Costello Divisibility: Exploration of a Comedic Division Algorithm

This paper offers two new division algorithms by digit recurrence. Compared to the standard radix-2 division algorithms with carry-save addition, the new division algorithms trade off a simpler selection logic for more alternatives in the basic repetition step. Our final division algorithm is potentially faster and more energy efficient than radix-2 division with carry-save addition, because the selection logic has less delay and the repetition steps on average perform fewer additions and subtractions.

This study examined 11 U.S. textbooks written for prospective teachers to investigate how standard decimal multiplication and division algorithms are presented, especially the rationale of both algorithms. Analytical frameworks of various methods used in different textbooks were developed. The findings suggest that half of the textbooks do not fully present the standard algorithms. The results indicate variations in textbooks’ explanations for the procedures involved in the standard algorithms. Some procedures are explained more frequently than others. The reason why moving the decimal point in both dividend and divisor keeps the quotient unchanged is provided in most of the textbooks. Half of the textbooks explain why adding the decimal places in the multiplication algorithm. However, only two books explain the rationale for placing the decimal point in the quotient. The findings suggest that textbook writers pay more attention to explaining the rationales of the standard algorithms for decimal multiplication and division.

Previous article A Generalization of the Bairstow ProcessA. A. GrauA. A. Grauhttps://doi.org/10.1137/0111036PDFPDF PLUSBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] L. Bairstow, Investigations relating to the stability of the aeroplane, Reports and Memoranda, 154, Advisory Committee for Aeronautics, 1914 Google Scholar[2] F. L. Hitchcock, Finding complex roots of algebraic equations, J. Math. Phys., 17 (1938), 55–58 0019.13203 CrossrefGoogle Scholar[3] Van A. McAuley, A method for the real and complex roots of a polynomial, J. Soc. Indust. Appl. Math., 10 (1962), 657–667 10.1137/0110050 MR0154407 0112.34506 LinkISIGoogle Scholar[4] Alston S. Householder, Principles of numerical analysis, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953, 139–144 MR0059056 Google Scholar[5] J. H. Wilkinson, The evaluation of the zeros of ill-conditioned polynomials. I, II, Numer. Math., 1 (1959), 150–180 10.1007/BF01386381 MR0109435 CrossrefGoogle Scholar[6] Peter Naur, Report on the algorithmic language ALGOL 60, Comm. ACM, 3 (1960), 299–314 10.1145/367236.367262 MR0134432 CrossrefISIGoogle Scholar Previous article FiguresRelatedReferencesCited ByDetails Volume 11, Issue 2| 1963Journal of the Society for Industrial and Applied Mathematics205-519 History Submitted:16 April 1962Published online:13 July 2006 InformationCopyright © 1963 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0111036Article page range:pp. 508-519ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics

This paper describes three algorithms for the digital computation of the frequency response of linear sys tems and discusses their advantages and disadvantages. The third method presented is a novel one which applies the standard polynominal quadratic synthetic division algorithm in a unique way to the problem. The method allows the forward and feedback-path transfer functions to be inputted in any form; it computes the open-loop and closed-loop frequency responses in one pass; and, most importantly, it does not require numerical factoring at any point. The method is easy to use and appears to run effi ciently, particularly for high-order systems. The paper concludes with FORTRAN and BASIC programs which implement the algorithm of Method Three.

In this paper, we propose a simple algorithmic solution to the best approximation problem of finding the nearest multivariate rational function, with a fixed separable denominator polynomial, from a given multivariate polynomial, where the numerator polynomial is desired to minimize the integral of the squared error over the distinguished boundary of the unit polydisc. The proposed algorithm does not require any numerical integration or numerical root finding technique because this is realized based on the standard multivariate division algorithm. A simple observation of the proposed algorithm leads to an ideal membership problem characterizing the solution to the problem. A relation of this characterization and a multivariate generalization of the Walsh's Theorem is also discussed with another ideal membership problem derived by applying a corollary of the Hilbert Nullstellensatz to the Walsh's Theorem. Although the discussion to derive the latter ideal membership problem seems to be roundabout, such a characterization would be useful for further generalization, for example to some weighted least-squares approximation. Numerical examples demonstrate the practical applicability of the proposed method to design problems of multidimensional IIR filters.

The major problem with most representations of the division algorithm, as opposed to the concept of division, is that students become lost in the mechanics of the manipulations and do not make the connection between the model and the process being modeled. The model described in thi article relates step by step to the standard division algorithm; the connection is immediate and obvious. It is a spin-off of a method presented by DiSpigno in the October 1971 issue of the Arithmetic Teacher. There are two major differences; one, the introduction of a story line; and two, the direct connection that is made between the representational model and the standard algorithm.

