Split-complex number

In linear algebra, a split-complex number is of the form z = x +y j where j² = +1 , and x and y are real numbers. Linear algebra is the branch of Mathematics concerned with In Mathematics, the real numbers may be described informally in several different ways The main geometric difference between these complex numbers and the ordinary ones, is that whereas multiplication of ordinary complex numbers respects the standard (square) Euclidean norm (x² + y²) on R², multiplication of split-complex numbers respects the (square) Minkowski norm (x² − y²). Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity

Algebraically the split-complex numbers have the interesting property, absent from the complex numbers, of containing nontrivial idempotents (other than 0 and 1). Idempotence ˌaɪdɨmˈpoʊtəns describes the property of operations in Mathematics and Computer science which means that multiple applications of the operation Furthermore, the collection of all split-complex numbers does not form a field, but instead this structure lies in the broader category of rings. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity and whose Morphisms

Split-complex numbers have many other names; see the synonyms section below. The name split comes from the fact that signatures of the form (p,p) are called split signatures. The signature of a Metric tensor (or more generally a nondegenerate Symmetric bilinear form, thought of as Quadratic form) is the number of positive In other words, the split-complex numbers are similar to complex numbers but in the split signature (1,1).

1 Definition
- 1.1 Conjugate, norm, and inner product
- 1.2 The diagonal basis
2 Geometry
3 Algebraic properties
4 Matrix representations
5 History
6 Synonyms
7 See also
8 References and external links

Definition

A split-complex number is one of the form

z = x + j y

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

j² = +1. In Mathematics, the real numbers may be described informally in several different ways

Choosing j² = −1 results in the complex numbers. It is this sign change which distinguishes the split-complex numbers from the 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) = (xu + yv) + j(xv + yu). Addition is the mathematical process of putting things together

This multiplication is commutative, associative and distributes over addition. In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, associativity is a property that a Binary operation can have In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law

Conjugate, norm, and inner product

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.

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

(z + w)* = z* + w*

(zw)* = z*w*

(z*)* = z.

These three properties imply that the split-complex conjugate is an automorphism of order 2. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself In Group theory, a branch of Mathematics, the term order is used in two closely related senses the order of a group is

The quadratic form of a split-complex number z = x + j y is given by

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

It has an important property that it is preserved by split-complex multiplication:

$\lVert z w \rVert = \lVert z \rVert \lVert w \rVert.$

However, this quadratic form is not positive-definite but rather has signature (1,1), so it is not a norm. In Mathematics, a quadratic form is a Homogeneous polynomial of degree two in a number of variables In Mathematics, a definite bilinear form is a Bilinear form B such that B ( x, x) has a fixed The signature of a Metric tensor (or more generally a nondegenerate Symmetric bilinear form, thought of as Quadratic form) is the number of positive In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length

The associated (1,1) inner product is given by

<z, w> = Re(zw*) = Re(z*w) = xu − yv

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

$\lVert z \rVert = \langle z, z \rangle.$

A split-complex number is invertible if and only if its norm is nonzero ( $\lVert z \rVert \ne 0$ ). ↔ The inverse of such an element is given by

$z^{-1} = z^* / \lVert z \rVert.$

Split-complex numbers which are not invertible are called null elements. These are all of the form (a ± j a) for some real number a.

The diagonal basis

There are two nontrivial idempotents given by e = (1 − j)/2 and e* = (1 + j)/2. Idempotence ˌaɪdɨmˈpoʊtəns describes the property of operations in Mathematics and Computer science which means that multiple applications of the operation 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 e and e* as an alternate basis for the split-complex plane. Basis vector redirects here For basis vector in the context of crystals see Crystal structure. 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

(a₁,b₁)(a₂,b₂) = (a₁a₂, b₁b₂).

In this basis, it becomes clear that the split-complex numbers are isomorphic to the direct sum R $\oplus$ R with addition and multiplication defined pairwise. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction

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

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

and the norm by

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

Geometry

Unit rectangular 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 R^1,1. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity Just as much of the geometry of the Euclidean plane R² can be described with complex numbers, the geometry of the Minkowski plane R^1,1 can be described with split-complex numbers. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position

The set of points

$\{ z : \lVert z \rVert = a^2 \}$

is a hyperbola for every nonzero a in R. In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions 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

$\{ z : \lVert z \rVert = -a^2 \}$

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:

$\{ z : \lVert z \rVert = 0 \}.$

These two lines (sometimes called the null cone) are perpendicular in R² and have slopes ±1. An asymptote of a real-valued function y=f(x is a curve which describes the behavior of f as either x or y goes to infinity In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent

Split-complex numbers z and w are said to be hyperbolic-orthogonal if <z, w> = 0. In Mathematics, two points in the Cartesian plane are hyperbolically orthogonal if the Slopes of their rays from the origin are Reciprocal to 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. In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity

