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In abstract algebra, the split-complex numbers (or hyperbolic numbers) are a two-dimensional commutative algebra over the real numbers different from the complex numbers. Every split-complex number has the form

x + y j,

where x and y are real numbers. The number j is similar to the imaginary unit i, except that

j2 = 1.

As an algebra over the reals, the split-complex numbers are related to the direct sum of algebras RR (under the isomorphism sending x + y j to (x + yx − y) ). The name split comes from this characterization: as a real algebra, the split-complex numbers split into the direct sum RR.

Geometrically, multiplication of split-complex numbers preserves the (square) Minkowski norm (x2 − y2) in the same way that multiplication of complex numbers preserves the (square) Euclidean norm (x2 + y2). Unlike the complex numbers, the split-complex numbers contain nontrivial idempotents (other than 0 and 1), as well as zero divisors, and therefore they do not form a field.

The split-complex number is one of the concepts necessary to read a 2 × 2 real matrix.

Split-complex numbers have many other names; see the synonyms section below.

Contents

Definition

A split-complex number is one of the form

z = x + y \jmath

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

\jmath^2 = +1.

Choosing \jmath^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 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).

This multiplication is commutative, associative and distributes over addition.

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Conjugate, modulus, and bilinear form

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* = xj 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.

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

\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 the modulus is not a norm.

The associated bilinear form is given by

<z, w> = Re(zw*) = Re(z*w) = xuyv

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

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

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

A split-complex number is invertible if and only if its modulus is nonzero (\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. 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. 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 = (xy)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\oplusR 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.

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 R1,1. Just as much of the geometry of the Euclidean plane R2 can be described with complex numbers, the geometry of the Minkowski plane R1,1 can be described with split-complex numbers.

The set of points

\{ z : \lVert z \rVert = a^2 \}

is a hyperbola for every nonzero a in R. The hyperbola consists of a right and left branch passing through (a, 0) and (−a, 0). The case a = 1 is called the unit hyperbola. The conjugate hyperbola is given by

\{ 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.

Split-complex numbers z and w are said to be hyperbolic-orthogonal if <z, w> = 0. While analogous to ordinary orthogonality, particularly as it is known with ordinary complex number arithmetic, this condition is more subtle. It forms the basis for the simultaneous hyperplane concept in spacetime.

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

\exp(j\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. 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. Numbers such as λ have been called hyperbolic versors.

Since λ has modulus 1, multiplying any split-complex number z by λ preserves the modulus of z and represents a hyperbolic rotation (also called a Lorentz boost or a squeeze mapping). Multiplying by λ preserves the geometric structure, taking hyperbolas to themselves and the null cone to itself.

The set of all transformations of the split-complex plane which preserve the modulus(or equivalently, the inner product) forms a group called the generalized orthogonal group O(1,1). 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}.\,

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

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).

The image of x in the quotient is the "imaginary" unit j. With this description, it is clear that the split-complex numbers form a commutative ring with characteristic 0. Moreover if we define scalar multiplication in the obvious manner, the split-complex numbers actually form a commutative and associative algebra over the reals of dimension two. The algebra is not a division algebra or field since the null elements are not invertible. In fact, all of the nonzero null elements are zero divisors. Since addition and multiplication are continuous operations with respect to the usual topology of the plane, the split-complex numbers form a topological ring.

The split-complex numbers do not form a normed algebra in the usual sense of the word since the "norm" is not positive-definite. 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.

The split-complex numbers are a special case of a Clifford 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 C1,0(R) = C01,1(R). This is an extension of the real numbers defined analogously to the complex numbers C = C0,1(R) = C02,0(R).

Matrix representations

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

z = x + j y

can be represented by the matrix

z \mapsto \begin{pmatrix}x & y \\ y & x\end{pmatrix}.

Addition and multiplication of split-complex numbers are then given by matrix addition and multiplication. The modulus of z is given by the determinant of the corresponding matrix. In this representation, split-complex conjugation corresponds to multiplying on both sides by the matrix

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

For any real number a, a hyperbolic rotation by through a hyperbolic angle a corresponds to multiplication by the matrix

\begin{pmatrix}\cosh a & \sinh a \\ \sinh a & \cosh a\end{pmatrix}.

