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.

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 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.

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.

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.

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.

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.

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.

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...

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.