Bibliography of material on Decimal Arithmetic [Index]

Decimal Arithmetic: Choice of base

think 10

Implementable Decimal Arithmetic Algorithms for Micro/Minicomputers, M. Ahmad, Microprocessing and Microprogramming, Vol. 19 #2, pp119–128, February 1987.
Abstract: The need for efficient decimal arithmetic and its ever increasing applications in micro/minicomputers and microprocessor based equipment and appliances has been emphasised. Some algorithms suitable for implementation for decimal arithmetic operations of BCD packed decimal numbers have been suggested. These algorithms employ comparatively faster instructions available on most of the microprocessors and provide efficient and faster decimal arithmetic.
An APL interpreter and system for a small computer, M. Alfonseca, M. L. Tavera, and R. Casajuana, IBM Systems Journal, Vol. 16 #1, pp18–40, IBM, 1977.
Abstract: The design and implementation of an experimental APL system on the small, sensor-based System/7 is described. Emphasis is placed on the solution to the problem of fitting a full APL system into a small computer.
   The system has been extended through an I/O auxiliary processor to make it possible to use APL in the management and control of the System/7 sensor-based I/O operations.
Where did all my decimals go?, Chuck Allison, Computing Sciences in Colleges, Vol. 21 #3, pp47–59, Consortium for Computing Sciences in Colleges, February 2006.
Abstract: It is tremendously ironic that computers were invented with number crunching in mind, yet nowadays most CS graduates leave school with little or no experience with the intricacies of numeric computation. This paper surveys what every CS graduate should know about floating-point arithmetic, based on experience teaching a recently-created course on modern numerical software development.
The Choice of Base, W. S. Brown and P. L. Richman, Communications of the ACM, Vol. 12 #10, pp560–561, ACM Press, October 1969.
Abstract: A digital computer is considered, whose memory words are composed of N r-state devices plus two sign bits (two state devices). The choice of base b for the internal representation of floating-point numbers on such a computer is discussed. It is shown that in a certain sense b = r is best.
Fingers or Fists? (The Choice of Decimal or Binary representation), Werner Buchholz, Communications of the ACM, Vol. 2 #12, pp3–11, ACM Press, December 1959.
Abstract: The binary number system offers many advantages over a decimal representation for a high-perfornmnee, general-purpose computer. The greater simplicity of a binary arithmetic unit and the greater compactness of binary numbers both contribute directly to arithmetic speed. Less obvious and perhaps more important is the way binary addressing and instruction formats can increase the overall performance. Binary addresses are also essential to certain powerful operations which are not practical with decimal instruction formats.
    On the other hand, decimal numbers are essential for communicating between man and the computer. In applications requiring the processing of a large volume of inherently decimal input and output data, the time for decimal-binary conversion needed by a purely binary computer may be significant. A slower decimal adder may take less time than a fast binary adder doing an addition and two conversions.
    A careful review ef the significance of decimal and binary number systems led to the adoption in the IBM STRETCH computer of binary addressing and both binary and decimal data arithmetic, supplemented by efficient conversion instructions.

Note: Letters to the edtor in response to this paper were published in CACM, Vol. 3, #3, March 1960.
Preliminary discussion of the logical design of an electronic computing instrument, Arthur W. Burks, Herman H. Goldstine, and John von Neumann, 42pp, Inst. for Advanced Study, Princeton, N. J., June 28, 1946.
Abstract: Inasmuch as the completed device will be a general-purpose computing machine it should contain certain main organs relating to arithmetic, memory-storage, control and connection with the human operator. It is intended that the machine be fully automatic in character, i.e. independent of the human operator after the computation starts...
Note: Reprinted in von Neumann’s Collected Works, Vol. 5, A. H. Taub, Ed. (Pergamon, London, 1963), pp 34-79, and also in Computer Structures: Reading and Examples, Bell & Newell, McGraw-Hill Inc., 1971. Now widely available on the Internet.
    Contract W-36-034-ORD-H81. R&D Service, Ordnance Department, US Army and Institute for Advanced Study, Princeton
Decimal Floating-Point: Algorism for Computers, Michael F. Cowlishaw, Proceedings of the 16th IEEE Symposium on Computer Arithmetic, ISBN 0-7695-1894-X, pp104–111, IEEE, June 2003.
Abstract: Decimal arithmetic is the norm in human calculations, and human-centric applications must use a decimal floating-point arithmetic to achieve the same results.
    Initial benchmarks indicate that some applications spend 50% to 90% of their time in decimal processing, because software decimal arithmetic suffers a 100× to 1000× performance penalty over hardware. The need for decimal floating-point in hardware is urgent.
    Existing designs, however, either fail to conform to modern standards or are incompatible with the established rules of decimal arithmetic. This paper introduces a new approach to decimal floating-point which not only provides the strict results which are necessary for commercial applications but also meets the constraints and requirements of the IEEE 854 standard.
    A hardware implementation of this arithmetic is in development, and it is expected that this will significantly accelerate a wide variety of applications.

