View source for Split-complex number

{{Short description|Reals with an extra square root of +1 adjoined}}
{{redirect|Double number|the computer number format|double-precision floating-point format}}

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

The collection {{mvar|D}} of all split-complex numbers <math>z=x+yj</math> for {{tmath|x,y \in \R}} forms an [[Algebra over a field|algebra over the field of real numbers]].  Two split-complex numbers {{mvar|w}} and {{mvar|z}} have a product {{mvar|wz}} that satisfies <math>N(wz)=N(w)N(z).</math>  This composition of {{mvar|N}} over the algebra product makes {{math|(''D'', +, ×, *)}} a [[Composition algebra|composition algebra]].

A similar algebra based on {{tmath|\R^2}} and component-wise operations of addition and multiplication, {{tmath|(\R^2, +, \times, xy),}} where {{mvar|xy}} is the [[Quadratic form|quadratic form]] on {{tmath|\R^2,}} 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 form#Quadratic space|quadratic spaces]].

Split-complex numbers have many other names; see ''{{section link||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

<math display=block>z = x + jy</math>

where {{mvar|x}} and {{mvar|y}} are [[Real number|real number]]s 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 {{ISBN|978-1-84816-858-9}}</ref> {{mvar|j}} satisfies

<math display=block>j^2 = +1</math>

In the field of [[Complex number|complex number]]s  the [[Imaginary unit|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 {{mvar|j}} is ''not'' a real number but an independent quantity.

The collection of all such {{mvar|z}} is called the '''split-complex plane'''. [[Addition]] and [[Multiplication|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 [[Distributive property|distributes]] over addition.

===Conjugate, modulus, and bilinear form===
Just as for complex numbers, one can define the notion of a '''split-complex conjugate'''. If

<math display=block> z = x + jy ~,</math>

then the conjugate of {{mvar|z}} is defined as

<math display=block> z^* = x - jy ~.</math>

The conjugate is an [[Involution (mathematics)|involution]] which satisfies similar properties to the [[Complex conjugate|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|isotropic quadratic form]]

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

It has the [[Composition algebra|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 [[Metric signature|signature]] {{math|(1, −1)}}, so the modulus is ''not'' a [[Norm (mathematics)|norm]].

The associated [[Bilinear form|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|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|if and only if]] its modulus is nonzero {{nowrap|(<math>\lVert z \rVert \ne 0</math>),}} thus numbers of the form {{math|''x'' ± ''j x''}} have no inverse. The [[Multiplicative inverse|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 vector|null vector]]s. These are all of the form {{math|(''a'' ± ''j a'')}} for some real number {{mvar|a}}.

===The diagonal basis===
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 {{mvar|e}} and {{mvar|e}}<sup>∗</sup> as an alternate [[Basis (linear algebra)|basis]] for the split-complex plane. This basis is called the '''diagonal basis''' or '''null basis'''. The split-complex number {{mvar|z}} can be written in the null basis as

<math display=block> z = x + jy = (x - y)e + (x + y)e^* ~.</math>

If we denote the number <math>z=ae+be^*</math> for real numbers {{mvar|a}} and {{mvar|b}} by {{math|(''a'', ''b'')}}, then zero is {{math|(0, 0)}}, one is {{math|(1, 1)}}, split-complex addition is given by
<math display=block>\left( a_1, b_1 \right) + \left( a_2, b_2 \right) = \left( a_1 + b_1, a_2 + b_2 \right) ~,</math>
and 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>

===Isomorphism===
[[File:Commutative diagram split-complex number 2.svg|right|200px|thumb|This [[commutative diagram]] relates the action of the hyperbolic versor on {{mvar|D}} to squeeze mapping {{mvar|σ}} applied to {{tmath|\R^2}}]]

On the basis {e, e*} it becomes clear that the split-complex numbers are ring-isomorphic to the direct sum {{tmath|\R \oplus \R}} with addition and multiplication defined pairwise.

