Split-complex number

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

The collection Template:Mvar of all split-complex numbers <math>z=x+yj</math> for Template:Tmath forms an algebra over the field of real numbers. Two split-complex numbers Template:Mvar and Template:Mvar have a product Template:Mvar that satisfies <math>N(wz)=N(w)N(z).</math> This composition of Template:Mvar over the algebra product makes Template:Math a composition algebra.

A similar algebra based on Template:Tmath and component-wise operations of addition and multiplication, Template:Tmath where Template:Mvar is the quadratic form on Template:Tmath also forms a quadratic space. The ring isomorphism <math display=block>\begin{align}

      D &\to \mathbb{R}^2 \\
 x + yj &\mapsto (x - y, x + y)

\end{align}</math> is an isometry of quadratic spaces.

Split-complex numbers have many other names; see Template:Section link below. See the article Motor variable for functions of a split-complex number.

DefinitionEdit

A split-complex number is an ordered pair of real numbers, written in the form

where Template:Mvar and Template:Mvar are real numbers and the hyperbolic unit<ref>Vladimir V. Kisil (2012) Geometry of Mobius Transformations: Elliptic, Parabolic, and Hyperbolic actions of SL(2,R), pages 2, 161, Imperial College Press Template:ISBN</ref> Template:Mvar satisfies

In the field of complex numbers the imaginary unit i satisfies <math>i^2 = -1 .</math> The change of sign distinguishes the split-complex numbers from the ordinary complex ones. The hyperbolic unit Template:Mvar is not a real number but an independent quantity.

The collection of all such Template:Mvar is called the split-complex plane. Addition and multiplication of split-complex numbers are defined by

<math display=block>\begin{align}

 (x + jy) + (u + jv) &= (x + u) + j(y + v) \\
    (x + jy)(u + jv) &= (xu + yv) + j(xv + yu).

\end{align}</math>

This multiplication is commutative, associative and distributes over addition.

Conjugate, modulus, and bilinear formEdit

Just as for complex numbers, one can define the notion of a split-complex conjugate. If

then the conjugate of Template:Mvar is defined as

The conjugate is an involution which satisfies similar properties to the complex conjugate. Namely,

<math display=block>\begin{align}

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

\end{align}</math>

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

<math display=block>\lVert z \rVert^2 = z z^* = z^* z = x^2 - y^2 ~.</math>

It has the composition algebra property:

<math display=block>\lVert z w \rVert = \lVert z \rVert \lVert w \rVert ~.</math>

However, this quadratic form is not positive-definite but rather has signature Template:Math, so the modulus is not a norm.

The associated bilinear form is given by

<math display=block>\langle z, w \rangle = \operatorname\mathrm{Re}\left(zw^*\right) = \operatorname\mathrm{Re} \left(z^* w\right) = xu - yv ~,</math>

where <math>z=x+jy</math> and <math>w=u+jv.</math> Here, the real part is defined by <math>\operatorname\mathrm{Re}(z) = \tfrac{1}{2}(z + z^*) = x</math>. Another expression for the squared modulus is then

<math display=block> \lVert z \rVert^2 = \langle z, z \rangle ~.</math>

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 Template:Nowrap thus numbers of the form Template:Math have no inverse. The multiplicative inverse of an invertible element is given by

<math display=block>z^{-1} = \frac{z^*}{ {\lVert z \rVert}^2} ~.</math>

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

The diagonal basisEdit

There are two nontrivial idempotent elements given by <math>e=\tfrac{1}{2}(1-j)</math> and <math>e^* = \tfrac{1}{2}(1+j).</math> Idempotency means that <math>ee=e</math> and <math>e^*e^*=e^*.</math> Both of these elements are null:

<math display=block>\lVert e \rVert = \lVert e^* \rVert = e^* e = 0 ~.</math>

It is often convenient to use Template:Mvar and Template:Mvar^∗ as an alternate basis for the split-complex plane. This basis is called the diagonal basis or null basis. The split-complex number Template:Mvar can be written in the null basis as

If we denote the number <math>z=ae+be^*</math> for real numbers Template:Mvar and Template:Mvar by Template:Math, then split-complex multiplication is given by

<math display=block>\left( a_1, b_1 \right) \left( a_2, b_2 \right) = \left( a_1 a_2, b_1 b_2 \right) ~.</math>

The split-complex conjugate in the diagonal basis is given by <math display=block>(a, b)^* = (b, a)</math> and the squared modulus by

<math display=block> \lVert (a, b) \rVert^2 = ab.</math>

IsomorphismEdit

File:Commutative diagram split-complex number 2.svg

This commutative diagram relates the action of the hyperbolic versor on Template:Mvar to squeeze mapping Template:Mvar applied to Template:Tmath

On the basis {e, e*} it becomes clear that the split-complex numbers are ring-isomorphic to the direct sum Template:Tmath with addition and multiplication defined pairwise.

