split-complex plane

In algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit satisfying $j^2=1$, where $j \neq \pm 1$. A split-complex number has two real number components and , and is written $z=x+yj .$ The ''conjugate'' of is $z^*=x-yj.$ Since $j^2=1,$ the product of a number with its conjugate is $N(z) := zz^* = x^2 - y^2,$ an isotropic quadratic form.

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 on also forms a quadratic space. The ring isomorphism
$\begin D &\to \mathbb^2 \\ x + yj &\mapsto (x - y, x + y) \end$
is an isometry of quadratic spaces.

Split-complex numbers have many other names; see ' below. See the article '' Motor variable'' 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 real numbers and the hyperbolic unit satisfies

$j^2 = +1$

In the field of complex numbers the imaginary unit i satisfies $i^2 = -1 .$ The change of sign distinguishes the split-complex numbers from the ordinary complex ones. The hyperbolic unit 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 is an involution which satisfies similar properties to the complex conjugate. Namely,

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

The squared modulus of a split-complex number $z=x+jy$ is given by the isotropic quadratic form

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

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 , so the modulus is ''not'' a norm.

The associated bilinear form is given by

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

where $z=x+jy$ and $w=u+jv.$ Here, the ''real part'' is defined by $\operatorname\mathrm(z) = \tfrac(z + z^*) = x$. 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; 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 thus numbers of the form have no inverse. The multiplicative inverse 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 $e^* = \tfrac(1+j).$ Idempotency 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

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

The split-complex conjugate in the diagonal basis is given by
$(a, b)^* = (b, a)$
and the squared modulus by

$\lVert (a, b) \rVert^2 = ab.$

Isomorphism

On the basis it becomes clear that the split-complex numbers are ring-isomorphic to the direct sum with addition and multiplication defined pairwise.

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 $e^ \!$ then corresponds under this linear transformation to a squeeze mapping

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

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 by . The dilation in particular has sometimes caused confusion in connection with areas of a hyperbolic sector. Indeed, hyperbolic angle corresponds to area of a sector in the plane with its "unit circle" given by $\.$ The contracted unit hyperbola $\$ 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 with the Minkowski inner product is called -dimensional Minkowski space, often denoted Just as much of the geometry 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 for every nonzero in The hyperbola consists of a right and left branch passing through and . The case is called the unit hyperbola. 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 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 for the split-complex numbers is

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

This formula 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 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 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 . 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

In abstract algebra terms, the split-complex numbers can be described as the quotient of the polynomial ring by the ideal generated by the polynomial $x^2-1,$

\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 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 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 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

Elements of the identity component in the group of units in D have four square roots.: say $p = \exp (q), \ \ q \in D. \text \pm \exp(\frac)$ are square roots of ''p''. Further, $\pm j \exp(\frac)$ are also square roots of ''p''.

The idempotents $\frac$ are their own square roots, and the square root of $s \frac, \ \ s > 0, \ \text \ \sqrt \frac$

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 squared modulus of is given by the determinant of the corresponding matrix.

In fact there are many representations of the split-complex plane in the four-dimensional ring of 2x2 real matrices. The real multiples of the identity matrix form a real line in the matrix ring M(2,R). Any hyperbolic unit ''m'' provides a basis element with which to extend the real line to the split-complex plane. The matrices

$m = \begina & c \\ b & -a \end$

which square to the identity matrix satisfy $a^2 + bc = 1 .$
For example, when ''a'' = 0, then (''b,c'') is a point on the standard hyperbola. More generally, there is a hypersurface in M(2,R) of hyperbolic units, any one of which serves in a basis to represent the split-complex numbers as a subring of M(2,R).

The number $z = x + jy$ can be represented by the matrix  $x\ I + y\ m .$

History

The use of split-complex numbers dates back to 1848 when James Cockle revealed his tessarines. James Cockle (1849
On a New Imaginary in Algebra
34:37–47, ''London-Edinburgh-Dublin Philosophical Magazine'' (3) 33:435–9, link from Biodiversity Heritage Library. 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.

