Split-complex number

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

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

A similar algebra based on ${\displaystyle \mathbb {R} ^{2}}$ and component-wise operations of addition and multiplication, ${\displaystyle (\mathbb {R} ^{2},+,\times ,xy),}$ where xy is the quadratic form on ${\displaystyle \mathbb {R} ^{2},}$ also forms a

relates proportional quadratic forms, but the mapping is not an isometry since the multiplicative identity (1, 1) of ${\displaystyle \mathbb {R} ^{2}}$ is at a distance ${\displaystyle {\sqrt {2}}}$ 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 hyperbolic unit^[1] j satisfies

In the field of complex numbers the imaginary unit i satisfies ${\displaystyle i^{2}=-1.}$ 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

This multiplication is

over addition.

Conjugate, modulus, and bilinear form

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

$${\displaystyle z=x+jy~,}$$

then the conjugate of z is defined as

$${\displaystyle z^{*}=x-jy~.}$$

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

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

It has the composition algebra property:

However, this quadratic form is not

.

The associated bilinear form is given by

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

Since it is not positive-definite, this bilinear form is not an

; 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 (${\displaystyle \lVert z\rVert \neq 0}$), 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

given by ${\displaystyle e={\tfrac {1}{2}}(1-j)}$ and ${\displaystyle e^{*}={\tfrac {1}{2}}(1+j).}$ Recall that idempotent means that ${\displaystyle ee=e}$ and ${\displaystyle e^{*}e^{*}=e^{*}.}$ 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 ${\displaystyle z=ae+be^{*}}$ 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

to the direct sum ${\displaystyle \mathbb {R} \oplus \mathbb {R} }$ 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 ${\displaystyle z=x+jy}$ and making the mapping

Now the quadratic form is ${\displaystyle uv=(x+y)(x-y)=x^{2}-y^{2}~.}$ Furthermore,

so the two parametrized hyperbolas are brought into correspondence with S.

The

${\displaystyle e^{bj}\!}$ then corresponds under this linear transformation to a squeeze mapping

Though lying in the same isomorphism class in the

of a sector in the ${\displaystyle \mathbb {R} \oplus \mathbb {R} }$ plane with its "unit circle" given by ${\displaystyle \{(a,b)\in \mathbb {R} \oplus \mathbb {R} :ab=1\}.}$ The contracted unit hyperbola ${\displaystyle \{\cosh a+j\sinh a:a\in \mathbb {R} \}}$ 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 ${\displaystyle \mathbb {R} \oplus \mathbb {R} }$.

Geometry

A two-dimensional real vector space with the Minkowski inner product is called (1 + 1)-dimensional Minkowski space, often denoted ${\displaystyle \mathbb {R} ^{1,1}.}$ Just as much of the geometry of the Euclidean plane ${\displaystyle \mathbb {R} ^{2}}$ can be described with complex numbers, the geometry of the Minkowski plane ${\displaystyle \mathbb {R} ^{1,1}}$ can be described with split-complex numbers.

The set of points

is a hyperbola for every nonzero a in ${\displaystyle \mathbb {R} .}$ 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

in ${\displaystyle \mathbb {R} ^{2}}$ and have slopes ±1.

Split-complex numbers z and w are said to be

concept in spacetime.

The analogue of Euler's formula for the split-complex numbers is

This formula can be derived from a

.

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

). 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

given by

$${\displaystyle z\mapsto \pm z}$$

and ${\displaystyle z\mapsto \pm z^{*}.}$

The exponential map

$${\displaystyle \exp \colon (\mathbb {R} ,+)\to \mathrm {SO} ^{+}(1,1)}$$

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.

Algebraic properties

In abstract algebra terms, the split-complex numbers can be described as the quotient of the polynomial ring ${\displaystyle \mathbb {R} [x]}$ by the ideal generated by the polynomial ${\displaystyle x^{2}-1,}$

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

.

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 ${\displaystyle \mathbb {R} [C_{2}]}$ of the cyclic group C₂ over the real numbers ${\displaystyle \mathbb {R} .}$

Matrix representations

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

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

element with which to extend the real line to the split-complex plane. The matrices

$${\displaystyle m={\begin{pmatrix}a&c\\b&-a\end{pmatrix}}}$$

which square to the identity matrix satisfy ${\displaystyle a^{2}+bc=1.}$ 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

better source needed

]

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

History

The use of split-complex numbers dates back to 1848 when

complex variable

.

Since the late twentieth century, the split-complex multiplication has commonly been seen as a

Mermin's feet

. The future corresponds to the quadrant of events {z : |y| < x}, which has the split-complex polar decomposition ${\displaystyle z=\rho e^{aj}\!}$. 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

when ${\displaystyle z^{*}w+zw^{*}=0.}$ 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

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]
• bireal numbers, U. Bencivenga (1946)
• real hyperbolic numbers, N. Smith (1949)^[17]
• 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),^[18] 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)^[19]
• 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.
  MR0021123
  .
• 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.
  ISBN 0-12-329650-1
  . 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
  MR2014924
• 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,
  ISBN 0-444-51123-7
  .
• 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.
  ISBN 978-3-319-07058-2
  .