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

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

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. This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 + In Mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular functions In Mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular functions For all real values of the hyperbolic angle θ the split-complex number λ = exp(jθ) has norm 1 and lies on the right branch of the unit hyperbola. A hyperbolic angle in standard position is the Angle at (0 0 between the ray to (1 1 and the ray to ( x, 1/ x) where x > 1

Since λ has norm 1, multiplying any split-complex number z by λ preserves the norm of z and represents a hyperbolic rotation (also called a Lorentz boost). In Physics, the Lorentz transformation converts between two different observers' measurements of space and time where one observer is in constant motion with respect to 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 norm (or equivalently, the inner product) forms a group called the generalized orthogonal group O(1,1). In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, the indefinite orthogonal group, O( p, q) is the Lie group of all Linear transformations of a n = p This group consists of the hyperbolic rotations — which form a subgroup denoted SO⁺(1,1) — combined with four discrete reflections given by

$z\mapsto\pm z$ and $z\mapsto\pm z^{*}.$

The exponential map

$\exp\colon(\mathbb R, +) \to \mathrm{SO}^{+}(1,1)$

sending θ to rotation by exp(jθ) is a group isomorphism since the usual exponential formula applies:

$e^{j(\theta+\phi)} = e^{j\theta}e^{j\phi}.\,$

Algebraic properties

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

R[x]/(x² − 1). In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete in the sense of not supporting or requiring the In Mathematics, a reflection (also spelled reflexion) is a map that transforms an object into its Mirror image. In Abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the In Mathematics, especially in the field of Abstract algebra, a polynomial ring is a ring formed from the set of Polynomials in one or more variables iDEAL is an Internet payment method in The Netherlands, based on online banking In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations

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. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's 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. In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive The algebra is not a division algebra or field since the null elements are not invertible. In the field of Mathematics called Abstract algebra, a division algebra is roughly speaking an Algebra over a field in which division is possible In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division If fact, all of the nonzero null elements are zero divisors. In Abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0 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 which is also a Topological space such that both the addition and the multiplication are

The split-complex numbers do not form a normed algebra in the usual sense of the word since the "norm" is not positive-definite. In Mathematics, especially Functional analysis, a Banach algebra, named after Stefan Banach, is an Associative algebra A over the However, if one extends the definition to include norms of general signature, they do form such an algebra. This follows from the fact that

$\lVert zw \rVert = \lVert z \rVert \lVert w \rVert.$

For an exposition of normed algebras in general signature, see the reference by Harvey.

From the definition it is apparent that the ring of split-complex numbers is isomorphic to the group ring R[C₂] of the cyclic group C₂ over the real numbers R. In Mathematics, a group ring is a ring R constructed from a ring R and a group G (written multiplicatively In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an

The split-complex numbers are a special case of a Clifford algebra. In Mathematics, Clifford algebras are a type of Associative algebra. Namely, they form a Clifford algebra over a one-dimensional vector space with a positive-definite quadratic form. Contrast this with the complex numbers which form a Clifford algebra over a one-dimensional vector space with a negative-definite quadratic form. (NB: some authors switch the signs in the definition of a Clifford algebra which will interchange the meaning of positive-definite and negative-definite). In mathematics, the split-complex numbers are members of the Clifford algebra Cℓ_1,0(R) = Cℓ⁰_1,1(R). Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, Clifford algebras are a type of Associative algebra. This is an extension of the real numbers defined analogously to the complex numbers C = Cℓ_0,1(R) = Cℓ⁰_2,0(R). In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

Matrix representations

One can easily represent split-complex numbers by matrices. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally The split-complex number

z = x + j y

can be represented by the matrix

$z \mapsto \begin{bmatrix}x & y \\ y & x\end{bmatrix}.$

Addition and multiplication of split-complex numbers are then given by matrix addition and multiplication. The norm of z is given by the determinant of the corresponding matrix. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n Split-complex conjugation corresponds to multiplying on both sides by the matrix

$C = \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}.$

A hyperbolic rotation by exp(jθ) corresponds to multiplication by the matrix

$\begin{bmatrix}\cosh\theta & \sinh\theta \\ \sinh\theta & \cosh\theta\end{bmatrix}.$

Working in the diagonal basis leads to a diagonal matrix representation

$z \mapsto \begin{bmatrix}x - y & 0 \\ 0 & x + y\end{bmatrix}.$

Hyperbolic rotations in this basis correspond to multiplication by

$\begin{bmatrix}e^{-\theta} & 0 \\ 0 & e^{\theta}\end{bmatrix}$

which shows that they are squeeze mappings. In Linear algebra, a squeeze mapping is a type of Linear map that preserves Euclidean Area of regions in the Cartesian plane, but is not a

History

The use of split-complex numbers dates back to 1848 when James Cockle revealed his Tessarines. Sir James Cockle FRS FRAS FCPS FMS ( 14 January 1819 &ndash 27 January 1895) was an English The tessarines are a mathematical idea introduced by James Cockle in 1848 William Kingdon Clifford used split-complex numbers to represent sums of spins. William Kingdon Clifford FRS ( May 4, 1845 &ndash March 3, 1879) was an English Mathematician and Clifford introduced the use of split-complex numbers as coefficients in a quaternion algebra now called split-biquaternions. In Mathematics, a split-biquaternion is a member of the Clifford algebra C �( R) He called its elements "motors", a term sometimes used in the study of split-complex numbers.

