Number Systems

In this study, the archaic counting systems of Mesopotamia as understood through the Neolithic tokens, numerical impressions, and proto-cuneiform notations were compared to the traditional number-words and counting methods of Polynesia as understood through contemporary and historical descriptions of vocabulary and behaviors. The comparison and associated analyses capitalized on the ability to understand well-known characteristics of Uruk-period numbers like object-specific counting, polyvalence, and context-dependence through historical observations of Polynesian counting methods and numerical language, evidence unavailable for ancient numbers. Similarities between the two number systems were then used to argue that archaic Mesopotamian numbers, like those of Polynesia, were highly elaborated and would have served as cognitively efficient tools for mental calculation. Their differences also show the importance of material technologies like tokens, impressions, and notations to developing mathematics. This project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No. 785793.

The residue number system (RNS) has been an important research field in computer arithmetic for many decades, mainly because of its carry-free nature, which can provide high-performance computing architectures with superior delay specifications. Recently, research on RNS has found new directions that have resulted in the introduction of efficient algorithms and hardware implementations for RNS with much better performance than previous ones. Furthermore, the applicability of RNS in many computer and digital signal processing (DSP) applications has simultaneously greatly increased. Hence, the need is evident for the development of a new and well-organized RNS teaching method with emphasis on recent achievements. In this paper, a step-by-step teaching process for RNS that describes RNS design in separate parts is presented. Each part is investigated in detail, taking into account the recent research performed on RNS. The compatibility of the proposed method with the new RNS research trends makes this method suitable for researchers as well as students.

Decimal Numbers:-Base 10 uses ten digits = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Binary Numbers:-Base 2 uses 2 digits = 0, 1 Octal Numbers:-Base 8 uses 8 digits = 0, 1, 2, 3, 4, 5, 6, 7 Hexadecimal Numbers:-Base 16 uses 16 digits = 0, 1, 2, 3, 4, ……., 14, 15-{ A=10 }, { B=11 }, { C=12 }, { D=13}, { E=14 }, { F=15 } > These conversion shows how to convert any base to decimal… • Converting Octal to Decimal…. • Example : 1234568 = (1x8 5) + (2x8 4) + (3x8 3) + (4x8 2) + (5x8 1) + (6x8 0) = 4279810 • Converting Hexadecimal to decimal…. • Example : ABC123 16 = (Ax16 5) + (Bx16 4) + (Cx16 3) + (1x16 2) + (2x16 1) + (3x16 0) = (10x16 5) + (11x16 4) + (12x16 3) + (1x16 2) + (2x16 1) + (3x16 0) = 17388410 • Converting Binary to decimal…. • Example : 110101 2 = (1x2 5) + (1x2 4) + (1x2 2) + (1x2 0) = 5310 *Tiny numbers that are written under each number system shows what number system is that ,for example: 2 means Binary, 8 means Octal, 10 means Decimal, 16 means Hexadecimal..

The Calculators in Primary Mathematics Project was a long-term investigation into the effects of the introduction of calculators on the learning and teaching of primary mathematics. The Australian project commenced with children who were in kindergarten and grade 1 in 1990, moving up through the schools to grade 4 level by 1993. Children were given their own calculators to use when they wished, while teachers were provided with some systematic professional support. Over 60 teachers and 1,000 children participated in the project. This paper gives an overview of the project, with particular emphasis on the ways in which teachers incorporated calculators into their classrooms and the resulting long-term learning outcomes for the students. It first reports on a survey of 700 primary, 7th-, and 8th-grade teachers which established that teachers now support calculator use, even in the first grades, but that actual use falls far behind the support .expressed. A brief description is given of the major ways in which the calculator was used in project schools--as a computational device, as a recording device, to count, and as an object to explore. Testing of 3rd-and 4th-grade students (n=225) established that children did understand the number system better after sustained calculator use and that they were better able to choose an appropriate operation in a word problem. A series of interviews showed that calculator use had assisted children to develop number sense and skills of mental comitation. Contains 20 references. (MKR)