Note: Softcopy is available in PDF.
A method of designing a decimal arithmetic processor, M. A. Gladshtein, Automatic Control and Computer Sciences, Vol. 25 #6, pp51–56, 1991.
Abstract: The advantages and drawbacks of binary numeric coding in digital computers have been considered. This type of coding has been shown ineffective in processing large data arrays especially when represented in the floating-point form. Also, the low efficiency of conventionally employed decimal computational procedures using the so-called corrections has been noted. It has been proposed, in designing digital computers, to renounce the principle of binary computations in favor of decimal operations on the basis of stored addition and multiplication tables using binary-decimal numeric coding. A version of circuit design for a decimal processor, algorithms and microprograms for addition and multiplication operations have been described. Advantages inherent in the method proposed have been analyzed.
Note: Translated from Avtomatika i Vychislitel’naya Tekhnika UDC 681.3.48.
Desirable Floating-Point Arithmetic and Elementary Functions for Numerical Computation, T. E. Hull, ACM Signum Newsletter, Vol. 14 #1 (Proceedings of the SIGNUM Conference on the Programming Environment for Development of Numerical Software), pp96–99, ACM Press, 1978.
Abstract: The purpose of this talk is to summarize proposed specifications for floating-point arithmetic and elementary functions. The topics considered are: the base of the number system, precision control, number representation, arithmetic operations, other basic operations, elementary functions, and exception handling. The possibility of doing without fixed-point arithmetic is also mentioned. The specifications are intended to be entirely at the level of a programming language such as Fortran. The emphasis is on convenience and simplicity from the user’s point of view. Conforming to such specifications would have obvious beneficial implications for the portability of numerical software, and for proving programs correct, as well as attempting to provide facilities which are most suitable for the user. The specifications are not complete in every detail, but it is intended that they be complete “in spirit” – some further details, especially syntactic details, would have to be provided, but the proposals are otherwise relatively complete.
Note: Also in Proceedings of the IEEE 4th Symposium on Computer Arithmetic pp63-69.
Principles, Preferences and Ideals for Computer Arithmetic, Thomas E. Hull, Christian H. Reinsch, and John R. Rice, CSD-TR-339, 13pp, Dept. of Computer Science, Purdue University, June 1980.
Abstract: This paper presents principles and preferences for the implementation of computer arithmetic and ideals for the arithmetic facilities in future programming languages. The implementation principles and preferences are for the current approaches to the design of arithmetic units. The ideals are for the long term development of programming languages, with the hope that arithmetic units will be built to support the requirements of programming languages.
The Universal History of Numbers, Georges Ifrah, ISBN 1-86046-324-X, 633pp, The Harvill Press Ltd., 1994.
Abstract: More than a history of counting and calculating from the caveman to the late twentieth century, this is the story of how the human race has learnt to think logically. The reader is taken through the whole art and science of numeration as it has developed all over the world, from Europe to China, via the Classical World, Mesopotamia, South America, and, above all, India and the Arab lands. ...
Note: Translated from the French by David Bellos, E. F. Harding, Sophie Wood, and Ian Monk.
    (Also published is a translation of an earlier edition – From One to Zero: A Universal History of Numbers. Translated by Lowell Bair. Viking, New York, 1985.)
Representational error in binary and decimal numbering systems, Paul Johnstone, Proceedings of the 20th annual ACM Southeast Regional Conference, pp85–88, ACM Press, 1982.
Abstract: The representation of a general rational number of the form A/B as a floating point number requires a conversion from the general form to a base specific form. This conversion often results in the generation of infinitely repeating non-zero strings of digits which are truncated to the size of the mantissa resulting in a loss of precision. It is shown that the proportion of repeating versus finite rational numbers specific to a base is expotentially related to the number of unique prime factors of the base. Simulation results are presented which show the relative proportions of finite representations for binary and decimal cases over a range of mantissa sizes.
