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{{Short description|
{{redirect|Double number|the computer number format|double-precision floating-point format}}
In [[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]] 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]].
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]] on {{tmath|\R^2,}} also forms a [[quadratic space]]. The [[ring isomorphism]]
<math display=
D &\to \mathbb{R}^2 \\
x + yj &\mapsto (x - y, x + y)
\end{align}</math>
is an [[quadratic space#isometry|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.
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==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]]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]]s the [[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]] 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).
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===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]]. 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]]
<math display=block>\lVert z \rVert^2 = z z^* = z^* z = x^2 - y^2 ~.</math>
It has the [[composition algebra]] property:
<math display=block>\lVert z w \rVert = \lVert z \rVert \lVert w \rVert ~.</math>
However, this quadratic form is not [[definite bilinear form|positive-definite]] but rather has [[metric signature|signature]] {{math|(1, −1)}}, so the modulus is ''not'' a [[norm (mathematics)|norm]].
The associated [[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]]; 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 {{nowrap|(<math>\lVert z \rVert \ne 0</math>),}} thus numbers of the form {{math|''x'' ± ''j x''}} have no inverse. The [[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]]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 element (ring theory)|idempotent element]]s given by <math>e=\tfrac{1}{2}(1-j)</math> and <math>e^* = \tfrac{1}{2}(1+j).</math>
<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 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=
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 isomorphism|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]]
<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]], 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]]. Indeed, [[hyperbolic angle]] corresponds to [[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]] <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}}.
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The set of points
<math display=block>\left\{ z : \lVert z \rVert^2 = a^2 \right\}</math>
is a [[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]]. The [[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]]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]] 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]] expansion using the fact that [[hyperbolic cosine|cosh]] has only even powers while that for [[hyperbolic sine|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]] {{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]].
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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]] 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]] 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]].
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==Algebraic properties==
In [[abstract algebra]] terms, the split-complex numbers can be described as the [[quotient ring|quotient]] of the [[polynomial ring]] {{tmath|\R[x]}} by the [[ideal (ring theory)|ideal]] generated by the [[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 (structure)|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]]s.
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The algebra of split-complex numbers forms a [[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]] {{tmath|\R[C_2]}} of the [[cyclic group]] {{math|C{{sub|2}}}} over the real numbers {{tmath|\R.}}
Elements of the [[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 [[idempotent element (ring theory)|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==
One can easily represent split-complex numbers by [[matrix (mathematics)|matrices]]. The split-complex number <math>z
Addition and multiplication of split-complex numbers are then given by matrix addition and multiplication. The squared modulus of {{mvar|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 (mathematics)|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 (linear algebra)|basis]] element with which to extend the real line to the split-complex plane. The matrices
<math display=block>m = \begin{pmatrix}a & c \\ b & -a \end{pmatrix}</math>
which square to the identity matrix satisfy <math>a^2 + bc = 1 .</math>
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).<ref>{{wikibooks-inline|Abstract Algebra/2x2 real matrices}}</ref>{{Better source needed|reason=Wikibooks not a reliable source ([[WP:USERGENERATED]]) |date=May 2023}}
The number <math>z = x + jy</math> can be represented by the matrix <math>x\ I + y\ m .</math>
== History ==
The use of split-complex numbers dates back to 1848 when [[James Cockle (lawyer)|James Cockle]] revealed his [[tessarine]]s.<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 [[Biodiversity Heritage Library]].</ref> [[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]]s. 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.<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>[[Louis Kauffman]] (1985) "Transformations in Special Relativity", [[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 [[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 [[frame of reference]] of [[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=
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
In 1933 [[Max Zorn]] was using the [[split-octonion]]s 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, {{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
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]], [[Argentina|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>
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In 1941 E.F. Allen used the split-complex geometric arithmetic to establish the [[nine-point hyperbola]] of a triangle inscribed in {{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. 253–257, {{MR|id=0081372}}</ref> [[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.
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* (''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]]
* ''double numbers'', [[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)
* ''
* ''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==
Line 236 ⟶ 266:
* 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.
* [[William Kingdon Clifford]] (1882) ''Mathematical Works'', A. W. Tucker editor, page 392, "Further Notes on Biquaternions"
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* 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.
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