The diagonal basis for the split-complex number plane can be invoked by using an ordered pair {{math|(''x'', ''y'')}} 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 [[One-parameter group|parametrized]] hyperbolas are brought into correspondence with {{mvar|S}}.

The [[Group action (mathematics)|action]] of [[Versor#Hyperbolic versor|hyperbolic versor]] <math>e^{bj} \!</math> then corresponds under this linear transformation to a [[Squeeze mapping|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|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 (metric space)|dilation]] by {{sqrt|2}}. The dilation in particular has sometimes caused confusion in connection with areas of a [[Hyperbolic sector|hyperbolic sector]]. Indeed, [[Hyperbolic angle|hyperbolic angle]] corresponds to [[Area|area]] of a sector in the {{tmath|\R \oplus \R}} plane with its "unit circle" given by <math>\{(a,b) \in \R \oplus \R : ab=1\}.</math> The contracted [[Unit hyperbola|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 {{tmath|\R \oplus \R}}.

==Geometry==
<!-- This section is linked from [[Lorentz transformation]] -->
[[Image:Drini-conjugatehyperbolas.svg|thumb|
{{legend-line|solid blue|Unit hyperbola: {{math|1=‖''z''‖ = 1}}}}
{{legend-line|solid green|Conjugate hyperbola: {{math|1=‖''z''‖ = −1}}}}
{{legend-line|solid red|Asymptotes: {{math|1=‖''z''‖ = 0}}}}]]

A two-dimensional real [[Vector space|vector space]] with the Minkowski inner product is called {{math|(1 + 1)}}-dimensional [[Minkowski space]], often denoted {{tmath|\R^{1,1}.}} Just as much of the [[Geometry|geometry]] of the Euclidean plane {{tmath|\R^2}} can be described with complex numbers, the geometry of the Minkowski plane {{tmath|\R^{1,1} }} 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|hyperbola]] for every nonzero {{mvar|a}} in {{tmath|\R.}} The hyperbola consists of a right and left branch passing through {{math|(''a'', 0)}} and {{math|(−''a'', 0)}}. The case {{math|1=''a'' = 1}} is called the [[Unit hyperbola|unit hyperbola]]. The [[Conjugate hyperbola|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 {{math|(0, ''a'')}} and {{math|(0, −''a'')}}. The hyperbola and conjugate hyperbola are separated by two diagonal [[Asymptote|asymptote]]s 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|perpendicular]] in {{tmath|\R^2}} and have slopes ±1.

Split-complex numbers {{mvar|z}} and {{mvar|w}} are said to be hyperbolic-orthogonal if {{math|1=⟨''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 [[Minkowski space#Causal structure|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|power series]] expansion using the fact that cosh has only even powers while that for sinh has odd powers.<ref>James Cockle (1848) [https://www.biodiversitylibrary.org/item/20157#page/452/mode/1up On a New Imaginary in Algebra], ''Philosophical Magazine'' 33:438</ref> For all real values of the [[Hyperbolic angle|hyperbolic angle]] {{mvar|θ}} the split-complex number {{math|1=''λ'' = exp(''jθ'')}} has norm 1 and lies on the right branch of the unit hyperbola. Numbers such as {{mvar|λ}} have been called [[Versor#Hyperbolic versor|hyperbolic versors]].

Since {{mvar|λ}} has modulus 1, multiplying any split-complex number {{mvar|z}} by {{mvar|λ}} preserves the modulus of {{mvar|z}} and represents a ''hyperbolic rotation'' (also called a Lorentz boost or a [[Squeeze mapping|squeeze mapping]]). Multiplying by {{mvar|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 (mathematics)|group]] called the generalized orthogonal group {{math|O(1, 1)}}. This group consists of the hyperbolic rotations, which form a [[Subgroup|subgroup]] denoted {{math|SO{{sup|+}}(1, 1)}}, combined with four [[Discrete mathematics|discrete]] [[Reflection (mathematics)|reflection]]s 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 {{mvar|θ}} to rotation by {{math|exp(''jθ'')}} is a [[Group isomorphism|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 {{mvar|z}} does not lie on one of the diagonals, then {{mvar|z}} has a [[Polar decomposition#Alternative planar decompositions|polar decomposition]].

