Split-complex product
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A portion of the split-complex number plane showing subsets with modulus zero (red), one (blue), and minus one (green).

In abstract algebra, the **split-complex numbers** (or **hyperbolic numbers**, also **perplex numbers**, and **double numbers**) are a two-dimensional commutative algebra over the real numbers different from the complex numbers. Every split-complex number has the form

*x* + *y* *j*,

where *x* and *y* are real numbers. The number *j* is similar to the imaginary unit *i*, except that

*j* ^{2} = +1.

As an algebra over the reals, the split-complex numbers are the same as the direct sum of algebras **R** ⊕ **R** (under the isomorphism sending *x* + *y* *j* to (*x* + *y*, *x* − *y*)). The name *split* comes from this characterization: as a real algebra, the split-complex numbers *split* into the direct sum **R** ⊕ **R**. It arises, for example, as the real subalgebra generated by an involutory matrix.

Geometrically, split-complex numbers are related to the **modulus** (*x*^{2} − *y*^{2}) in the same way that complex numbers are related to the square of the Euclidean norm (*x*^{2} + *y*^{2}). Unlike the complex numbers, the split-complex numbers contain nontrivial idempotents (other than 0 and 1), as well as zero divisors, and therefore they do not form a field.

In interval analysis, a split complex number *x* + *y* *j* represents an interval with midpoint *x* and radius *y*. Another application involves using split-complex numbers, dual numbers, and ordinary complex numbers, to interpret a 2 × 2 real matrix as a complex number.

Split-complex numbers have many other names; see the synonyms section below. See the article Motor variable for functions of a split-complex number.

## Definition[edit]

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; that is, it is not equal to ±1.

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

- (
*x* + *j* *y*) + (*u* + *j* *v*) = (*x* + *u*) + *j* (*y* + *v*)
- (
*x* + *j* *y*)(*u* + *j* *v*) = (*xu* + *yv*) + *j* (*xv* + *yu*).

This multiplication is commutative, associative and distributes over addition.

### Conjugate, modulus, and bilinear form[edit]

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

*z* = *x* + *j* *y*

the conjugate of *z* is defined as

*z*^{∗} = *x* − *j* *y*.

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

- (
*z* + *w*)^{∗} = *z*^{∗} + *w*^{∗}
- (
*zw*)^{∗} = *z*^{∗}*w*^{∗}
- (
*z*^{∗})^{∗} = *z*.

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

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

It has an important property that it is preserved by split-complex multiplication:

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

- 〈
*z*, *w*〉 = Re(*zw*^{∗}) = Re(*z*^{∗}*w*) = *xu* − *yv*,

where *z* = *x* + *j* *y* and *w* = *u* + *j* *v*. 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* ± *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 elements**. These are all of the form (*a* ± *j* *a*) for some real number *a*.

### The diagonal basis[edit]

There are two nontrivial idempotents given by *e* = (1 − *j*)/2 and *e*^{∗} = (1 + *j*)/2. Recall that idempotent means that *ee* = *e* and *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

*z* = *x* + *j* *y* = (*x* − *y*)*e* + (*x* + *y*)*e*^{∗}.

If we denote the number *z* = *ae* + *be*^{∗} for real numbers *a* and *b* by (*a*, *b*), then split-complex multiplication is given by

- (
*a*_{1}, *b*_{1})(*a*_{2}, *b*_{2}) = (*a*_{1}*a*_{2}, *b*_{1}*b*_{2}).

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

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

- (
*a*, *b*)^{∗} = (*b*, *a*)

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 hyperbolic sectors. Indeed, hyperbolic angle corresponds to area of sectors in the plane with its "unit circle" given by The contracted "unit circle" 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[edit]

Unit hyperbola with ||

*z*||=1 (blue),

conjugate hyperbola with ||

*z*||=−1 (green),

and asymptotes ||

*z*||=0 (red)

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

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 *x*^{2} − 1,

**R**[*x*]/(*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 commutative ring with characteristic 0. Moreover if we define scalar multiplication in the obvious manner, the split-complex numbers actually form a commutative and associative algebra over the reals of dimension two. The algebra is *not* a division algebra or field since the null elements are not invertible. In fact, 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

- for any numbers
*z* and *w*.