Mastering the digit-based division algorithm remains relevant in contemporary mathematics education, as it enhances algorithmic, multiplicative and relational thinking. This study examines the various practices used by Slovenian sixth-grade students to solve a multi-digit division calculation. Addressing a largely underexplored area in mathematics education, the research draws on video recordings of 27 students, offering a fine-grained analysis of their written procedures and reasoning. Four main types of division practices were identified, ranging from long digitbased to short digit-based division algorithm. Most of the participating students employed a long digit-based division algorithm with short recordings of partial dividend and partial difference and side calculation for multiplication and subtraction. The students’ practices differed in determining partial quotients, intermediate products and intermediate differences. However, practices without side calculations proved to be more time efficient. The findings highlight the need for explicit instruction in recording formats and strategy selection. By providing empirical insights into students’ actual division practices, the study contributes to both theory and classroom practice, informing the design of textbooks, curricula and teacher education programmes.

The untimely death of Michael Robert Herman in November 2000 deprived the scientific community of one of its deepest mathematical minds, who had a profound impact on the theory of dynamical systems over the last 30 years. Born in New York, he was educated in France. He was a student at École Polytechnique before being one of the first members of the Centre de Mathématiques created there by Laurent Schwartz. For more than 20 years, his seminar had a major influence worldwide and was the main vector of the development of the theory of dynamical systems in France. All of his students remember with thankfulness and emotion the passion with which he led them into the wonderful mathematical world. He maintained through the years strong connections with the Instituto de Matemática Pura e Aplicada in Rio de Janeiro. His interests covered most aspects of the modern theory of dynamical systems and much beyond that, from economics to arts and philosophy. However, it is fair to say that from the start the so-called small divisors problems, related in particular to the stability of quasiperiodic motions, were closest to his heart. His epoch-making theorem on the linearization of circle diffeomorphisms [1–4], his two volumes [5, 6] on invariant curves for twist diffeomorphisms, which are still the standard reference 20 years later, his very many deep contributions on the existence and geometry of invariant tori all bear witness to that interest [7–10].

The implementation of quantum computing processors for scientific applications includes quantum floating points circuits for arithmetic operations. This work adopts the standard division algorithms for floating-point numbers with restoring, non-restoring, and Goldschmidt division algorithms for single-precision inputs. The design proposals are carried out while using the quantum Clifford+T gates set, and resource estimates in terms of numbers of qubits, T-count, and T-depth are provided for the proposed circuits. By improving the leading zero detector (LZD) unit structure, the proposed division circuits show a significant reduction in the T-count when compared to the existing works on floating-point division.

For students to find algebra conceptually meaningful, as well as useful in modeling and analyzing real-world problems, they must be able to reflect on, make sense of, and communicate about general numerical procedures (Kieran 1992). Such procedures consist of set sequences of arithmetic operations performed on numbers. Examples include computing an average and performing the standard division algorithm. Thinking about numerical procedures starts in the elementary grades and continues in successive grades until students can eventually express and reflect on the procedures using algebraic symbolism. This article outlines how such thinking can progress to algebraic reasoning and illustrates how computers used to promote this progression.

The Common Core State Standards of Mathematics (CCSSM), a set of US educational standards which has recently been adopted by 45 states, creates a more rigorous and coherent set of standards for American students, making elementary math anything but elementary. The adoption of these new standards formulates the research questions for this study: How well do current curricula match the CCSSM and how well do current curricula support teacher knowledge to implement the standards? In this study, three diverse curricula used in the United States, Prentice Hall, Singapore Math, and CK-12, are examined with three evaluation tools. These tools measure (a) the cognitive demands of the mathematical tasks in each curricula, (b) the mathematical coherency of an instructional unit, and (c) the resources in each curricula that support teachers’ understanding of mathematics. Division of fractions is the topic of analysis because of its frequent occurrence in algebra which is the foundation for higher-level math. I find that Singapore Math’s problems reach higher-level cognitive demands more often than Prentice Hall and CK-12. Prentice Hall and CK-12’s reliance on using the standard division algorithm inhibits conceptual thinking for both students and teachers. From a Curriculum Review Tool, which focuses on teacher knowledge, I find that Singapore Math is the closest to reach the division of fraction CCSSM when compared to Prentice Hall and CK-12. Resource tools for teachers can be developed that better support students’ learning by combining characteristics from each curriculum such as word problems, manipulatives/pictures, and samples of students’ work.

Of the four basic arithmetic operations, division is perhaps the most poorly learned and the first forgotten. There is no denying that the compactness of the standard division algorithm makes it a desirable method to be learned. However, we are seeing a whole generation of people who learned it but forgot it as quickly as they could. I have developed a “duplation division” (so-called because of a doubling process) that is a viable alternative for many students who have difficulties with the traditional method of long division.