Higher Radix Floating Point Representations, P. Johnstone and F. Petry, Proceedings of the 9th Symposium on Computer Arithmetic, ISBN 0-8186-8963-3, pp128–135, IEEE Computer Society Press, September 1989.
Abstract: This paper examines the feasibility of higher radix floating point representations, and in particular, decimal based representations. Traditional analyses of such representations have assumed the format of a floating point datum to be roughly identical to that of traditional binary floating point encodings such as the IEEE P754 task group standard representations. We relax this restriction and propose a method of encoding higher radix floating point data with range, precision, and storage requirements comparable to those exhibited by traditional binary representations. Results from McKeeman’s Maximum and Average Relative Representational Error (MRRE and ARRE) analyses, Brent’s RMS error evaluation, Matula’s ratio of significance space and gap functions, and Brown and Richman’s exponent range estimates are extended to accomodate the proposed representation. A decimal alternative to traditional binary representations is proposed, and the behavior of such a system is contrasted with that of a comparable binary system.
The child-engineering of arithmetic in ToonTalk, Ken Kahn, Proceedings of the 2004 conference on Interaction Design and Children, ISBN 1-58113-791-5, pp141–142, ACM Press, 2004.
Abstract: Providing a child-appropriate interface to an arithmetic package with large numbers and exact fractions is surprisingly challenging. We discuss solutions to problems ranging from how to present fractions such as 1/3 to how to deal with numbers with tens of thousands of digits. As with other objects in ToonTalkŪ, we strive to make the enhanced numbers work in a concrete and playful manner.
The Art of Computer Programming, Vol 2, Donald E. Knuth, ISBN 0-201-89684-2, 762pp, Addison Wesley Longman, 1998.
Abstract: The chief purpose of this chapter [4] is to make a careful study of the four basic processes of arithmetic: addition, subtraction, multiplication, and division. Many people see arithmetic as a trivial thing that children learn and computers do, but we will see that arithmetic is a fascinating topic with many interesting facets. ...
Note: Third edition. See especially sections 4.1 through 4.4.
Numeric types, representations, and other fictions, T. Ochs, Computer Language, Vol. 8 #8, pp93–101, August 1991.
Abstract: Only rational numbers are explicitly representable in computers. Any explicit representation has a zero measure. Both rational and BCD arbitrary precision meet [author’s] initial requirements [of precision and range]. Floating-point numbers have a strange distribution.
Supporting packed decimal in Ada, David A. Rosenfeld, Proceedings of the conference on TRI-Ada '91, ISBN 0-89791-445-7, pp187–190, ACM Press, 1991.
Abstract: One of the principal barriers to Ada in the Information Systems (IS) marketplace is that Ada compilers do not support decimal arithmetic and a packed decimal representation for numbers. An Ada apologist could argue that Ada as a language does support these featurtx, but such arguments do little to help a COBOL programmer, accustomed to manipulating decimal quantities in a straightforward way. Our project, under contract to the Army, is addressing the problem directly, by implementing packed decimal numbers in its MVS Ada Compiler. T his paper will discuss the possible approaches to the problem, and explain the approach selected, comparing it briefly with other solutions...
Decimal arithmetic in applications and hardware, Akira Shibamiya, 2pp, pers. comm., 14 June 2000.
Abstract: (None)
A Study of DataBase 2 Customer Queries, Annie Tsang and Manfred Olschanowsky, IBM Technical Report TR 03.413, 25pp, IBM Santa Teresa Laboratory, San Jose, CA, April 1991.
Abstract: Over 200 Database 2 read-only and update queries were collected from 30 major DB2 customers during 1989 and 1990. These queries were considered representative of customers using DB2. Analysis of these queries were made in order to determine their characteristics and also to determine which SQL funetions were commonly used and how frequently they were used by these customers.
    Results of this study can be used in various ways, induding:
  • provide input to the planning and development organizations to enable them to implement new functions, enhance existing functions, and improve performance of functions that are most commonly used by customers.
  • allow developers to develop realistic workloads for benchmarking.

The 19 references listed on this page are selected from the bibliography on Decimal Arithmetic collected by Mike Cowlishaw. Please see the index page for more details and other categories.

Last updated: 10 Mar 2011
Some elements Copyright © IBM Corporation, 2002, 2009. All rights reserved.