The idea the New Zealand Māori once counted by elevens has been viewed as a cultural misunderstanding originating with a mid-nineteenth-century dictionary of their language. Yet this “remarkable singularity” had an earlier, Continental origin, the details of which have been lost over a century of transmission in the literature. The affair is traced to a pair of scientific explorers, René-Primevère Lesson and Jules Poret de Blosseville, as reconstructed through their publications on the 1822–1825 circumnavigational voyage of the Coquille, a French corvette. Possible explanations for the affair are briefly examined, including whether it might have been a prank by the Polynesians or a misunderstanding or hoax on the part of the Europeans. Reasons why the idea of counting by elevens remains topical are discussed. First, its very oddity has obscured the counting method actually used—setting aside every tenth item as a tally. This “ephemeral abacus” is examined for its physical and mental efficiencies and its potential to explain aspects of numerical structure and vocabulary (e.g., Mangarevan binary counting; the Hawaiian number word for twenty, iwakalua), matters suggesting material forms have a critical if underappreciated role in realising concepts like exponential value. Second, it provides insight into why it can be difficult to appreciate highly elaborated but unwritten numbers like those found throughout Polynesia. Finally, the affair illuminates the difficulty of categorising number systems that use multiple units as the basis of enumeration, like Polynesian pair-counting; potential solutions are offered. This project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No. 785793.

Many children have significant mathematical learning disabilities (MLD, or dyscalculia) despite adequate schooling. The current study hypothesizes that MLD partly results from a deficiency in the Approximate Number System (ANS) that supports nonverbal numerical representations across species and throughout development. In this study of 71 ninth graders, it is shown that students with MLD have significantly poorer ANS precision than students in all other mathematics achievement groups (low, typically, and high achieving), as measured by psychophysical assessments of ANS acuity (w) and of the mappings between ANS representations and number words (cv). This relation persists even when controlling for domain-general abilities. Furthermore, this ANS precision does not differentiate low-achieving from typically achieving students, suggesting an ANS deficit that is specific to MLD.

Seven papers presented at a research conference on number and measurement are presented in this volume. The first paper provides an overview of research concerning number and measurement, and suggests directions for future research. The second paper discusses the relationships between measurement and number concepts, and psychological and instructional issues related to transfer. Two papers are devoted to synthesizing and analyzing research on measurement, and the delineation of questions about which research is needed. Two papers concern fractions; the first of these analyzes the foundations of the rational numbers from mathematical, cognitive, and instructional points of view, while the second reviews and synthesizes educational research related to fractions. The final paper concerns children's development of cardinal and ordinal number concepts. (SD) Education. These reports fall into three brbad categories. Research reviews summarize and analyze recent research in specific areas of mathematics education. Resource guides identify and analyze materials and references for use by mathematics teachers at all levels. Special bibliographies announce the availability of documents and review the literature in selected interest areas of mathematics education. Reports in each of these categories may also be targeted for specific subpopulations of the mathematics education community.

This paper is a response to Olderroge 1984 that Efik does not have a system of number beyond one thousand, since according to the claim, the word for thousand in Efik is tosin, a recent borrowing from English and assimilated phonologically to synchronize with Efik. We dispute this position and argue that Efik can count up to 999,999 internally, and that borrowing is not meant to enrich the internal resourcefulness of the language in this case, but merely for ease of pronunciation. In this paper, we examine the structure of the system of cardinal numbers and the processes of forming complex numbers in Efik. Complex numbers are formed from addition, subtraction or multiplication of the basic units which are 1-5, 10, 15 and 20 or from a combination of addition and multiplication and/or subtraction. Ye is the particle for addition, -suk is the subtraction marker and multiplication is indicated by a zero allomorph. The general pattern of numeration in Efik is iterative. Our assumption is that the analysis of the number system in Efik has some linguistic implications as it triggers certain grammatical phenomena in the language.

