Arithmetic Of Natural Numbers Research Articles

Shaping Mathematical Thinking Skills In Primary Schools

The arithmetic material forms the main content of the course. The core of the elementary course consists of arithmetic of natural numbers and basic quantities. In addition, the basic concepts of geometry and algebra are combined in this course.

Natural number bias in arithmetic operations with missing numbers – A reaction time study

When reasoning about numbers, students are susceptible to a natural number bias (NNB): When reasoning about non-natural numbers they use properties of natural numbers that do not apply. The present study examined the NNB when students are asked to evaluate the validity of algebraic equations involving multiplication and division, with an unknown, a given operand, and a given result; numbers were either small or large natural numbers, or decimal numbers (e.g., 3 × _ = 12, 6 × _ = 498, 6.1 × _ = 17.2). Equations varied on number congruency (unknown operands were either natural or rational numbers), and operation congruency (operations were either consistent – e.g., a product is larger than its operand – or inconsistent with natural number arithmetic). In a response-time paradigm, 77 adults viewed equations and determined whether a number could be found that would make the equation true. The results showed that the NNB affects evaluations in two main ways: a) the tendency to think that missing numbers are natural numbers; and b) the tendency to associate each operation with specific size of result, i.e., that multiplication makes bigger and division makes smaller. The effect was larger for items with small numbers, which is likely because these number combinations appear in the multiplication table, which is automatized through primary education. This suggests that students may count on the strategy of direct fact retrieval from memory when possible. Overall the findings suggest that the NNB led to decreased student performance on problems requiring rational number reasoning.

Teoretyczne podstawy dzielenia z resztą liczb naturalnych wraz z uwagami o dzieleniu z resztą liczb całkowitych, wymiernych i rzeczywistych

In this article, I analyze the theoretical foundations of the division with remainder in the arithmetic of natural numbers. As a result of this analysis I justify that the notation a:b=c r s, where a, b, c, s are natural numbers and r denotes, is correct at school mathematics level and does not lead to a contrediction suggested by the author of the article (Semadeni, 1978). As a generalization of the division with remainder of natural numbers, I consider the division with remainder of integers, rational and real numbers.

The Zahl-Anzahl Distinction in Gottlob Frege: Arithmetic of Natural Numbers with Anzahl as a Primitive Term

The starting point is Peano’s expression of the axiomatics of natural numbers in the framework of Leśniewski’s elementary ontology. The author enriches elementary ontology with the so-called Frege’s predication scheme and goes on to propose the formulations of this axiomatic, in which the original natural number (N) term is replaced by the term Anzahl (A). The functor of the successor (S) is defined in it.

On Certain Axiomatizations of Arithmetic of Natural and Integer Numbers

The systems of arithmetic discussed in this work are non-elementary theories. In this paper, natural numbers are characterized axiomatically in two different ways. We begin by recalling the classical set P of axioms of Peano’s arithmetic of natural numbers proposed in 1889 (including such primitive notions as: set of natural numbers, zero, successor of natural number) and compare it with the set W of axioms of this arithmetic (including the primitive notions like: set of natural numbers and relation of inequality) proposed by Witold Wilkosz, a Polish logician, philosopher and mathematician, in 1932. The axioms W are those of ordered sets without largest element, in which every non-empty set has a least element, and every set bounded from above has a greatest element. We show that P and W are equivalent and also that the systems of arithmetic based on W or on P, are categorical and consistent. There follows a set of intuitive axioms PI of integers arithmetic, modelled on P and proposed by B. Iwanuś, as well as a set of axioms WI of this arithmetic, modelled on the W axioms, PI and WI being also equivalent, categorical and consistent. We also discuss the problem of independence of sets of axioms, which were dealt with earlier.

O niektórych inspiracjach Hilbertowskiego programu formalizmu

Hilbert’s program of formalism was undoubtedly a result of many mathematical, logical, and philosophical factors. The aim of this paper is to indicate some rarely highlighted contexts. One important goal of Hilbert’s program was to prove the consistency of arithmetic. The paper shows that Hilbert did not begin the study of this issue only after the discovery of Russell’s paradox. The issue of the consistency of the arithmetic of real numbers was associated with the discovery — by Burali-Forti and Cantor — of the first set-theoretical antinomy, i.e. the antinomy of the greatest ordinal number. Hilbert, already in 1899, asked whether the set of real numbers — to use Cantor’s terminology — was a consistent collection. He then raised the issue of the consistency of the arithmetic of natural numbers in 1904, after the discovery of Russell’s paradox. Fundamental for Hilbert’s mature program of formalism was the distinction between the finitistic and the infinitisticmathematics. The paper points out that the source of this distinction can be found in Brouwer’s proof-theoretical and constructivist criticism of certain theorems of the classical logic. So significant was the criticism that Hilbert had to take it into account in his formalistic reconstruction of classical mathematics. The result was precisely his distinction between the finitistic and the infinitistic mathematics.