In the twentieth-century the split-complex numbers became a common platform to describe the Lorentz boosts of special relativity, in a spacetime plane, because a velocity change between frames of reference is nicely expressed by a hyperbolic rotation. In Physics, the Lorentz transformation converts between two different observers' measurements of space and time where one observer is in constant motion with respect to Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial

In 1935 J. C. Vignaux and A. Durañona y Vedia developed the split-complex geometric algebra and function theory in four articles in Contribucion a las Ciencias Fisicas y Matematicas, National University of La Plata, República Argentina (in Spanish). The National University of La Plata ( Spanish: Universidad Nacional de La Plata, UNLP is an Argentine National university and the most important For a topic outline on this subject see List of basic Argentina topics. See the article on functions of a motor variable for details. The function of a motor variable is a concept developed in Germany Argentina and Russia (see references

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) and G. Sobczyk (1995)
countercomplex or hyperbolic numbers from Musean hypernumbers
double numbers, I.M. Yaglom (1968) and Hazewinkel (1990)
anormal-complex numbers, W. Musean hypernumbers are an algebraic concept envisioned by Charles A Isaak Moiseevich Yaglom (Иссак Моисеевич Яглом ( 6 March 1921, Kharkov — 17 April 1988, Moscow) was Benz (1973)
perplex numbers, P. Fjelstad (1986) [see De Boer (1987) for the identification]
Lorentz numbers, F. R. Harvey (1990)
dual numbers, J. Hucks (1993)
split-complex numbers, B. Rosenfeld (1997)

Split-complex numbers and their higher-dimensional relatives (coquaternions / split-quaternions and split-octonions) were at times referred to as "Musean numbers", since they are a subset of the hypernumber program developed by Charles Musès. In Abstract algebra, the split-quaternions or coquaternions are elements of an Associative algebra introduced by James Cockle in 1849 under the In Mathematics, the split-octonions are a Nonassociative extension of the Quaternions (or the Split-quaternions. Musean hypernumbers are an algebraic concept envisioned by Charles A Charles A Muses (1919&ndash2000 a figure who wrote articles and books under various pseudonyms (including Musès Musaios Kyril Demys Arthur Fontaine Kenneth Demarest and Carl

References and external links

Cockle, James (1848) "A New Imaginary in Algebra", London-Edinburgh-Dublin Philosophical Magazine (3) 33:345-9. In Physics (and mathematics the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity In Abstract algebra, the split-quaternions or coquaternions are elements of an Associative algebra introduced by James Cockle in 1849 under the In Mathematics, the split-octonions are a Nonassociative extension of the Quaternions (or the Split-quaternions. In Mathematics, Clifford algebras are a type of Associative algebra. The term hypercomplex number has been used in Mathematics for the elements of algebras that extend or go beyond Complex number arithmetic In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie
Clifford, W. K. ,Mathematical Works (1882) edited by A. W. Tucker,pp. 392,"Further Notes on Biquaternions"
Vignaux, J. In Mathematics, a split-biquaternion is a member of the Clifford algebra C �( R) (1935) "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.
Benz, W. (1973)Vorlesungen uber Geometrie der Algebren, Springer
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.
Fjelstadt, P. (1986) "Extending Special Relativity with Perplex Numbers", American Journal of Physics 54:416.
De Boer, R. (1987) "An also known as list for perplex numbers", American Journal of Physics 55(4):296.
K. Carmody, Circular and hyperbolic quaternions, octonions, and sedenions, Appl. Math. Comput. 28:47–72 (1988)
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) editor Encyclopaedia of Mathematics Soviet/AMS/Kluwer, Dordrect. The Encyclopaedia of Mathematics is a large reference work in Mathematics.
Hucks, J. (1993) "Hyperbolic Complex Structures in Physics", Journal of Mathematical Physics 34:5986.
Introduction to Algebraic Motors
Sobczyk, G. (1995) Hyperbolic Number Plane (PDF)
Rosenfeld, B. (1997) Geometry of Lie Groups Kluwer Academic Pub.
K. Carmody, Circular and hyperbolic quaternions, octonions, and sedenions— further results, Appl. Math. Comput. 84:27–48 (1997)
Yaglom, I. (1968) Complex Numbers in Geometry,translated by E. Primrose from 1963 Russian original, Academic Press, N. Y. , pp. 18-20.

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Contents

Definition

Conjugate, norm, and inner product

The diagonal basis

Geometry

Algebraic properties

Matrix representations

History

Synonyms

See also

References and external links