Split-complex number: Difference between revisions

Revision as of 14:02, 15 March 2023

Revision as of 02:02, 24 March 2023

Line 167:

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>Fjelstadt,{{cite P.journal |mode=cs2 |last=Fjelstad |first=Paul (|year=1986) "[http://scitation.aip.org/content/aapt/journal/ajp/54/5/10.1119/1.14605 |title=Extending Specialspecial Relativityrelativity withvia Perplexthe Numbers]",perplex [[numbers |journal=American Journal of Physics]] |volume=54 :416|issue=5 |page=416–422 |doi=10.1119/1.14605 }}</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 nanoseconds and {{mvar|y}} in [[David Mermin#Word and phrase coinages|Mermin's feet]]. 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>