It has been hypothesized that developmental dyscalculia (DD) is either due to a defect of the approximate number system (ANS) or to an impaired access between that system and symbolic numbers. Several studies have tested these two hypotheses in children with DD but none has dealt with adults who had experienced DD as children. This study aimed to compare these two hypotheses in an adult population in order to investigate which deficits still persist at that age. To that aim, numerical estimation tasks were given to adults who had or had not experienced DD as a child. Three of the estimation tasks required a mapping between the ANS and symbolic numbers: participants had to estimate the number of same or different-sized dots presented by producing the corresponding Arabic number or, conversely, to produce the number of dots corresponding to a presented Arabic number. A fourth task did not require any processing of symbolic numbers; participants had to produce a collection of dots of the same numerosity as another one previously presented. Consistently, in all the four numerical tasks and irrespective of whether the tasks used symbolic numbers or not, the estimates of DD participants were less accurate than those of the control participants. These results indicate that adults who had experienced DD as children continue to demonstrate a less precise magnitude representation.

The aim of this paper is to investigate the ways in which the number line can function in solving mathematical tasks by first graders (6 year olds). The main research question was whether the number line functioned as an auxiliary means or as an obstacle for these students. Through analysis of the 32 students‟ answers it appears that the number line functions both as an auxiliary means and as an obstacle with the latter occurring in the majority of cases.

Seven papers presented at a research conference on number and measurement are presented in this volume. The first paper provides an overview of research concerning number and measurement, and suggests directions for future research. The second paper * * via the ERIC Document Reproduction Service (EDRS). EDRS is not * responsible for the quality of the original document. Reproductions * * supplied by EDES are the best that can be made from the original.

This paper examines how Ciulistet, a group of Yup'ik Eskimo elders, teachers, aides, and university collaborators, has slowly begun transforming education in southwest Alaska. Specifically, this paper shows how this indigenous group has produced, interpreted, and applied ancient Yup'ik wisdom to the modern context of schooling.

In this article, I present research on some traditional numeracy practices of the Gusilay people group in Senegal and make recommendations for developing a numeracy programme for women. Based on a strong foundation of traditional knowledge and practices, the programme will aim to meet felt needs of women who are faced with new numeracy related challenges due to changes in society. My research is placed in the framework of social practice theory, which emphasizes the fact that numeracy is not a set of skills that are learned and used in isolation, but rather practices that happen in context and vary with it. After a brief outline of social practice theory and the methodology I have chosen for my research, I analyze my findings from that perspective and suggest some practical implications for developing a numeracy programme for Gusilay women.

This article confronts the issue of why secondary and post-secondary students resist accepting the equality of 0.999… and 1, even after they have seen and understood logical arguments for the equality. In some sense, we might say that the equality holds by definition of 0.999…, but this definition depends upon accepting properties of the real number system, especially the Archimedean property and formal definitions of limits. Students may be justified in rejecting the equality if they decide to work in another system—namely the non-standard analysis of hyperreal numbers—but then they need to understand the consequences of that decision. This review of arguments and consequences holds implications for how we introduce real numbers in secondary school mathematics.

A complex number α is said to satisfy the height reducing property if there is a finite subset, say F, of the ring Z of the rational integers such that Z[α] = F [α]. This problem has been considered by several authors, especially in contexts related to self affine tilings, and expansions of real numbers in non-integer bases. We continue, in this paper, the description of the numbers satisfying the height reducing property, and we specify a related characterization of the roots of integer polynomials with dominant term.

Abstract: The Calculators in Primary Mathematics Project was a long-term investigation into the effects of the introduction of calculators on the learning and teaching of primary mathematics. The Australian project commenced with children who were in kindergarten ...

