Split-complex number

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In abstract algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components x and y, and is written z = x + yj, where j2 = 1. The conjugate of z is z = xyj. Since j2 = 1, the product of a number z with its conjugate is zz = x2y2, an isotropic quadratic form, N(z) = x2y2.

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The collection D of all split complex numbers z = x + yj for x, yR forms an algebra over the field of real numbers. Two split-complex numbers w and z have a product wz that satisfies N(wz) = N(w)N(z). This composition of N over the algebra product makes (D, +, ×, *) a composition algebra.

A similar algebra based on R2 and component-wise operations of addition and multiplication, (R2, +, ×, xy), where xy is the quadratic form on R2, also forms a quadratic space. The ring isomorphism

relates proportional quadratic forms, but the mapping is not an isometry since the multiplicative identity (1, 1) of R2 is at a distance 2 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 quantity j satisfies

Choosing 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.

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

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

the conjugate of z is defined as

The conjugate satisfies similar properties to usual complex conjugate. Namely,

These three properties imply that the split-complex conjugate is an automorphism of order 2.

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

It has the composition algebra property:

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

where z = x + jy and w = u + jv. Another expression for the 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 x ± jx 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 ± ja) for some real number a.

The diagonal basis

There are two nontrivial idempotent elements given by e = (1 − j)/2 and e = (1 + j)/2. Recall that idempotent means that ee = e and ee = e. 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 z = ae + be for real numbers a and b by (a, b), then split-complex multiplication is given by

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

The split-complex conjugate in the diagonal basis is given by

and the modulus by

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 RR plane with its "unit circle" given by {(a, b) ∈ RR : ab = 1}. The contracted unit hyperbola {cosh a + j sinh a : aR} 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 RR.

Geometry

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Unit hyperbola with ||z|| = 1,

conjugate hyperbola with ||z|| = -1, and

asymptotes ||z|| = 0. Drini-conjugatehyperbolas.svg
  Unit hyperbola with z‖ = 1,
  conjugate hyperbola with z‖ = −1, and
  asymptotes z‖ = 0.

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

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

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 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

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() 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

and

The exponential map

sending θ to rotation by exp() 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.

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 commmutative 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.

The split-complex numbers are isomorphic as an algebra to the direct product of R by itself. It is thus a reduced Artinian ring.

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

 for any numbers z and w.

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.

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 this representation, split-complex conjugation corresponds to multiplying on both sides by the matrix

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

This commutative diagram relates the action of the hyperbolic versor on D to squeeze mapping s applied to R Commutative diagram split-complex number 2.svg
This commutative diagram relates the action of the hyperbolic versor on D to squeeze mapping σ applied to R

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.

The action of hyperbolic versor then corresponds under this linear transformation to a squeeze mapping

There is a great number of different representations of split-complex numbers into the 2×2 real matrices. In fact, every matrice whose square is the identity matrix gives such a representation.

The above diagonal representation represents the Jordan canonical form of the matrix representation of the split-complex numbers. For a split-complex number z = (x, y) given by the following matrix representation:

its Jordan canonical form is given by:

where and

History

The use of split-complex numbers dates back to 1848 when James Cockle revealed his tessarines. [1] 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. [2] [3] [4] [5] [6] [7] In that model, the number z = x + yj represents an event in a spacio-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 rapidity a 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 zw + zw = 0. 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. [8] 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." [9] 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. [10]

In 1941 E.F. Allen used the split-complex geometric arithmetic to establish the nine-point hyperbola of a triangle inscribed in zz = 1. [11]

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. [12] 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:

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

Related Research Articles

Complex number Element of a number system in which –1 has a square root

In mathematics, a complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a symbol called the imaginary unit, and satisfying the equation i2 = -1. Because no "real" number satisfies this equation, i was called an imaginary number by René Descartes. For the complex number a + bi, a is called the real part and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols or C. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.

Imaginary unit Principal square root of −1

The imaginary unit or unit imaginary number is a solution to the quadratic equation x2 + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of i in a complex number is 2 + 3i.