==Algebraic properties==
In [[Abstract algebra|abstract algebra]] terms, the split-complex numbers can be described as the [[Quotient ring|quotient]] of the [[Polynomial ring|polynomial ring]] {{tmath|\R[x]}} by the [[Ideal (ring theory)|ideal]] generated by the [[Polynomial|polynomial]] <math>x^2-1,</math>

<math display=block>\R[x]/(x^2-1 ).</math>

The image of {{mvar|x}} in the quotient is the "imaginary" unit {{mvar|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 (mathematics)|field]] since the null elements are not invertible. All of the nonzero null elements are [[Zero divisor|zero divisor]]s.

Since addition and multiplication are continuous operations with respect to the usual topology of the plane, the split-complex numbers form a [[Topological ring|topological ring]].

The algebra of split-complex numbers forms a [[Composition algebra|composition algebra]] since

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

for any numbers {{mvar|z}} and {{mvar|w}}.

From the definition it is apparent that the ring of split-complex numbers is isomorphic to the [[Group ring|group ring]] {{tmath|\R[C_2]}} of the [[Cyclic group|cyclic group]] {{math|C{{sub|2}}}} over the real numbers {{tmath|\R.}}

Elements of the [[Identity component|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 representations==
Using the concepts of [[Matrix (mathematics)|matrices]]  and [[Matrix multiplication|matrix multiplication]], split-complex numbers can be represented in linear algebra. The real unit {{math|1}} and hyperbolic unit {{mvar|j}} can be represented by any pair of matrices {{mvar|I}} and {{mvar|J}} satisfying {{math|1=''I''{{isup|2}} = ''J''{{isup|2}} = ''I''}} and {{math|1=''IJ'' = ''JI'' = ''J''}} with {{math|''J'' ≠ ±''I''.}} Then a split-complex number {{math|''a'' + ''bj''}} can be represented by the matrix {{math|''aI'' + ''bJ'',}} and all of the ordinary rules of split-complex arithmetic can be derived from the rules of matrix arithmetic.

The most common choice is to represent {{math|1}} and {{mvar|j}} by the {{math|2&hairsp;×&hairsp;2}} [[Identity matrix|identity matrix]] {{mvar|I}} and the matrix {{mvar|J}},

<math display=block>
I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad
J = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.</math>

Then an arbitrary split-complex number {{math|''a'' + ''bj''}} can be represented by:

<math display=block>aI + bJ = \begin{pmatrix} a & b \\ b & a \end{pmatrix}.</math>

More generally, any real-valued {{math|2&hairsp;×&hairsp;2}} matrix with a [[Trace (linear algebra)|trace]] of zero and a [[Determinant|determinant]] of negative one squares to {{mvar|I}}, so could be chosen for {{mvar|J}}. Regardless, {{math|1=det(''aI'' + ''bJ'') = ''a''{{sup|2}} − ''b''{{sup|2}}}}, the squared modulus of the split-complex number.

Larger matrices could also be used; for example, {{math|1}} could be represented by the {{math|4&hairsp;×&hairsp;4}} identity matrix and {{mvar|j}} could be represented by the {{math|''γ''{{sup|0}}}} Dirac matrix.

