus (1956), for use in

interval analysis Interval arithmetic (also known as interval mathematics, interval analysis, or interval computation) is a mathematical technique used to Floating point error mitigation, put bounds on rounding errors and measurement errors in numerical analysis ...

* ''countercomplex'' or ''hyperbolic'' numbers from Musean hypernumbers * ''double numbers'', I.M. Yaglom (1968), Kantor and Solodovnikov (1989),

Hazewinkel The Hazewinkel is a 2,000 m rowing and regatta course belonging to Bloso in Heindonk, municipality of Willebroek, near Mechelen, Belgium. The site consists of a finishing tower, boathouses, a cafeteria and eight basic huts that house athl ...

(1990), Rooney (2014) * ''anormal-complex numbers'', W. Benz (1973) * ''perplex numbers'', P. Fjelstad (1986) and Poodiack & LeClair (2009) * ''Lorentz numbers'', F.R. 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 (1997)Rosenfeld, B. (1997) ''Geometry of Lie Groups'', page 30,

Kluwer Academic Publishers Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...

* ''spacetime numbers'', N. Borota (2000) * ''Study numbers'', P. Lounesto (2001) * ''twocomplex numbers'', S. Olariu (2002) Split-complex numbers and their higher-dimensional relatives (

split-quaternion In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They form an associative algebra of dimension four over the real numbers. After introduction in th ...

s / coquaternions and

s) were at times referred to as "Musean numbers", since they are a subset of the hypernumber program developed by

Charles Musès Charles Arthur Muses (; 28 April 1919 – 26 August 2000), was a mathematician, cyberneticist and an esoteric philosopher who wrote articles and books under various pseudonyms (including ''Musès'', ''Musaios'', ''Kyril Demys'', ''Arthur Fontai ...