The class of composition algebras extends the normed algebras class which also has this composition property.

From the definition it is apparent that the ring of split-complex numbers is isomorphic to the group ring **R**[C_{2}] of the cyclic group C_{2} over the real numbers **R**.

The split-complex numbers are a particular case of a Clifford algebra. Namely, they form a Clifford algebra over a one-dimensional vector space with a *positive-definite* quadratic form. Contrast this with the complex numbers which form a Clifford algebra over a one-dimensional vector space with a *negative-definite* quadratic form. (NB: some authors switch the signs in the definition of a Clifford algebra which will interchange the meaning of positive-definite and negative-definite). In mathematics, the **split-complex numbers** are members of the Clifford algebra *C*ℓ_{1,0}(**R**) = *C*ℓ^{0}_{1,1}(**R**) (the superscript 0 indicating the even subalgebra). This is an extension of the real numbers defined analogously to the complex numbers **C** = *C*ℓ_{0,1}(**R**) = *C*ℓ^{0}_{2,0}(**R**).

## Matrix representations[edit]

One can easily represent split-complex numbers by matrices. The split-complex number

*z* = *x* + *j* *y*

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 σ applied to R

^{2}
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

Note that in the context of 2 × 2 real matrices there are in fact a great number of different representations of split-complex numbers. 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[edit]

The use of split-complex numbers dates back to 1848 when James Cockle revealed his Tessarines. 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 early twentieth century, the split-complex multiplication has commonly been seen as a Lorentz boost of a spacetime plane. In that model, the number *z* = *x* + *y* *j* 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 *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 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 *zz*^{∗} = 1.

In 1956 Mieczyslaw Warmus published "Calculus of Approximations" in *Bulletin de l’Academie Polanaise 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[edit]

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)
*bireal numbers*, U. Bencivenga (1946)
*approximate numbers*, Warmus (1956), for use in interval analysis
*countercomplex* or *hyperbolic* numbers from Musean hypernumbers
*double numbers*, I.M. Yaglom (1968), Kantor and Solodovnikov (1989), Hazewinkel (1990), Rooney (2014)
*anormal-complex numbers*, W. Benz (1973)
*perplex numbers*, P. Fjelstad (1986) and Poodiack & LeClair (2009)
*Lorentz numbers*, F.R. Harvey (1990)
*hyperbolic numbers*, G. Sobczyk (1995)
*semi-complex numbers*, F. Antonuccio (1994)
*split-complex numbers*, B. Rosenfeld (1997)
*spacetime numbers*, N. Borota (2000)
*Study numbers*, P. Lounesto (2001)
*twocomplex numbers*, S. Olariu (2002)

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[edit]

Higher-order derivatives of split-complex numbers, obtained through a modified Cayley–Dickson construction:

In Lie theory, a more abstract generalization occurs:

Enveloping algebras and number programs:

## References and external links[edit]

- Francesco Antonuccio (1994) Semi-complex analysis and mathematical physics
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*Atti della real academie della scienze e belle-lettre di Napoli*, Ser (3) v.2 No7. MR 0021123.
- Benz, W. (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.
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- 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.
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*Geometry of Minkowski Space-Time*. Springer Science & Business Media. ISBN 978-3-642-17977-8.
- 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,
*Mathematical Works* (1882) edited by A.W.Tucker,pp. 392,"Further Notes on Biquaternions"
- De Boer, R. (1987) "An also known as list for perplex numbers",
*American Journal of Physics* 55(4):296.
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*American Journal of Physics* **54**:416.
- Anthony A. Harkin & Joseph B. Harkin (2004) Geometry of Generalized Complex Numbers, Mathematics Magazine 77(2):118–29.
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*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.
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*Complex Numbers in N Dimensions*, Chapter 1: Hyperbolic Complex Numbers in Two Dimensions, pages 1–16, North-Holland Mathematics Studies #190, Elsevier ISBN 0-444-51123-7.
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*Contribucion al Estudio de las Ciencias Fisicas y Matematicas*, Universidad Nacional de la Plata, Republica Argentina.
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*Bulletin de l'Academie Polonaise de Sciences*, Vol. 4, No. 5, pp. 253–257, MR 0081372
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