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The subject is introduced by a reflection on the different aspects of numbers and a review of a paper written by Andreas Nieder: Representing Something Out of Nothing: The Dawning of Zero. This is followed by a reflection on the precise nature of the transformation of zero from a signifier of quantity to a number. Edmund Husserl called this the constitution (Stiftung) of the zero as a new ideal object and he described the process in The Origin of Geometry. Zero as a number is an element of a self-contained symbolic deductive system that does not refer to the life world and has no sense at all; it is, in the words of the analytic philosophical tradition, nonsense. This implies that no meaning can possibly be transferred from zero as a quantity to zero as a number. The history of zero as a number coincides with the history of abstract mathematics and the development of calculation systems, the tremendous usefulness of which is precisely based on its very meaninglessness. All speculation about any eventual progress of the concept of zero as a quantity to zero as a number is therefore unsubstantiated. Marc Richir showed that the constitution of an ideal object is a leap; it leaves a gap between its new sense and its historical basis. This chapter will conclude with the question whether there is a link between this historical basis of zero as a quantity and the concept of emptiness in Buddhism; this supposed relation has been and still is the pet of quite a few Buddhologists. [ABSTRACT FROM AUTHOR]

This is an exploration of the mathematics, history, and philosophy of zero. It looks at zero and its development from three perspectives: purely as a sign (syntactic); its meaning (semantics); and its social contexts, roles, and uses (pragmatic). Adopting the meaning as use theory, it argues that the meaning of zero was not fully established until the time of Brahmagupta, at around 628 CE, for Brahmagupta established the role of zero within the network of number relations in the integers. Most notably, zero is the additive identity. This synthesis is analogous for number theory with Euclid's geometry. Thus, it is posited that there are stages in the development of the concept of zero. First, its predecessor serves as a placeholder in an empty column. Second, zero is understood as a sign for nothing (nought), resting on the concept of the empty void. Third, its role as additive identity is understood and accepted. The conjecture that the idea of void is a prerequisite for the development of the number zero is confirmed positively in Indian and Mayan cultures. But the case of Ancient Egypt, which has the concept of void but not zero shows that having the concept of void appears, not of itself, sufficient. [ABSTRACT FROM AUTHOR]

IN THIS PAPPER WE WILL PRESENT SOME METHODS AND PROCEDURES USED IN FORMING MATHEMATICAL REPRESENTATIONS AT PRE-SCHOOL AGE. WE REFER IN PARTICULAR TO HOW TO PERCEIVE THE SETS AND THE RELATIONSHIPS BETWEEN THE ELEMENTS OF THE SETS AND TO ACQUIRE THE COUNTING OPERATION. [ABSTRACT FROM AUTHOR]

This article explains a model of teaching computer programming applied to the solution of a specific problem from functional programming paradigm establishing a relationship to the student, academically among the topics covered in mathematics and possibility of capitalizing computer technology to solve it. This research used the method of study and solution of a specific case from the perspective of mathematical logic implementation developed in the first course of computer programming in Systems Engineering. The results show an assimilation, appropriation, application and feedback conducted so that it has been applied, by students in similar exercises from the perspective of autonomous learning and active participation. The conclusion is that the solution of mathematical problems from programming is an area that could be exploited much more, from the teaching, to reach learning goals of student and that this strategy enable selflearning and active learning. [ABSTRACT FROM AUTHOR]

This article presents a new theory of simplex numerals that incorporates a slight revision of Chomsky's (2008) set-theoretic conception of natural number, which assumes that the notion of natural number is innate. The new theory makes it possible to account for the behavior of numerals in counting as well as the developmental stages that children go through in learning numerals. The key idea is that set-theoretic objects corresponding to natural number notions are subject to operations that apply when a syntactic object is converted to phonological form. These operations provide a crucial link that connects the meaning of a numeral with the count list consisting of numerals. A notable feature of the proposed analysis is that the Cardinal Principle is derived by recruiting linguistic computation and therefore is no longer stipulated as such. [ABSTRACT FROM AUTHOR]

Multilevel inverters have received considerable attention from industries and researchers for its high power and voltage applications. Various switching techniques have been suggested for improving the quality and performance of inverters. One of the conventional techniques for implementing the switching algorithm in these inverters is optimized harmonic-stepped waveform. However, this technique involves the major problem of solving non-linear and complex equations, which indicates a possibility of multiple solutions. This paper describes a novel technique that uses the simple arithmetic sequence of natural numbers to determine the optimized switching angles of a single-phase cascaded multilevel inverter of any number of levels and fed by equal dc sources. The basic objective was to avoid the laborious process of solving the non-linear equations using complex switching algorithm for finding the optimal solution of the switching angles. This technique is implemented to calculate the optimized switching angles of a nine-level cascaded inverter that reduces the total harmonic distortion to below 9%. [ABSTRACT FROM AUTHOR]

Natural numbers all of whose prime divisors (even or odd in number) belong to special sets are considered. It is proved that numbers with an odd number of different prime divisors predominate; more precisely, the difference between these numbers not exceeding a given x tends to infinity with increasing x. [ABSTRACT FROM AUTHOR]

The article discusses generating function for σ (3n − 1). Topics include the formula obtain which contains the generating functions for p(5n − 1), the Euler pentagonal theorem, and the Jacobi formula for Eulerian product cube. Also mentioned the use of theta functions apparatus, elliptic functions theory for alternative bases, and the argument of the function regarding the sum of the divisors is equal to a fractional number.