The diagonal basis for the split-complex number plane can be invoked by using an ordered pair Template:Math for <math>z = x + jy</math> and making the mapping

<math display=block> (u, v) = (x, y) \begin{pmatrix}1 & 1 \\1 & -1\end{pmatrix} = (x, y) S ~. </math>

Now the quadratic form is <math>uv = (x + y)(x - y) = x^2 - y^2 ~.</math> Furthermore,

<math display=block> (\cosh a, \sinh a) \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} = \left(e^a, e^{-a}\right) </math>

so the two parametrized hyperbolas are brought into correspondence with Template:Mvar.

The action of hyperbolic versor <math>e^{bj} \!</math> then corresponds under this linear transformation to a squeeze mapping

<math display=block> \sigma: (u, v) \mapsto \left(ru, \frac{v}{r}\right),\quad r = e^b ~. </math>

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 Template:Sqrt. 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 Template:Tmath plane with its "unit circle" given by <math>\{(a,b) \in \R \oplus \R : ab=1\}.</math> The contracted unit hyperbola <math>\{\cosh a+j\sinh a : a \in \R\}</math> 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 Template:Tmath.

GeometryEdit

File:Drini-conjugatehyperbolas.svg

Template:Legend-line Template:Legend-line Template:Legend-line

A two-dimensional real vector space with the Minkowski inner product is called Template:Math-dimensional Minkowski space, often denoted Template:Tmath Just as much of the geometry of the Euclidean plane Template:Tmath can be described with complex numbers, the geometry of the Minkowski plane Template:Tmath can be described with split-complex numbers.

The set of points

<math display=block>\left\{ z : \lVert z \rVert^2 = a^2 \right\}</math>

is a hyperbola for every nonzero Template:Mvar in Template:Tmath The hyperbola consists of a right and left branch passing through Template:Math and Template:Math. The case Template:Math is called the unit hyperbola. The conjugate hyperbola is given by

<math display=block>\left\{ z : \lVert z \rVert^2 = -a^2 \right\}</math>

with an upper and lower branch passing through Template:Math and Template:Math. The hyperbola and conjugate hyperbola are separated by two diagonal asymptotes which form the set of null elements:

<math display=block>\left\{ z : \lVert z \rVert = 0 \right\}.</math>

These two lines (sometimes called the null cone) are perpendicular in Template:Tmath and have slopes ±1.

Split-complex numbers Template:Mvar and Template:Mvar are said to be hyperbolic-orthogonal if Template:Math. 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

<math display=block>\exp(j\theta) = \cosh(\theta) + j\sinh(\theta).</math>

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.<ref>James Cockle (1848) On a New Imaginary in Algebra, Philosophical Magazine 33:438</ref> For all real values of the hyperbolic angle Template:Mvar the split-complex number Template:Math has norm 1 and lies on the right branch of the unit hyperbola. Numbers such as Template:Mvar have been called hyperbolic versors.

Since Template:Mvar has modulus 1, multiplying any split-complex number Template:Mvar by Template:Mvar preserves the modulus of Template:Mvar and represents a hyperbolic rotation (also called a Lorentz boost or a squeeze mapping). Multiplying by Template:Mvar 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 Template:Math. This group consists of the hyperbolic rotations, which form a subgroup denoted Template:Math, combined with four discrete reflections given by

<math display=block>z \mapsto \pm z</math> and <math>z \mapsto \pm z^*.</math>

The exponential map

<math display=block>\exp\colon (\R, +) \to \mathrm{SO}^{+}(1, 1)</math>

sending Template:Mvar to rotation by Template:Math is a group isomorphism since the usual exponential formula applies:

<math display=block>e^{j(\theta + \phi)} = e^{j\theta}e^{j\phi}.</math>

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

Algebraic propertiesEdit

In abstract algebra terms, the split-complex numbers can be described as the quotient of the polynomial ring Template:Tmath by the ideal generated by the polynomial <math>x^2-1,</math>

The image of Template:Mvar in the quotient is the "imaginary" unit Template:Mvar. 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

<math display=block>\lVert zw \rVert = \lVert z \rVert \lVert w \rVert ~</math>

for any numbers Template:Mvar and Template:Mvar.

From the definition it is apparent that the ring of split-complex numbers is isomorphic to the group ring Template:Tmath of the cyclic group Template:Math over the real numbers Template:Tmath

Elements of the identity component in the group of units in D have four square roots.: say <math>p = \exp (q), \ \ q \in D. \text{then} \pm \exp(\frac{q}{2}) </math> are square roots of p. Further, <math>\pm j \exp(\frac{q}{2})</math> are also square roots of p.