The goal of this study was to investigate understanding of inservice elementary school teachers in Taiwan about number sense, teaching strategies of number sense and the development of number sense of students; and the profile of integrating number sense into mathematical instruction , and teaching practice. Data wase gathered through interviews of two elementary mathematics teachers regarding their understanding about number sense followed by observations of the teachers instructing in their mathematics classes. The data included the categorization and comparison of these teachers’ understanding and teaching practices. The conclusions are as follows: The common point shared by two teachers was that in the teaching of four fundamental operations of fraction, they tended to ask the students to repeat and memorize the four fundamental operations of arithmetic or the arithmetic rules of addition, subtraction, multiplication and division of fraction. It was only the instruction valuing ...

Talk presented to "SCRIBO" (the INSCRIBE ERC Project), University of Bologna, 19 May 2021. This talk examines the labels “abstract” and “concrete” in terms of what they mean in philosophy, psychology, and mathematics. These labels and the purported distinction between the two have been applied and, arguably, misapplied to Mesopotamian numbers, particularly the Neolithic tokens. The conceptual change they refer to can be understood, not as attaining a new state of abstractness from a previous state of concreteness, but rather, as a prototypical process of conceptual change, or numerical realization and elaboration, by means of material devices like tokens and writing. Four ways in which writing changed numbers will be mentioned, noting that some of the changes commonly associated with writing were already well underway in conjunction with the Neolithic tokens, while others were still in progress as late as the European Middle Ages. Finally, by drawing on case studies from Africa and Polynesia, new ways of understanding highly elaborated but non-notationally mediated number systems will be offered.

Graduate students in an English-as-a-Second-Language education program at the University of Maryland were surveyed concerning suggested methods for incorporating languages other than English into the curriculum in a multicultural classroom. Respondents were master's and doctoral students from a variety of language backgrounds. The article details the resulting suggestions for these languages and cultures: Korean; Caribbean Island Nations; Hindi; Thai; and Turkish. Classroom behaviors and attitudes for teachers to both incorporate and anticipate include those addressing specific kinds of teacher-parent and teacher-student communication, respect for parents, language usage, classroom questions or lack of them, grading and other classroom teaching techniques, classroom environment, nonverbal behaviors and body language, and educational values. Teachers are also advised explicitly of things not to do in the classroom for each of the cultures. (Contains 9 references.) (MSE)

It has been hypothesized that developmental dyscalculia (DD) is either due to a defect of the approximate number system (ANS) or to an impaired access between that system and symbolic numbers. Several studies have tested these two hypotheses in children with DD but none has dealt with adults who had experienced DD as children.This study aimed to compare these two hypotheses in an adult population in order to investigate which deficits still persist at that age. To that aim, numerical estimation tasks were given to adults who had or had not experienced DD as a child. Three of the estimation tasks required a mapping between the ANS and symbolic numbers: participants had to estimate the number of same or different-sized dots presented by producing the corresponding Arabic number or, conversely, to produce the number of dots corresponding to a presented Arabic number. A fourth task did not require any processing of symbolic numbers; participants had to produce a collection of dots of the same numerosity as another one previously presented.Consistently, in all the four numerical tasks and irrespective of whether the tasks used symbolic numbers or not, the estimates of DD participants were less accurate than those of the control participants. These results indicate that adults who had experienced DD as children continue to demonstrate a less precise magnitude representation.

Drawing on results from psychology and from cultural and linguistic studies, we argue for an increased focus on developing quantity sense in school mathematics. We explore the notion of "feeling number," a phrase that we offer in a two-fold sense-resisting tendencies to feel numb-er (more numb) by developing a feeling for numbers and the quantities they represent. First, we distinguish between quantity sense and the relatively vague notion of number sense. Second, we consider the human capacity for quantity sense and place that in the context of related cultural issues, including verbal and symbolic representations of number. Third, and more pragmatically, we offer teaching strategies that seem helpful in the development of quantity sense coupled with number sense. Finally, we argue that there is a moral imperative to connect number sense with such a quantity sense that allows students to feel the weight of numbers. It is important that learners develop a feeling for number, which includes a sense of what numbers are and what they can do.

