Multiplication Of Natural Numbers Research Articles

Logic, Game Theory, and Social Choice: What Do They Have in Common?

The answer to the question above is that in all these domains axiomatic characterizations are given of, respectively, mathematical reasoning, certain notions from game theory, and certain social choice rules. The meaning of the completeness theorem in logic is that mathematical reasoning can be characterized by a handful of certain (logical) axioms and rules. If we apply mathematical reasoning to elementary arithmetic, i.e., the addition and multiplication of natural numbers, it turns out that almost all true arithmetical statements, for instance, ∀x∀y[x+y=y+x], can be logically deduced from the axioms of Peano. However, in 1931 Kurt Gődel showed that the axioms of Peano do not (fully) characterize the addition and multiplication of the natural numbers, more precisely, that there are certain special self-referential arithmetical sentences that, although true, cannot be deduced from Peano’s axioms. There are axiomatic characterizations of several social choice and ranking rules that say that a given rule is the only one satisfying a particular set of axioms. Arrow’s impossibility theorem in social choice theory tells us that a certain set of, at first sight, quite reasonable axioms for a social ranking rule turns out to be inconsistent. Consequently, a social ranking rule that satisfies the axioms in question cannot exist. Finally, many notions from game theory, such as the Shapley–Shubik and the Banzhaf index, may also be characterized by a set of axioms.

The effects of computer-assisted instruction on the achievement, attitudes and retention of fourth grade mathematics students in North Cyprus

The purpose of this study was to examine the effects of the educational software Frizbi Mathematics 4 on 4th grade student's mathematics achievement, retention, attitudes toward mathematics and attitude toward computer assisted learning. Two groups (experimental and control) of students from the state primary school in Gazimagusa, North Cyprus were used in this study. The control group was taught using a lecture-based traditional instruction and experimental group was taught using educational software, namely Frizbi Mathematics 4. The control group consisted of 26 students while the experimental group consisted of 29 students. The groups were compared on achievement of mathematics, retention, and attitude toward mathematics and computer assisted learning. The study included three units, Multiplication of Natural Numbers, Division of Natural Numbers, and Fractions. Scores on achievement tests were collected three times; at the beginning of the study, immediately after the intervention, and 4 months later. The mathematics attitude scale and computer assisted learning attitude scale were administrated only two times; at the beginning of the study and immediately after the completion of the study. A series of ANOVAs for repeated measures revealed significant difference between the groups on the post achievement tests and attitude scales in favor of experimental group. However, statistically significant differences in favor of the treatment group, on the retention tests were attained on the multiplication and division units but not on fractions. The evidence indicates that Frizbi Mathematics 4 for learning and teaching mathematics at the primary school level in North Cyprus is an effective tool.

Strategy Independent Reduction Lengths in Rewriting and Binary Arithmetic

In this paper we give a criterion by which one can conclude that every reduction of a basic term to normal form has the same length. As a consequence, the number of steps to reach the normal form is independent of the chosen strategy. In particular this holds for TRSs computing addition and multiplication of natural numbers, both in unary and binary notation.

Finite quantifier hierarchies in relational algebras

For a given structure of finite signature, one can construct a hierarchy of classes of relations definable in this structure according to the number of quantifier alternations in the formulas expressing the relations. In ordinary examples, this hierarchy is either infinite (as in the arithmetic of addition and multiplication of natural numbers) or stabilizes very rapidly (in structures with decidable theories, such as the field of real numbers). In the present paper, we construct a series of examples showing that the above-mentioned hierarchy may have an arbitrary finite length. The proof employs a modification of the Ehrenfeucht game.

On ordering and multiplication of natural numbers

Even if the ordering of all natural number is (known to be) not definable from multiplication of natural numbers and ordering of primes, there is a simple axiom system in the language $(\times,<,1)$ such that the multiplicative structure of positive integers has a unique expansion by a linear order coinciding with the standard order for primes and satisfying the axioms – namely the standard one.

Decidability and undecidability of theories with a predicate for the primes

AbstractIt is shown, assuming the linear case of Schinzel's Hypothesis, that the first-order theory of the structure 〈ω; +, P〉, where P is the set of primes, is undecidable and, in fact, that multiplication of natural numbers is first-order definable in this structure. In the other direction, it is shown, from the same hypothesis, that the monadic second-order theory of 〈ω S, P〉 is decidable, where S is the successor function. The latter result is proved using a general result of A. L. Semënov on decidability of monadic theories, and a proof of Semënov's result is presented.