Implicit Learning of Arithmetic Regularities Is Facilitated by Proximal Contrast

Natural number arithmetic is a simple, powerful and important symbolic system. Despite intense focus on learning in cognitive development and educational research many adults have weak knowledge of the system. In current study participants learn arithmetic principles via an implicit learning paradigm. Participants learn not by solving arithmetic equations, but through viewing and evaluating example equations, similar to the implicit learning of artificial grammars. We expand this to the symbolic arithmetic system. Specifically we find that exposure to principle-inconsistent examples facilitates the acquisition of arithmetic principle knowledge if the equations are presented to the learning in a temporally proximate fashion. The results expand on research of the implicit learning of regularities and suggest that contrasting cases, show to facilitate explicit arithmetic learning, is also relevant to implicit learning of arithmetic.

Review of Classical Mathematical Logic: The Semantic Foundations of Logic, by Richard L. Epstein, With contributions by Leslaw W. Szczerba

The main purpose of the book is a detailed exposition of methods used in semantical and deductive analysis of ordinary mathematical reasoning by means of classical mathematical logic. The book contains both a statement of modern mathematical logic and many its applications in various fields of mathematics. Introduction gives a brief review of the trends in mathematics in the 19th century which have resulted in a renewed interest in formal logic and its rapid development in the 20th century. Chapters I–XI contain a rather detailed exposition of basic facts in modern mathematical logic including classical propositional and predicate logics, their semantics and axiomatizations. Chapters XII–XVII present applications of mathematical logic to mathematics. In this connection formalizations of group theory, linear orderings, arithmetic of natural numbers, integers, rationals, and real numbers are considered. First-order and second-order arithmetic theories are discussed. Chapters XVIII and XIX written in collaboration with Lesław Szczerba give...

Arithmetizations of Syllogistic à la Leibniz

ABSTRACT Two models of the Aristotelian syllogistic in arithmetic of natural numbers are built as realizations of an old Leibniz idea. In the interpretation, called Scholastic, terms are replaced by integers greater than 1, and s.Ap (“Every s is a p”) is translated as “s is a divisor of p”, sIp (“Some s is a p”) as “g.c.d. (s, p) > 1” (the same letters are used for the replacing numbers as well as for the terms). In the interpretation, called Leibnizian, terms are replaced by proper divisors of a special “Universe number” u < 1 (i.e., s < u, p < u), and sAp is translated as “s is divisible by p”, sIp as ‘l.c.m. (s, p) < u”. Both interpretations are proved to be adequate to the Aristotelian syllogistic. They are extended to syllogistic including term negation and term conjunction as well (and, therefore, all Boolean operations with terms).

The Algebraic Specifications do not have the Tennenbaum Property

It is commonly believed that a programmable model satisfying the axioms of a given algebraic specification guarantees good properties and is a correct implementation of the specification. This convinction might be related to the Tennenbaum's property[Ten] of the arithmetic: every computable model of the Peano arithmetic of natural numbers is isomorphic to the standard model. Here, on the example of stacks, we show a model satisfying all axioms of the algebraic specification of stacks which can not be accepted as a good model in spite of the fact that it is defined by a program. For it enables to ”pop” a stack infinitely many times.

PAL – Propositional Algorithmic Logic

The aim of propositional algorithmic logic is to investigate the properties of program connectives. Complete axiomatic systems for deterministic as well as for nondeterministic interpretations of program variables are presented. They constitute basic sets of tools useful in the practice of proving the properties of program schemes. Propositional theories of data structures, e.g. the arithmetic of natural numbers and stacks, are constructed. This shows that in many aspects PAL is close to first-order algorithmic logic. Tautologies of PAL become tautologies of algorithmic logic after replacing program variables by programs and propositional variables by formulas. Another corollary to the completeness theorem asserts that it is possible to eliminate nondeterministic program variables and replace them by schemes with deterministic atoms.