Hyperbolic functions Mathematical functions for hyperbolas similar to trigonometric functions for circles

In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola. Also, just as the derivatives of sin(t) and cos(t) are cos(t) and –sin(t), the derivatives of sinh(t) and cosh(t) are cosh(t) and +sinh(t).

In mathematics, de Moivre's formula states that for any real number x and integer n it holds that

Hyperboloid Unbounded quadric surface

In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation.

In mathematics, the indefinite orthogonal group, O(p, q) is the Lie group of all linear transformations of an n-dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature (p, q), where n = p + q. The dimension of the group is n(n − 1)/2.

In mathematics, a linear fractional transformation is, roughly speaking, a transformation of the form

Hyperbolic orthogonality

In geometry, the relation of hyperbolic orthogonality between two lines separated by the asymptotes of a hyperbola is a concept used in special relativity to define simultaneous events. Two events will be simultaneous when they are on a line hyperbolically orthogonal to a particular time line. This dependence on a certain time line is determined by velocity, and is the basis for the relativity of simultaneity.

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Hyperbolic angle

In mathematics, a hyperbolic angle is a geometric figure that defines a hyperbolic sector. The relationship of a hyperbolic angle to a hyperbola parallels the relationship of an "ordinary" angle to a circle.

In abstract algebra, the biquaternions are the numbers w + xi + yj + zk, where w, x, y, and z are complex numbers, or variants thereof, and the elements of {1, i, j, k} multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof:

In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They form an associative algebra of dimension four over the real numbers.

In mathematics, a function of a motor variable is a function with arguments and values in the split-complex number plane, much as functions of a complex variable involve ordinary complex numbers. William Kingdon Clifford coined the term motor for a kinematic operator in his "Preliminary Sketch of Biquaternions" (1873). He used split-complex numbers for scalars in his split-biquaternions. Motor variable is used here in place of split-complex variable for euphony and tradition.

In mathematics, a versor is a quaternion of norm one.

In relativity, rapidity is commonly used as a measure for relativistic velocity. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being associated with distance and time coordinates.

Hyperboloid model

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In abstract algebra, a bicomplex number is a pair (w, z) of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate , and the product of two bicomplex numbers as

Unit hyperbola geometric figure

In geometry, the unit hyperbola is the set of points (x,y) in the Cartesian plane that satisfy the implicit equation In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an alternative radial length

Exponential map (Lie theory)

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Homersham Cox (1857–1918) was an English mathematician.

References

  1. 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.
  2. Francesco Antonuccio (1994) Semi-complex analysis and mathematical physics
  3. 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. ISBN   978-3-7643-8613-9.
  4. Francesco Catoni; Dino Boccaletti; Roberto Cannata; Vincenzo Catoni; Paolo Zampetti (2011). "Chapter 2: Hyperbolic Numbers". Geometry of Minkowski Space-Time. Springer Science & Business Media. ISBN   978-3-642-17977-8.
  5. Fjelstadt, P. (1986) "Extending Special Relativity with Perplex Numbers", American Journal of Physics 54 :416.
  6. Louis Kauffman (1985) "Transformations in Special Relativity", International Journal of Theoretical Physics 24:223–36.
  7. Sobczyk, G.(1995) Hyperbolic Number Plane, also published in College Mathematics Journal 26:268–80.
  8. Robert B. Brown (1967)On Generalized Cayley-Dickson Algebras, Pacific Journal of Mathematics 20(3):415–22, link from Project Euclid.
  9. N.H. McCoy (1942) Review of "Quadratic forms permitting composition" by A.A. Albert, Mathematical Reviews #0006140
  10. 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
  11. Allen, E.F. (1941) "On a Triangle Inscribed in a Rectangular Hyperbola", American Mathematical Monthly 48(10): 675–681
  12. M. Warmus (1956) "Calculus of Approximations", Bulletin de l'Académie polonaise des sciences, Vol. 4, No. 5, pp. 253–257, MR 0081372
  13. Rosenfeld, B. (1997) Geometry of Lie Groups, page 30, Kluwer Academic Publishers ISBN   0-7923-4390-5