The idempotents <math>\frac{1 \pm j}{2}</math> are their own square roots, and the square root of <math>s \frac{1 \pm j}{2}, \ \ s > 0, \ \text{is} \ \sqrt{s} \frac{1 \pm j}{2}</math>

Matrix representationsEdit

One can easily represent split-complex numbers by matrices. The split-complex number <math>z = x + jy</math> can be represented by the matrix <math>z \mapsto \begin{pmatrix}x & y \\ y & x\end{pmatrix}.</math>

Addition and multiplication of split-complex numbers are then given by matrix addition and multiplication. The squared modulus of Template:Mvar 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

<math display=block>m = \begin{pmatrix}a & c \\ b & -a \end{pmatrix}</math>

which square to the identity matrix satisfy <math>a^2 + bc = 1 .</math> 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).<ref>Template:Wikibooks-inline</ref>Template:Better source needed

The number <math>z = x + jy</math> can be represented by the matrix <math>x\ I + y\ m .</math>

HistoryEdit

The use of split-complex numbers dates back to 1848 when James Cockle revealed his tessarines.<ref name=JC>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.</ref> 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.<ref>Francesco Antonuccio (1994) Semi-complex analysis and mathematical physics</ref><ref>F. Catoni, D. Boccaletti, R. Cannata, V. Catoni, E. Nichelatti, P. Zampetti. (2008) The Mathematics of Minkowski Space-Time, Birkhäuser Verlag, Basel. Chapter 4: Trigonometry in the Minkowski plane. Template:Isbn.</ref><ref>Template:Cite book</ref><ref>Template:Cite journal</ref><ref>Louis Kauffman (1985) "Transformations in Special Relativity", International Journal of Theoretical Physics 24:223–36.</ref><ref>Sobczyk, G.(1995) Hyperbolic Number Plane, also published in College Mathematics Journal 26:268–80.</ref> In that model, the number Template:Math represents an event in a spatio-temporal plane, where x is measured in seconds and Template:Mvar in light-seconds. The future corresponds to the quadrant of events Template:Math, which has the split-complex polar decomposition <math>z = \rho e^{aj} \!</math>. The model says that Template:Mvar can be reached from the origin by entering a frame of reference of rapidity Template:Mvar and waiting Template:Mvar nanoseconds. The split-complex equation

expressing products on the unit hyperbola illustrates the additivity of rapidities for collinear velocities. Simultaneity of events depends on rapidity Template:Mvar;

<math display=block>\{ z = \sigma j e^{aj} : \sigma \isin \R \}</math>

is the line of events simultaneous with the origin in the frame of reference with rapidity a.

Two events Template:Mvar and Template:Mvar are hyperbolic-orthogonal when <math>z^*w+zw^* = 0.</math> Canonical events Template:Math and Template:Math are hyperbolic orthogonal and lie on the axes of a frame of reference in which the events simultaneous with the origin are proportional to Template:Math.

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, Template:Mvar) to construct other composition algebras including the split-octonions. His innovation was perpetuated by Adrian Albert, Richard D. Schafer, and others.<ref>Robert B. Brown (1967)On Generalized Cayley-Dickson Algebras, Pacific Journal of Mathematics 20(3):415–22, link from Project Euclid.</ref> The gamma factor, with Template:Math 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."<ref>N.H. McCoy (1942) Review of "Quadratic forms permitting composition" by A.A. Albert, Mathematical Reviews #0006140</ref> Taking Template:Math and Template:Math 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.<ref>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</ref>

In 1941 E.F. Allen used the split-complex geometric arithmetic to establish the nine-point hyperbola of a triangle inscribed in Template:Math.<ref>Allen, E.F. (1941) "On a Triangle Inscribed in a Rectangular Hyperbola", American Mathematical Monthly 48(10): 675–681</ref>

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.<ref>M. Warmus (1956) "Calculus of Approximations" Template:Webarchive, Bulletin de l'Académie polonaise des sciences, Vol. 4, No. 5, pp. 253–257, Template:MR</ref> 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.

SynonymsEdit

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)<ref>Template:Cite thesis</ref>
bireal numbers, U. Bencivenga (1946)
real hyperbolic numbers, N. Smith (1949)<ref>Template:Cite thesis</ref>
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),<ref>Template:Cite journal</ref> 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)<ref>Rosenfeld, B. (1997) Geometry of Lie Groups, page 30, Kluwer Academic Publishers Template:Isbn</ref>
spacetime numbers, N. Borota (2000)
Study numbers, P. Lounesto (2001)
twocomplex numbers, S. Olariu (2002)
split binarions, K. McCrimmon (2004)

ReferencesEdit

Template:Reflist