Abstract
Here we consider the most important metrics on the classical number systems: the semiring \(\mathbb{N}\) of natural numbers, the ring \(\mathbb{Z}\) of integers, and the fields \(\mathbb{Q}\), \(\mathbb{R}\), \(\mathbb{C}\) of rational, real, complex numbers, respectively. We consider also the algebra \(\mathcal{Q}\) of quaternions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Barbaresco F. Information Geometry of Covariance Matrix: Cartan-Siegel Homogenous Bounded Domains, Mostow-Berger Fibration and Fréchet Median, in Matrix Information Geometry, Bhatia R. and Nielsen F. (eds.) Springer, 2012.
Copson E.T. Metric Spaces, Cambridge Univ. Press, 1968.
Ernvall S. On the Modular Distance, IEEE Trans. Inf. Theory, Vol. 31-4, pp. 521–522, 1985.
Giles J.R. Introduction to the Analysis of Metric Spaces, Australian Math. Soc. Lecture Series, Cambridge Univ. Press, 1987.
Hamilton W.R. Elements of Quaternions, second edition 1899–1901 enlarged by C.J.Joly, reprinted by Chelsea Publ., New York, 1969.
Higham N.J. Matrix Nearness Problems and Applications, in Applications of Matrix Theory, Gover M.J.C. and Barnett S. (eds.), pp. 1–27. Oxford University Press, 1989.
Huber K. Codes over Gaussian Integers, IEEE Trans. Inf. Theory, Vol. 40-1, pp. 207–216, 1994.
Huber K. Codes over Eisenstein-Jacobi Integers, Contemporary Math., Vol. 168, pp. 165–179, 1994.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Deza, M.M., Deza, E. (2016). Distances on Numbers, Polynomials, and Matrices. In: Encyclopedia of Distances. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-52844-0_12
Download citation
DOI: https://doi.org/10.1007/978-3-662-52844-0_12
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-52843-3
Online ISBN: 978-3-662-52844-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)