Talk presented to "Numerous Numerosity," Society for Multidisciplinary and Fundamental Research (SEMF), University of Edinburgh, 26 May 2021. Numbers are perhaps the longest-lived cultural system the world has ever known, but we still do not know how old numbers might be, or where and why they might have emerged, or in what form. Investigating these matters is necessarily an inferential endeavor. However, archaeological evidence alone is likely insufficient. Insights from psychology, linguistics, and ethnography have implications for how we might examine and interpret the archaeological record for signs of prehistoric numeracy, perhaps even calling into question some of the assumptions and methods currently in use. Some of the most pressing issues are reviewed, and recommendations are offered for incorporating interdisciplinary data and insights into archaeological investigations and interpretations.

Efforts to meet the needs of children's learning in arithmetic has led to an increased emphasis on the teaching of mental calculation strategies in England. This has included the adoption of didactical tools such as the empty number line (ENL) that was developed as part of the realistic mathematics movement in the Netherlands. It has been claimed that the English use of the ENL differs from that of the Dutch. In this study curriculum guidance materials are examined to investigate these differences. Two examples of teaching sequences are provided to illustrate them further. It is proposed that the different uses stem from different pedagogical principles. The implications of the English use are discussed in relation to the presentation of mental calculation strategies in an algorithmic way.

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A dedicated, non-symbolic, system yielding imprecise representations of large quantities (approximate number system, or ANS) has been shown to support arithmetic calculations of addition and subtraction. In the present study, 5-7-year-old children without formal schooling in multiplication and division were given a task requiring a scalar transformation of large approximate numerosities, presented as arrays of objects. In different conditions, the required calculation was doubling, quadrupling, or increasing by a fractional factor (2.5). In all conditions, participants were able to represent the outcome of the transformation at above-chance levels, even on the earliest training trials. Their performance could not be explained by processes of repeated addition, and it showed the critical ratio signature of the ANS. These findings provide evidence for an untrained, intuitive process of calculating multiplicative numerical relationships, providing a further foundation for formal arithmetic instruction.

A dedicated, non-symbolic, system yielding imprecise representations of large quantities (approximate number system, or ANS) has been shown to support arithmetic calculations of addition and subtraction. In the present study, 5-7-year-old children without formal schooling in multiplication and division were given a task requiring a scalar transformation of large approximate numerosities, presented as arrays of objects. In different conditions, the required calculation was doubling, quadrupling, or increasing by a fractional factor (2.5). In all conditions, participants were able to represent the outcome of the transformation at above-chance levels, even on the earliest training trials. Their performance could not be explained by processes of repeated addition, and it showed the critical ratio signature of the ANS. These findings provide evidence for an untrained, intuitive process of calculating multiplicative numerical relationships, providing a further foundation for formal arithm...

The word space is used in many ways and most of these applications give this word a different meaning. This makes the notion of space very obscure. Already in the common life of humans takes the word space many different uses. Especially philosophers, mathematicians, and physicists have attributed a huge number of interpretations of the noun “space”. This has led to a huge number of different forms of space. Humans live in an environment that is characterized by space and time. This paper focuses on the most elemental meanings that mathematicians and physicists attribute to the word “space”. Next, the immediate extensions of this elementary space are investigated. Since physicists investigate our physical reality, the paper also investigates how physical reality treats the notion of space.

We present a research report on addition and subtraction conducted with Down syndrome students between the ages of 12 and 31. We interviewed a group of students with Down syndrome who executed algorithms and solved problems using specific materials and ...

Abstract: This two-volume teacher's guide is to be used with a first-semester, third-grade mathematics program. The program is in large part the outgrowth of involvement with children and was influenced by information gathered in an extended pilot project ...