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

(u,v) = (x,y) \begin{pmatrix}1 & 1 \\1 & -1\end{pmatrix}.

Now the quadratic form is uv = (x + y)(xy) = x2y2. Furthermore,

(\cosh a, \sinh a)\begin{pmatrix}1 & 1\\1 & -1\end{pmatrix} = (e^a, e^{-a})

so the two parametrized hyperbolas are brought into correspondence. The action of hyperbolic versor e^{bj} \! then corresponds under this linear transformation to a squeeze mapping

(u,v) \mapsto (r u, v/r) ,\quad r = e^b .
Commutative square.svg

The commutative diagram interpretation of this correspondence has A = B = {split-complex number plane}, C = D = R2, f is the action of a hyperbolic versor, g & h are the linear transformation by the matrix of ones, and k is the squeeze mapping.

Note that in the context of 2 × 2 real matrices there are in fact a great number of different representations of split-complex numbers.

History

The use of split-complex numbers dates back to 1848 when James Cockle revealed his Tessarines. William Kingdon Clifford used split-complex numbers to represent sums of spins. Clifford introduced the use of split-complex numbers as coefficients in a quaternion algebra now called split-biquaternions. He called its elements "motors", a term in parallel with the "rotor" action of an ordinary complex number taken from the circle group. Extending the analogy, functions of a motor variable contrast to functions of an ordinary complex variable.

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 essentially the action of a hyperbolic versor in a space of mixed signature.

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 of a triangle inscribed in zz* = 1.

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)
  • bireal numbers, U. Bencivenga (1946)
  • countercomplex or hyperbolic numbers from Musean hypernumbers
  • double numbers, I.M. Yaglom (1968) and Hazewinkel (1990)
  • anormal-complex numbers, W. Benz (1973)
  • dual numbers, L. Kauffman (1985) and J. Hucks (1993)
  • perplex numbers, P. Fjelstad (1986) [see De Boer (1987) for the identification]
  • Lorentz numbers, F.R. Harvey (1990)
  • split-complex numbers, B. Rosenfeld (1997)

Split-complex numbers and their higher-dimensional relatives (split-quaternions / coquaternions and split-octonions) were at times referred to as "Musean numbers", since they are a subset of the hypernumber program developed by Charles Musès.

See also

Higher-order derivatives of split-complex numbers, obtained through a modified Cayley-Dickson construction:

Enveloping algebras and number programs:

References and external links

  • F. Catoni, D. Boccaletti, R. Cannata, V. Catoni, E. Nichelatti, P. Zampetti. (2008) The Mathematics of Minkowski Space-Time, Birkhäuser, Basel. Chapter 4: Trigonometry in the Minkowski plane. ISBN 978-3-7643-8613-9.
  • Cockle, James (1848) "A New Imaginary in Algebra", London-Edinburgh-Dublin Philosophical Magazine (3) 33:345-9.
  • Clifford, W.K.,Mathematical Works (1882) edited by A.W.Tucker,pp. 392,"Further Notes on Biquaternions"
  • Vignaux, J.(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.
  • Bencivenga, Uldrico (1946) "Sulla rappresentazione geometrica della algebra doppie dotate di modulo", Atti della real academie della scienze e belle-lettre di Napoli, Ser (3) v.2 No7. See MR0021123.
  • 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) "Double and dual numbers", Encyclopaedia of Mathematics, Soviet/AMS/Kluwer, Dordrect.
  • Hucks, J. (1993) "Hyperbolic Complex Structures in Physics", Journal of Mathematical Physics 34:5986.
  • Introduction to Algebraic Motors archived by WebCite.
  • Louis Kauffman (1985) "Transformations in Special Relativity", International Journal of Theoretical Physics 24:223-36.
  • Rosenfeld, B. (1997) Geometry of Lie Groups Kluwer Academic Pub.
  • Sobczyk, G.(1995) "Hyperbolic Number Plane", College Mathematics Journal 26:268–80.
  • 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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