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 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 ...

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 ...

. 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. 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 ...

on also forms a quadratic space. The ring isomorphism

\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'' ...

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 (mathematics), 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 coi ...

'' 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. 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, 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 Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...

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 composition algebra property:

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

However, this quadratic form is not positive-definite but rather has

signature A signature (; from la, signare, "to sign") is a Handwriting, handwritten (and often Stylization, 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 ...

, so the modulus is ''not'' a norm. The associated bilinear form 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 bic ...

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 multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/' ...

of an invertible element is given by

z^ = \frac ~.

Split-complex numbers which are not invertible are called null vectors. These are all of the form for some real number .

The diagonal basis

There are two nontrivial idempotent elements 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 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, 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 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 surgi ...

by . The dilation in particular has sometimes caused confusion in connection with areas of a hyperbolic sector. Indeed, hyperbolic angle 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 or planar lamina, while '' surface area'' refers to the area of an op ...

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 ra ...

\

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, 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, ca ...

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 asymptotes 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 c ...

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

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 con ...

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 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 ...

or a squeeze mapping). 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 called the

generalized orthogonal group In mathematics, the indefinite orthogonal group, is the Lie group of all linear transformations of an ''n''-dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature , where . It is also called the p ...

. This group consists of the hyperbolic rotations, which form a subgroup denoted , combined with four discrete reflections 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.

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 ter ...

terms, the split-complex numbers can be described as the quotient 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 considered ...

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 exampl ...

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 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 ze ...

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 composition algebra 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 of the cyclic group over the real numbers

Matrix representations

One can easily represent split-complex numbers by matrices. 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 a ...

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 hyperbolic angle 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 of

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 W ...

e^ \!

then corresponds under this linear transformation to a squeeze mapping

\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 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 Charterh ...

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-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 In mathematics, a function of a motor variable is a function (mathematics), 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 coi ...

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 algebra ...

. 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 differ ...

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 math ...

of rapidity 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 hyperbolic-orthogonal 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 m ...

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 composition algebra property. He realized that the Cayley–Dickson construction, 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 Abraham Adrian Albert (November 9, 1905 – June 6, 1972) was an American mathematician. In 1939, he received the American Mathematical Society's Cole Prize in Algebra for his work on Riemann matrices. He is best known for his work on the A ...

, 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, 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, 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 Derrick Henry "Dick" Lehmer (February 23, 1905 – May 22, 1991), almost always cited as D.H. Lehmer, was an American mathematician significant to the development of computational number theory. Lehmer refined Édouard Lucas' work in the 1930s and ...

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.

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 * ''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 athletes ...

(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 * ''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

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 ...