== History ==
The use of split-complex numbers dates back to 1848 when James Cockle revealed his tessarines.<ref name=JC>James Cockle (1849) [https://www.biodiversitylibrary.org/item/20121#page/51/mode/1up On a New Imaginary in Algebra] 34:37–47, ''London-Edinburgh-Dublin Philosophical Magazine'' (3) '''33''':435–9, link from [[Social:Biodiversity Heritage Library|Biodiversity Heritage Library]].</ref> [[Biography:William Kingdon Clifford|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-biquaternion|split-biquaternion]]s. He called its elements "motors", a term in parallel with the "rotor" action of an ordinary complex number taken from the [[Circle group|circle group]]. Extending the analogy, functions of a [[Motor variable|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 [[Physics:Spacetime|spacetime]] plane.<ref>Francesco Antonuccio (1994) [https://arxiv.org/abs/gr-qc/9311032 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. {{isbn|978-3-7643-8613-9}}.</ref><ref>{{cite book |author1=Francesco Catoni|author2=Dino Boccaletti |author3=Roberto Cannata |author4=Vincenzo Catoni |author5=Paolo Zampetti|title=Geometry of Minkowski Space-Time |year=2011 |publisher=Springer Science & Business Media |isbn=978-3-642-17977-8 |chapter=Chapter 2: Hyperbolic Numbers}}</ref><ref>{{cite journal |mode=cs2 |last=Fjelstad |first=Paul |year=1986 |title=Extending special relativity via the perplex numbers |journal=American Journal of Physics |volume=54 |issue=5 |pages=416–422 |doi=10.1119/1.14605 |bibcode=1986AmJPh..54..416F }}</ref><ref>[[Biography:Louis Kauffman|Louis Kauffman]] (1985) "Transformations in Special Relativity", [[Physics:International Journal of Theoretical Physics|International Journal of Theoretical Physics]] 24:223–36.</ref><ref>Sobczyk, G.(1995) [http://garretstar.com/secciones/publications/docs/HYP2.PDF Hyperbolic Number Plane], also published in ''College Mathematics Journal'' 26:268–80.</ref> In that model, the number {{math|1=''z'' = ''x'' + ''y'' ''j''}} represents an event in a spatio-temporal plane, where ''x'' is measured in seconds and {{mvar|y}} in [[Astronomy:Light-second|light-second]]s. The future corresponds to the quadrant of events {{math| {''z'' : {{abs|''y''}} < ''x''}<nowiki/>}}, which has the split-complex polar decomposition <math>z = \rho e^{aj} \!</math>. The model says that {{mvar|z}} can be reached from the origin by entering a [[Astronomy:Frame of reference|frame of reference]] of [[Physics:Rapidity|rapidity]] {{mvar|a}} and waiting {{mvar|ρ}} nanoseconds. The split-complex equation
<math display=block>e^{aj} \ e^{bj} = e^{(a + b)j}</math>
expressing products on the unit hyperbola illustrates the additivity of rapidities for collinear velocities. Simultaneity of events depends on rapidity {{mvar|a}};
<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 {{mvar|z}} and {{mvar|w}} are hyperbolic-orthogonal when {{tmath|1= z^*w+zw^* = 0 }}. Canonical events {{math| exp(''aj'')}} and {{math|''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 {{math|''j'' exp(''aj'')}}.

