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. 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).

Algebraically the split-complex numbers have the interesting property, absent from the complex numbers, of containing nontrivial idempotents (other than 0 and 1). Furthermore, the collection of all split-complex numbers does not form a field, but instead form a ring.

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. In other words, the split-complex numbers are similar to complex numbers but in the split signature (1,1).

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.

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).

This multiplication is commutative, associative and distributes over addition.

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.

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 (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 modulus is then

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

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 = (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.

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

A two-dimensional real vector space with the Minkowski inner product is called 1+1 dimensional Minkowski space, often denoted R^1,1. 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.

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

$exp(jtheta)\; =\; cosh(theta)\; +\; jsinh(theta).,$

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

$zmapstopm\; z$ and $zmapstopm\; z^\{*\}.$

The exponential map

$expcolon(mathbb\; R,\; +)\; to\; mathrm\{SO\}^\{+\}(1,1)$

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

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

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. If 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 Cℓ_1,0(R) = Cℓ⁰_1,1(R). This is an extension of the real numbers defined analogously to the complex numbers C = Cℓ_0,1(R) = Cℓ⁰_2,0(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

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. Split-complex conjugation corresponds to multiplying on both sides by the matrix

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

Working in the diagonal basis leads to a diagonal matrix representation

Hyperbolic rotations in this basis correspond to multiplication by

which shows that they are squeeze mappings.

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 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 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). See the article on functions of a motor variable for details.

In 1941 E.F. Allen used the split-complex geometric arithmetic to establish the nine-point hyperbola of at 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)
• 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

• Lorentz group
• Minkowski space

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

• Split-quaternion (or coquaternion)
• Split-octonion

Enveloping algebras and number programs:

• Clifford algebra
• Hypercomplex numbers
• Lie algebra

References and external links

• 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.
• 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.
• Hucks, J. (1993) "Hyperbolic Complex Structures in Physics", Journal of Mathematical Physics 34:5986.
• Introduction to Algebraic Motors
• 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 (PDF)
• 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.