Abstract
Complex numbers can be considered as a two components quantity, as the plane vectors. Following Gauss complex numbers are also used for representing vectors in Euclidean plane. As a difference with vectors the multiplication of two complex numbers is yet a complex number. By means of this property complex numbers can be generalized and hyperbolic numbers that have properties corresponding to Lorentz group of two-dimensional Special Relativity are introduced.
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Notes
- 1.
The name versor has been firstly introduced by Hamilton for the unitary vectors of his quaternions [3]. This name derives from the property of the imaginary unity “\(\hbox{i}\)” since, as can be seen from Euler formulas, multiplying by “\(\hbox{i}\)” is equivalent to “rotate”, in a Cartesian representation, the complex number of \(\pi/2\). Since this property also holds for the hyperbolic numbers that we are going to introduce, we use this name that also states the difference with the unitary vectors of linear algebra.
- 2.
We use \(\vartriangleleft \ldots \vartriangleright\) to identify material that reports the original author’s words or is a literal translation.
- 3.
Here and in the following we use the symbol \(\tilde{.}\) for indicating the hyperbolic conjugate.
- 4.
In all the problems which refer to Special Relativity (in particular in Chap. 6) we change the symbols by indicating the variables with letters reflecting their physical meaning \(x, y \Rightarrow t, x\), i.e., \(t\) is a normalized time variable (light velocity \(c=1\)) and \(x\) a space variable.
- 5.
Within the limits of our knowledge, the first description of Special Relativity, directly by these numbers was introduced by I. M. Yaglom [6].
- 6.
Bombelli writes: \(\vartriangleleft\) … also if this introduction can appear as an extravagant idea and I considered it, for some time, as sophistical rather than true, I have found the demonstration that it works well in the operations. \(\vartriangleright\)
References
F. Catoni, D. Boccaletti, R. Cannata, V. Catoni, E. Nichelatti, P. Zampetti, The Mathematics of Minkowski Space–Time (Birkhäuser Verlag, Basel, 2008)
F. Catoni, R. Cannata, V. Catoni, P. Zampetti, N-dimensional geometries generated by hypercomplex numbers. Adv. Appl. Clifford Al. 15(1), 1 (2005)
C.C. Silva, R. de Andrade Martins, Polar and axial vectors versus quaternions. Am. J. Phys. 70(9), 958 (2002)
M. Lavrentiev, B. Chabat, Effets Hydrodynamiques et modèles mathématiques (Mir, Moscou, 1980)
I.M. Yaglom, Complex Numbers in Geometry (Academic Press, New York, 1968)
I.M. Yaglom, A Simple Non-Euclidean Geometry and its Physical Basis (Springer, New York, 1979)
G.L. Naber, The Geometry of Minkowski Spacetime. An Introduction to the Mathematics of the Special Theory of Relativity, Sect. 1.4 (Springer, New York, 1992)
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© 2011 Francesco Catoni
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Catoni, F., Boccaletti, D., Cannata, R., Catoni, V., Zampetti, P. (2011). Hyperbolic Numbers. In: Geometry of Minkowski Space-Time. SpringerBriefs in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17977-8_2
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DOI: https://doi.org/10.1007/978-3-642-17977-8_2
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