In 1933 Max Zorn was using the [[Split-octonion|split-octonion]]s and noted the [[Composition algebra|composition algebra]] property. He realized that the [[Cayley–Dickson construction]], used to generate division algebras, could be modified (with a factor gamma, {{mvar|γ}}) 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)[http://projecteuclid.org/euclid.pjm/1102992693 On Generalized Cayley-Dickson Algebras], [[Pacific Journal of Mathematics]] 20(3):415–22, link from Project Euclid.</ref> The gamma factor, with {{math|'''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 2<sup>''e''</sup> 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 {{math|1=''F'' = '''R'''}} and {{math|1=''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.<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|nine-point hyperbola]] of a triangle inscribed in&nbsp;{{math|1=''zz''{{sup|∗}} = 1}}.<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) [http://www.cs.utep.edu/interval-comp/warmus.pdf "Calculus of Approximations"] {{webarchive |url=https://web.archive.org/web/20120309164421/http://www.cs.utep.edu/interval-comp/warmus.pdf |date=2012-03-09 }}, ''Bulletin de l'Académie polonaise des sciences'', Vol. 4, No. 5, pp.&nbsp;253–257, {{MR|id=0081372}}</ref> [[Biography:D. H. Lehmer|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)<ref>{{cite thesis |last=Cree |first=George C. |title=The Number Theory of a System of Hyperbolic Complex Numbers |type=MA thesis |publisher=McGill University |year=1949 |url=https://escholarship.mcgill.ca/concern/theses/1v53k125p}}</ref>
* ''bireal numbers'', U. Bencivenga (1946)
* ''real hyperbolic numbers'', N. Smith (1949)<ref>{{cite thesis |last=Smith |first=Norman E. |title=Introduction to Hyperbolic Number Theory |type=MA thesis |publisher=McGill University |year=1949 |url=https://escholarship.mcgill.ca/concern/theses/1544bs68g }}</ref>
* ''approximate numbers'', Warmus (1956), for use in [[Interval analysis|interval analysis]]
* ''double numbers'', [[Biography:Isaak Yaglom|I.M. Yaglom]] (1968), Kantor and Solodovnikov (1989), [[Michiel Hazewinkel|Hazewinkel]] (1990), Rooney (2014)
* ''hyperbolic numbers'', W. Miller & R. Boehning (1968),<ref>{{cite journal |last1=Miller |first1=William |last2=Boehning |first2=Rochelle |title=Gaussian, parabolic, and hyperbolic numbers |journal=The Mathematics Teacher |volume=61 |number=4 |year=1968 |pages=377–382 |doi=10.5951/MT.61.4.0377 |jstor=27957849 }}</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 {{isbn|0-7923-4390-5}}</ref>
* ''spacetime numbers'', N. Borota (2000)
* ''Study numbers'', P. Lounesto (2001)
* ''twocomplex numbers'', S. Olariu (2002)
* ''split binarions'', K. McCrimmon (2004)

==See also==
{{wikibooks|Associative Composition Algebra|Binarions|Split binarions}}
* [[Minkowski space]]
* [[Split-quaternion]]
* [[Hypercomplex number]]

==References==
{{reflist|22em}}

==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. {{MathSciNet|id=0021123}}.
* [[Biography:Walter Benz|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) [https://heyokatc.com/pdfs/MISC/Circular_and_Hyperbolic_Quaternions_Octonions_and_Sedenions_-_carmody-amac-1988.pdf "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.
* [[Biography:William Kingdon Clifford|William Kingdon Clifford]] (1882) ''Mathematical Works'', A. W. Tucker editor, page 392, "Further Notes on Biquaternions"
* V.Cruceanu, P. Fortuny & P.M. Gadea (1996) [http://www.projecteuclid.org/euclid.rmjm/1181072105 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) [http://people.rit.edu/harkin/research/articles/generalized_complex_numbers.pdf 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 {{isbn|0-387-95447-3}} {{mr|id=2014924}}
* 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, [[Company:Elsevier|Elsevier]] {{isbn|0-444-51123-7}}.
* Poodiack, Robert D. & Kevin J. LeClair (2009) "Fundamental theorems of algebra for the perplexes", [[Organization:The College Mathematics Journal|The College Mathematics Journal]] 40(5):322–35.
* [[Biography:Isaak Yaglom|Isaak Yaglom]] (1968) ''Complex Numbers in Geometry'', translated by E. Primrose from 1963 Russian original, [[Company:Academic Press|Academic Press]], pp.&nbsp;18–20.
* {{cite book|editor=Marco Ceccarelli and Victor A. Glazunov|title=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|year=2014 | publisher=Springer | isbn=978-3-319-07058-2|chapter=Generalised Complex Numbers in Mechanics|author=J. Rooney|series=Mechanisms and Machine Science | volume=22|pages=55–62|doi=10.1007/978-3-319-07058-2_7}}

{{Number systems}}

{{DEFAULTSORT:Split-Complex Number}}
[[Category:Composition algebras]]
[[Category:Linear algebra]]
[[Category:Hypercomplex numbers]]

{{Sourceattribution|Split-complex number}}