Since the late twentieth century, the split-complex multiplication has commonly been seen as a Lorentz boost of a spacetime plane. In that model, the number represents an event in a spatio-temporal plane, where ''x'' is measured in seconds and in light-seconds. 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 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 was using the split-octonions 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, 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 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 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), G. Cree (1949)
* ''bireal numbers'', U. Bencivenga (1946)
* ''real hyperbolic numbers'', N. Smith (1949)
* ''approximate numbers'', Warmus (1956), for use in interval analysis
* ''double numbers'', I.M. Yaglom (1968), Kantor and Solodovnikov (1989), Hazewinkel (1990), Rooney (2014)
* ''hyperbolic numbers'', W. Miller & R. Boehning (1968), G. Sobczyk (1995)
* ''anormal-complex numbers'', W. Benz (1973)
* ''perplex numbers'', P. Fjelstad (1986) and Poodiack & LeClair (2009)
* ''countercomplex'' or ''hyperbolic'', Carmody (1988)
* ''Lorentz numbers'', F.R. Harvey (1990)
* ''semi-complex numbers'', F. Antonuccio (1994)
* ''paracomplex numbers'', Cruceanu, Fortuny & Gadea (1996)
* ''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 binarions'', K. McCrimmon (2004)

See also

* Minkowski space
* Split-quaternion
* Hypercomplex number

References

Further reading

* Bencivenga, Uldrico (1946) "Sulla rappresentazione geometrica delle algebre doppie dotate di modulo", ''Atti della Reale Accademia delle Scienze e Belle-Lettere di Napoli'', Ser (3) v.2 No7. .
* Walter Benz (1973) ''Vorlesungen uber Geometrie der Algebren'', Springer
* N. A. Borota, E. Flores, and T. J. Osler (2000) "Spacetime numbers the easy way", Mathematics and Computer Education 34: 159–168.
* N. A. Borota and T. J. Osler (2002) "Functions of a spacetime variable", ''Mathematics and Computer Education'' 36: 231–239.
* K. Carmody, (1988
"Circular and hyperbolic quaternions, octonions, and sedenions"
Appl. Math. Comput. 28:47–72.
* K. Carmody, (1997) "Circular and hyperbolic quaternions, octonions, and sedenions – further results", Appl. Math. Comput. 84:27–48.
* William Kingdon Clifford (1882) ''Mathematical Works'', A. W. Tucker editor, page 392, "Further Notes on Biquaternions"
* V.Cruceanu, P. Fortuny & P.M. Gadea (1996
A Survey on Paracomplex Geometry
Rocky Mountain Journal of Mathematics 26(1): 83–115, link from Project Euclid.
* De Boer, R. (1987) "An also known as list for perplex numbers", ''American Journal of Physics'' 55(4):296.
* Anthony A. Harkin & Joseph B. Harkin (2004
Geometry of Generalized Complex Numbers
Mathematics Magazine 77(2):118–29.
* F. Reese Harvey. ''Spinors and calibrations.'' Academic Press, San Diego. 1990. . 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.
* Kevin McCrimmon (2004) ''A Taste of Jordan Algebras'', pp 66, 157, Universitext, 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.
* Olariu, Silviu (2002) ''Complex Numbers in N Dimensions'', Chapter 1: Hyperbolic Complex Numbers in Two Dimensions, pages 1–16, North-Holland Mathematics Studies #190, Elsevier .
* Poodiack, Robert D. & Kevin J. LeClair (2009) "Fundamental theorems of algebra for the perplexes", The College Mathematics Journal 40(5):322–35.
* Isaak Yaglom (1968) ''Complex Numbers in Geometry'', translated by E. Primrose from 1963 Russian original, Academic Press, pp. 18–20.
*

{{DEFAULTSORT:Split-Complex Number
Composition algebras
Linear algebra
Hypercomplex numbers