It has been hypothesized that developmental dyscalculia (DD) is either due to a defect of the approximate number system (ANS) or to an impaired access between that system and symbolic numbers. Several studies have tested these two hypotheses in children with DD but none has dealt with adults who had experienced DD as children.This study aimed to compare these two hypotheses in an adult population in order to investigate which deficits still persist at that age. To that aim, numerical estimation tasks were given to adults who had or had not experienced DD as a child. Three of the estimation tasks required a mapping between the ANS and symbolic numbers: participants had to estimate the number of same or different-sized dots presented by producing the corresponding Arabic number or, conversely, to produce the number of dots corresponding to a presented Arabic number. A fourth task did not require any processing of symbolic numbers; participants had to produce a collection of dots of th...

The Calculators in Primary Mathematics Project in Australia was a long-term investigation into the effects of the introduction of calculators on the learning and teaching of primary mathematics. The Australian project commenced with children who were in kindergarten and grade 1 in 1990, moving up through the schools to grade 4 level by 1993. Children were given their own calculators to use when they wished, while teachers were provided with some systematic professional support. Over 60 teachers and 1,000 children participated in the project. This paper describes some critical features of project classrooms which supported the development of number sense and reports on the results of interviews with 4th-grade children (n=58), approximately half of whom had long-term experience with calculators. Children with long-term experience with calculators performed better on the 12 mental computation interview items overall, the 24 number knowledge items overall, and the 3 estimation items tak...

Many children have significant mathematical learning disabilities (MLD, or dyscalculia) despite adequate schooling. The current study hypothesizes that MLD partly results from a deficiency in the Approximate Number System (ANS) that supports nonverbal numerical representations across species and throughout development. In this study of 71 ninth graders, it is shown that students with MLD have significantly poorer ANS precision than students in all other mathematics achievement groups (low, typically, and high achieving), as measured by psychophysical assessments of ANS acuity (w) and of the mappings between ANS representations and number words (cv). This relation persists even when controlling for domain-general abilities. Furthermore, this ANS precision does not differentiate low-achieving from typically achieving students, suggesting an ANS deficit that is specific to MLD.

Number systems differ cross-culturally in characteristics like how high counting extends and which number is used as a productive base. Some of this variability can be linked to the way the hand is used in counting. The linkage shows that devices like the hand used as external representations of number have the potential to influence numerical structure and organization, as well as aspects of numerical language. These matters suggest that cross-cultural variability may be, at least in part, a matter of whether devices are used in counting, which ones are used, and how they are used.

Much research supports the existence of an Approximate Number System (ANS) that is recruited by infants, children, adults, and non-human animals to generate coarse, non-symbolic representations of number. This system supports simple arithmetic operations such as addition, subtraction, and ordering of amounts. The current study tests whether an intuition of a more complex calculation, division, exists in an indigene group in the Amazon, the Mundurucu, whose language includes no words for large numbers. Mundurucu children were presented with a video event depicting a division transformation of halving, in which pairs of objects turned into single objects, reducing the array's numerical magnitude. Then they were tested on their ability to calculate the outcome of this division transformation with other large-number arrays. The Mundurucu children effected this transformation even when non-numerical variables were controlled, performed above chance levels on the very first set of test trials, and exhibited performance similar to urban children who had access to precise number words and a surrounding symbolic culture. We conclude that a halving calculation is part of the suite of intuitive operations supported by the ANS.

Contents: List of tables. List of figures. Introduction. MATLAB quick start. Linear equations. Sparse matrices. Non-linear equations. Optimization. Integration and differentiation. Ordinary differential equations. Mathematical functions and operators. Symbolic computations. Graphing points and curves. Graphing surfaces and volumes. Animation. Graphics properties. Graphical user-interface functions. Programming in MATLAB. File input/output. Debugging MATLAB programs. Advanced data structures. Object-oriented programming. Miscellaneous. MATLAB resources. Index.