The numerical abilities of industrial trainee apprentices

Three tests were devised to assess in some depth elementary numerical abilities in a sample of 95 industrial trainee apprentices many of whom had passed CSE mathematics. The tests covered basic aspects of a variety of topics, including arithmetic manipulation of natural numbers, decimals and fractions; approximation; magnitude; ratio; and percentage. The results supported the following general conclusions: (i) arithmetic with natural numbers was performed relatively well, but decimals and especially fractions were performed badly; (ii) there were elementary conceptual difficulties with decimals and fractions, which were compounded in the case of fractions by difficulties of selecting and applying special rules during manipulation; (iii) proficiency in natural number arithmetic was not strongly associated with ability in other basic skills; (iv) frequency of pocket calculator usage as assessed by questionnaire was not significantly related to performance in any of the tests. These results have implications for both the selection and training of school‐leavers. It is also suggested that the tests themselves may be useful for the assessment and diagnosis of basic numeracy.

Theories incomparable with respect to relative interpretability

The present paper concerns the relation of relative interpretability introduced in [8], and arises from a question posed by Tarski: are there two finitely axiomatizable subtheories of the arithmetic of natural numbers neither of which is relatively interpretable in the other? The question was answered affirmatively (without proof) in [3], and the answer was generalized in [4]: for any positive integer n, there exist n finitely axiomatizable subtheories of arithmetic such that no one of them is relatively interpretable in the union of the remainder. A further generalization was announced in [5] and is proved here: there is an infinite set of finitely axiomatizable subtheories of arithmetic such that no one of them is relatively interpretable in the union of the remainder. Several lemmas concerning the existence of self-referential and mutually referential formulas are given in Section 1, and will perhaps be of interest on their own account.

Karel Louis de Bouvère. A method in proofs of undefinability, with applications to functions in the arithmetic of natural numbers. Dissertation Amsterdam 1959. North-Holland Publishing Company, Amsterdam1959, XV + 64 pp. - Karel Louis de Bouvère. Stellingen. Leaflet distributed with the foregoing, 4 pp. unnumbered.

Karel Louis de Bouvère. A method in proofs of undefinability, with applications to functions in the arithmetic of natural numbers. Dissertation Amsterdam 1959. North-Holland Publishing Company, Amsterdam1959, XV + 64 pp. - Karel Louis de Bouvère. Stellingen. Leaflet distributed with the foregoing, 4 pp. unnumbered. - Volume 25 Issue 3

Reduction of arithmetic to logic based on the theory of types without the axiom of infinity and the typical ambiguity of arithmetical constants

The reduction of arithmetic of natural numbers, to logic based on the theory of types meets with two main difficulties: (1) the typical ambiguity of constants determining natural numbers ; (2) the necessity of assuming the axiom of infinity. In this article I show in what way arithmetic of natural numbers may be reduced ? without these defects ? to the system of logic based on the theory of types and extended in the following manner :

Concatenation as a basis for arithmetic

General syntax, the formal part of the general theory of signs, has as its basic operation the operation of concatenation, expressed by the connective ‘⌢’ and understood as follows : where x and y are any expressions, x⌢y is the expression formed by writing the expression x immediately followed by the expression y. E.g., where ‘alpha’ and ‘beta’ are understood as names of the respective signs ‘α’ and ‘β’, the syntactical expression ‘alpha⌢beta’ is a name of the expression ‘αβ’.Tarski and Hermes have presented axioms for concatenation, and definitions of derivative syntactical concepts. Hermes has also related concatenation theory to the arithmetic of natural numbers, constructing a model of the latter within the former. Conversely, Gödel's proof of the impossibility of a complete consistent systematization of arithmetic depended on constructing a model of concatenation theory within arithmetic.

On undecidable statements in enlarged systems of logic and the concept of truth

It is my intention in this paper to add somewhat to the observations already made in my earlier publications on the existence of undecidable statements in systems of logic possessing rules of inference of a “non-finitary” (“non-constructive”) character (§§1–4).I also wish to emphasize once more the part played by the concept of truth in relation to problems of this nature (§§5–8).At the end of this paper I shall give a result which was not touched upon in my earlier publications. It seems to be of interest for the reason (among others) that it is an example of a result obtained by a fruitful combination of the method of constructing undecidable statements (due to K. Gödel) with the results obtained in the theory of truth.1. Let us consider a formalized logical system L. Without giving a detailed description of the system we shall assume that it possesses the usual “finitary” (“constructive”) rules of inference, such as the rule of substitution and the rule of detachment (modus ponens), and that among the laws of the system are included all the postulates of the calculus of statements, and finally that the laws of the system suffice for the construction of the arithmetic of natural numbers. Moreover, the system L may be based upon the theory of types and so be the result of some formalization of Principia mathematica. It may also be a system which is independent of any theory of types and resembles Zermelo's set theory.