A digression about the knowledge and mathematical knowledge focusing on the concept of natural numbers and the decimal numbering system. This is an invitation to teachers of basic education, mainly for teachers from early years of elementary school. One discusses issues assigned to the mathematical knowledge; to the construction of the concept of natural number based on conceptions of evolutionary epistemology and of genetic epistemology. Epistemological conceptions of mathematics are applied to an example based on experiments with undergraduates about the construction of the natural numbers and the decimal numbering system using simple materials resources. [ABSTRACT FROM AUTHOR]

The semiotic approach to mathematics education introduces the notion of 'semiotic system' as a tool to describe mathematical activity. The semiotic system is formed by the set of signs, the production rules of signs and the underlying meaning structures. In this paper, we present the notions of system of practices and configuration of objects and processes that complement the notion of semiotic system and help to understand the complex nature of mathematical objects. We also show in what sense these notions facilitate the description and comprehension of building and communicating mathematical knowledge, by applying them to analyze semiotic systems involved in the teaching and learning of some elementary arithmetic concepts. [ABSTRACT FROM AUTHOR]

We prove that almost all natural numbers n satisfying the congruence n ≡ 3 (mod 24), n ≢ 0 (mod 5), can be expressed as the sum of three squares of primes, at least one of which can be written as 1 + x + y. [ABSTRACT FROM AUTHOR]

The construction of a numerical sequence requires an ordinal model of conceptual support that can be defined by the given relationships in a system of progressions. Taking into account the ordinal model of conceptual support leads us to a coherent conceptual/interpretative system which takes into consideration conceptions and beliefs regarding numerical sequencing that immediately takes us back to epistemological and didactic considerations. Epistemological approaches are confined to looking at the nature, origin and mode of existence of the natural number and elementary arithmetic, so that construction of the numerical sequence relies, at this point, on the conclusions that are drawn in regard to the said problem. [ABSTRACT FROM AUTHOR]

This paper deals with triangular norms and conorms defined on the extended set N of natural numbers ordered by divisibility. From the fundamental theorem of arithmetic, N can be identified with a lattice of functions from the set of primes to the complete chain {0, 1, 2, ..., +∞}, thus our knowledge about (divisible) t-norms on this chain can be applied to the study of t-norms on N. A characterization of those t-norms on N which are a direct product of t-norms on {0, 1, 2, ..., +∞} is given and, after introducing the concept of T-prime (prime with respect to a t-norm T), a theorem about the existence of a T-prime decomposition is obtained. This result generalizes the fundamental theorem of arithmetic. [ABSTRACT FROM AUTHOR]

Abstract: This study compared 2- to 4-year-olds who understand how counting works (cardinal-principle-knowers) to those who do not (subset-knowers), in order to better characterize the knowledge itself. New results are that (1) Many children answer the question “how many” with the last word used in counting, despite not understanding how counting works; (2) Only children who have mastered the cardinal principle, or are just short of doing so, understand that adding objects to a set means moving forward in the numeral list whereas subtracting objects mean going backward; and finally (3) Only cardinal-principle-knowers understand that adding exactly 1 object to a set means moving forward exactly 1 word in the list, whereas subset-knowers do not understand the unit of change. [Copyright &y& Elsevier]

In this paper, two natural production inventory models based on fuzzy total production inventory cost with the preference of a decision maker are introduced, and combined by natural number parameters in which values are linguistic values in natural language, crisp real number variables, and fuzzy number variables. These are the one natural production inventory model for crisp production quantity, and the other natural production inventory model for fuzzy production quantity. The natural arithmetical operations of both natural numbers and fuzzy numbers by Function Principle are used to compute fuzzy total production inventory cost of each natural production inventory model. Graded k-preference integration representation method is discussed for defuzzifying fuzzy total production inventory cost by preference of decision maker. Furthermore, Extension of the Lagrangean method is used to solve inequality constrain problem in our natural production inventory model. We find out that our optimal solutions can be specified to the classical production inventory model when preference 0.5 of decision maker and natural numbers and fuzzy numbers in our models are crisp real numbers. [ABSTRACT FROM AUTHOR]

Abstract: According to one theory about how children learn the meaning of the words for the positive integers, they first learn that “one,” “two,” and “three” stand for appropriately sized sets. They then conclude by inductive inference that the next numeral in the count sequence denotes the size of sets containing one more object than the size denoted by the preceding numeral. We have previously argued, however, that the conclusion of this Induction does not distinguish the standard meaning of the integers from nonstandard meanings in which, for example, “ten” could mean set sizes of 10, 20, 30,… elements. Margolis and Laurence [Margolis, E., & Laurence, S. (2008). How to learn the natural numbers: Inductive inference and the acquisition of number concepts. Cognition, 106, 924–939] believe that our argument depends on attributing to children “radically indeterminate” concepts. We show, first, that our conclusion is compatible with perfectly determinate meanings for “one” through “three.” Second, although the inductive inference is indeed indeterminate – which is why it is consistent with nonstandard meanings – making it determinate presupposes the constraints that the inference is supposed to produce. [Copyright &y& Elsevier]