In
algebra, a split complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unitj satisfying A split-complex number has two
real number components x and y, and is written The conjugate of z is Since the product of a number z with its conjugate is an
isotropic quadratic form.
The collection D of all split complex numbers for forms an
algebra over the field of real numbers. Two split-complex numbers w and z have a product wz that satisfies This composition of N over the algebra product makes (D, +, ×, *) a
composition algebra.
relates proportional quadratic forms, but the mapping is not an
isometry since the multiplicative identity (1, 1) of is at a distance from 0, which is normalized in D.
Split-complex numbers have many other names; see § Synonyms 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
where x and y are
real numbers and the hyperbolic unit[1]j satisfies
In the field of
complex numbers the
imaginary unit i satisfies The change of sign distinguishes the split-complex numbers from the ordinary complex ones. The hyperbolic unit j is not a real number but an independent quantity.
The collection of all such z is called the split-complex plane.
Addition and
multiplication of split-complex numbers are defined by
where and Here, the real part is defined by . Another expression for the squared modulus is then
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 x ± j x have no inverse. The
multiplicative inverse of an invertible element is given by
Split-complex numbers which are not invertible are called
null vectors. These are all of the form (a ± j a) for some real number a.
The diagonal basis
There are two nontrivial
idempotent elements given by and Recall that idempotent means that and Both of these elements are null:
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
If we denote the number for real numbers a and b by (a, b), then split-complex multiplication is given by
The split-complex conjugate in the diagonal basis is given by
and the squared modulus by
Isomorphism
On the basis {e, e*} 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 (x, y) for and making the mapping
Now the quadratic form is Furthermore,
so the two
parametrized hyperbolas are brought into correspondence with S.
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 √2. 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 (1 + 1)-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
is a
hyperbola for every nonzero a in 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
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:
These two lines (sometimes called the null cone) are
perpendicular in 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
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.[2] For all real values of the
hyperbolic angleθ 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 groupO(1, 1). This group consists of the hyperbolic rotations, which form a
subgroup denoted SO+(1, 1), combined with four
discretereflections given by
and
The exponential map
sending θ to rotation by exp(jθ) is a
group isomorphism since the usual exponential formula applies:
If a split-complex number z does not lie on one of the diagonals, then z has a
polar decomposition.
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 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.
From the definition it is apparent that the ring of split-complex numbers is isomorphic to the
group ring of the
cyclic groupC2 over the real numbers
Matrix representations
One can easily represent split-complex numbers by
matrices. The split-complex number 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.
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
which square to the identity matrix satisfy
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).[3][better source needed]
The number can be represented by the matrix
History
The use of split-complex numbers dates back to 1848 when
James Cockle revealed his
tessarines.[4]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.[5][6][7][8][9][10] In that model, the number z = x + yj represents an event in a spatio-temporal plane, where x is measured in nanoseconds and y in
Mermin's feet. The future corresponds to the quadrant of events {z : |y| < x}, which has the split-complex polar decomposition . The model says that z can be reached from the origin by entering a
frame of reference of
rapiditya and waiting ρ 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 a;
is the line of events simultaneous with the origin in the frame of reference with rapidity a.
Two events z and w are
hyperbolic-orthogonal when Canonical events exp(aj) and j exp(aj) are hyperbolic orthogonal and lie on the axes of a frame of reference in which the events simultaneous with the origin are proportional to j exp(aj).
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, γ) to construct other composition algebras including the split-octonions. His innovation was perpetuated by
Adrian Albert, Richard D. Schafer, and others.[11] The gamma factor, with R 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 2e over F generalizing Cayley–Dickson algebras."[12] Taking F = R and e = 1 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.[13]
In 1941 E.F. Allen used the split-complex geometric arithmetic to establish the
nine-point hyperbola of a triangle inscribed in zz∗ = 1.[14]
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.[15]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)[16]
^Vladimir V. Kisil (2012) Geometry of Mobius Transformations: Elliptic, Parabolic, and Hyperbolic actions of SL(2,R), pages 2, 161, Imperial College Press
ISBN978-1-84816-858-9
^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.
ISBN978-3-7643-8613-9.
^Francesco Catoni; Dino Boccaletti; Roberto Cannata; Vincenzo Catoni; Paolo Zampetti (2011). "Chapter 2: Hyperbolic Numbers". Geometry of Minkowski Space-Time. Springer Science & Business Media.
ISBN978-3-642-17977-8.
^N.H. McCoy (1942) Review of "Quadratic forms permitting composition" by A.A. Albert,
Mathematical Reviews #0006140
^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 delle algebre doppie dotate di modulo", Atti della Reale Accademia delle Scienze e Belle-Lettere di Napoli, Ser (3) v.2 No7.
MR0021123.
Walter Benz (1973) Vorlesungen uber Geometrie der Algebren, Springer
F. Reese Harvey. Spinors and calibrations. Academic Press, San Diego. 1990.
ISBN0-12-329650-1. Contains a description of normed algebras in indefinite signature, including the Lorentz numbers.
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,
ElsevierISBN0-444-51123-7.
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.
J. Rooney (2014). "Generalised Complex Numbers in Mechanics". In Marco Ceccarelli and Victor A. Glazunov (ed.). Advances on Theory and Practice of Robots and Manipulators: Proceedings of Romansy 2014 XX CISM-IFToMM Symposium on Theory and Practice of Robots and Manipulators. Mechanisms and Machine Science. Vol. 22. Springer. pp. 55–62.
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10.1007/978-3-319-07058-2_7.
ISBN978-3-319-07058-2.