This Article 
 Bibliographic References 
 Add to: 
Redundant Radix Representations of Rings
November 1999 (vol. 48 no. 11)
pp. 1153-1165

Abstract—This paper presents an analysis of radix representations of elements from general rings; in particular, we study the questions of redundancy and completeness in such representations. Mappings into radix representations, as well as conversions between such, are discussed, in particular where the target system is redundant. Results are shown valid for normed rings containing only a finite number of elements with a bounded distance from zero, essentially assuring that the ring is “discrete.” With only brief references to the more usual representations of integers, the emphasis is on various complex number systems, including the “classical” complex number systems for the Gaussian integers, as well as the Eisenstein integers, concluding with a summary on properties of some low-radix representations of such systems.

[1] J.-P. Allouche, E. Cateland, W. Gilbert, H.-O. Peitgen, J. Shallit, and G. Skordev, “Automatic Maps in Exotic Number Systems,” Theory of Computing Systems, vol. 30, pp. 285-331, 1997.
[2] T. Aoki, H. Amada, and T. Higuchi, “Real/Complex Reconfigurable Arithmetic Using Redundant Complex Number Systems,” Proc. 13th IEEE Symp. Computer Arithmetic, pp. 200-207, 1997.
[3] A. Avizienis, “Signed-Digit Number Representations for Fast Parallel Arithmetic,” IRE Trans. Electronic Computers, vol. 10, pp. 389-400, Sept. 1961.
[4] J. Duprat, Y. Herreros, and S. Kla, “New Representation of Complex Numbers and Vectors,” Proc. 10th IEEE Symp. Computer Arithmetic, pp. 2-9, 1991.
[5] W. Gilbert, “Radix Representations of Quadratic Fields,” J. Math. Analysis and Applications, vol. 83, pp. 264-274, 1981.
[6] I. Kátai, “Number Systems in Imaginary Quadratic Fields,” Ann. Univ. Budapest, Sect. Comp., vol. 14, pp. 91-103, 1994.
[7] I. Kátai and B. Kovács, “Canonical Number Systems in Imaginary Quadratic Fields,” Acta Math. Acad. Sci. Hungaricae, vol. 37, pp. 1-3, 1981.
[8] I. Kátai and J. Szabo, “Canonical Number Systems for Complex Integers,” Acta Sci. Math. (Szeged), vol. 37, pp. 255-260, 1975.
[9] D.E. Knuth, “An Imaginary Number System,” Comm. ACM, vol. 3, no. 4, pp. 245-247, Apr. 1960.
[10] P. Kornerup, “Digit-Set Conversions: Generalizations and Applications,” IEEE Trans. Computers, vol. 43, pp. 622-629, 1994.
[11] D.W. Matula, “Radix Arithmetic: Digital Algorithms for Computer Architecture,” Applied Computation Theory: Analysis, Design, Modeling, R.T. Yeh, ed., chapter 9, pp. 374-448, Englewood Cliffs, N.J.: Prentice Hall, 1976.
[12] D. Matula, “Basic Digit Sets for Radix Representation,” J. ACM, vol. 29, pp. 1,131-1,143, 1982.
[13] A. Munk Nielsen and J.-M. Muller, “Borrow-Save Adders for Real and Complex Number Systems,” Proc. Second Conf. Real Numbers and Computers, Marseille, France, Apr. 1996.
[14] A. Munk Nielsen and P. Kornerup, “Generalized Base and Digit-Set Conversion, Extended Abstract,” Proc. SCAN 97, Lyon, pp. XII-8-11, Sept. 1997.
[15] A. Munk Nielsen and P. Kornerup, “On Radix Representation of Rings,” Proc. 13th IEEE Symp. Computer Arithmetic, pp. 34-43, 1997.
[16] W. Penney, “A `Binary' System for Complex Numbers,” J. ACM, vol. 12, no. 2, pp. 247-248, Apr. 1965.
[17] B. Parhami,"On the Implementation of Arithmetic Support Functions for Generalized Signed-Digit Number Systems," IEEE Trans. Computers, vol. 42 no. 3, pp. 379-384, Mar. 1993.
[18] L.N. Steward and D.O. Toll, Algebraic Number Theory. London: Chapman and Hall, 1979.
[19] B. Wei, H. Du, and H. Chen, “A Complex-Number Multiplier Using Radix-4 Digits,” Proc. 12th Symp. Computer Arithmetic, pp. 84-90, 1995.

Index Terms:
Radix representation of rings, integer and computer radix number systems, redundancy, number system conversion, computer arithmetic.
Asger Munk Nielsen, Peter Kornerup, "Redundant Radix Representations of Rings," IEEE Transactions on Computers, vol. 48, no. 11, pp. 1153-1165, Nov. 1999, doi:10.1109/12.811100
Usage of this product signifies your acceptance of the Terms of Use.