scholarly journals PASSAGEM DOS NÚMEROS NATURAIS AOS NÚMEROS RACIONAIS EM LIVROS DIDÁTICOS. PASSAGE OF NATURAL NUMBERS TO RATIONAL NUMBERS IN TEXTBOOKS. PASO DE LOS NÚMEROS NATURALES HASTA LOS NÚMEROS RACIONALES EN LIBROS DIDÁCTICOS.

2018 ◽  
Vol 12 (01) ◽  
pp. 1-11
Author(s):  
ELOAR BARRETO FEITOZA SA
Keyword(s):  
Rational Numbers ◽  
Natural Numbers

scholarly journals Avoiding Fractional Powers over the Natural Numbers

10.37236/6678 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Lara Pudwell ◽  
Eric Rowland
Keyword(s):  
Fixed Point ◽  
Rational Numbers ◽  
Finite Alphabet ◽  
Its Sequence ◽  
Natural Numbers ◽  
Fractional Powers ◽  
Free Word

We study the lexicographically least infinite $a/b$-power-free word on the alphabet of non-negative integers. Frequently this word is a fixed point of a uniform morphism, or closely related to one. For example, the lexicographically least $7/4$-power-free word is a fixed point of a $50847$-uniform morphism. We identify the structure of the lexicographically least $a/b$-power-free word for three infinite families of rationals $a/b$ as well many "sporadic" rationals that do not seem to belong to general families. To accomplish this, we develop an automated procedure for proving $a/b$-power-freeness for morphisms of a certain form, both for explicit and symbolic rational numbers $a/b$. Finally, we establish a connection to words on a finite alphabet. Namely, the lexicographically least $27/23$-power-free word is in fact a word on the finite alphabet $\{0, 1, 2\}$, and its sequence of letters is $353$-automatic.

Classical Arithmetization

2019 ◽  
pp. 26-50
Author(s):  
John Stillwell
Keyword(s):  
Number System ◽  
Rational Numbers ◽  
Continuous Functions ◽  
Complex Numbers ◽  
Natural Numbers ◽  
Infinite Sums ◽  
Why Questions

This chapter describes how one proceeds from natural to rational numbers, then to real and complex numbers, and to continuous functions—thus arithmetizing the foundations of analysis and geometry. The definitions of integers and rational numbers show why questions about them can, in principle, be reduced to questions about natural numbers and their addition and multiplication. This is what it means to say that the natural numbers are a foundation for the integer and rational numbers. But the next steps in the arithmetization project go beyond algebra. By admitting sets of rational numbers, one can enlarge the number system to one that admits certain infinite operations, such as forming infinite sums. This is crucial to building a foundation for analysis. As such, the chapter turns to the foundations of the natural numbers themselves, the “Peano axioms,” which gives a first glimpse of the logic underlying the arithmetization project.

scholarly journals Spiking Neural P Systems with Weights

2010 ◽  
Vol 22 (10) ◽  
pp. 2615-2646 ◽  
Author(s):  
Jun Wang ◽  
Hendrik Jan Hoogeboom ◽  
Linqiang Pan ◽  
Gheorghe Păun ◽  
Mario J. Pérez-Jiménez
Keyword(s):  
Rational Numbers ◽  
P Systems ◽  
Natural Numbers ◽  
Open Problems ◽  
Spiking Neural P Systems ◽  
Semilinear Sets ◽  
The Family ◽  
Neuron Fire ◽  
Hard Problems

A variant of spiking neural P systems with positive or negative weights on synapses is introduced, where the rules of a neuron fire when the potential of that neuron equals a given value. The involved values—weights, firing thresholds, potential consumed by each rule—can be real (computable) numbers, rational numbers, integers, and natural numbers. The power of the obtained systems is investigated. For instance, it is proved that integers (very restricted: 1, −1 for weights, 1 and 2 for firing thresholds, and as parameters in the rules) suffice for computing all Turing computable sets of numbers in both the generative and the accepting modes. When only natural numbers are used, a characterization of the family of semilinear sets of numbers is obtained. It is shown that spiking neural P systems with weights can efficiently solve computationally hard problems in a nondeterministic way. Some open problems and suggestions for further research are formulated.

A pair of rational double sequences

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Serik Altynbek ◽  
Heinrich Begehr
Keyword(s):  
Natural Number ◽  
Complex Plane ◽  
Rational Numbers ◽  
The Other ◽  
Number Sequence ◽  
Double Sequences ◽  
Natural Numbers ◽  
Reflection Process ◽  
Natural Way

Abstract Double sequences appear in a natural way in cases of iteratively given sequences if the iteration allows to determine besides the successors from the predecessors also the predecessors from their followers. A particular pair of double sequences is considered which appears in a parqueting-reflection process of the complex plane. While one end of each sequence is a natural number sequence, the other consists of rational numbers. The natural numbers sequences are not yet listed in OEIS Wiki. Complex versions from the double sequences are provided.

Not so rational: A more natural way to understand the ANS

2021 ◽  
Vol 44 ◽  
Author(s):  
Eli Hecht ◽  
Tracey Mills ◽  
Steven Shin ◽  
Jonathan Phillips
Keyword(s):  
Number System ◽  
Rational Numbers ◽  
Approximate Number System ◽  
Natural Numbers ◽  
Wide Range ◽  
Natural Way

Abstract In contrast to Clarke and Beck's claim that that the approximate number system (ANS) represents rational numbers, we argue for a more modest alternative: The ANS represents natural numbers, and there are separate, non-numeric processes that can be used to represent ratios across a wide range of domains, including natural numbers.

scholarly journals The problem with percentages

2018 ◽  
Vol 373 (1740) ◽  
pp. 20160519 ◽  
Author(s):  
Jennifer A. Jacobs Danan ◽  
Rochel Gelman
Keyword(s):  
Elementary School ◽  
Elementary School Students ◽  
Research University ◽  
Rational Numbers ◽  
Core Knowledge ◽  
Natural Numbers ◽  
Major Research ◽  
School Students ◽  
Simple Arithmetic ◽  
Numerical Abilities

A great many students at a major research university make basic conceptual mistakes in responding to simple questions about two successive percentage changes. The mistakes they make follow a pattern already familiar from research on the difficulties that elementary school students have in coming to terms with fractions and decimals. The intuitive core knowledge of arithmetic with the natural numbers makes learning to count and do simple arithmetic relatively easy. Those same principles become obstacles to understanding how to operate with rational numbers. This article is part of a discussion meeting issue ‘The origins of numerical abilities’.

scholarly journals Hypertranscendental elements of a formal power-series ring of positive characteristic

1992 ◽  
Vol 125 ◽  
pp. 93-103 ◽  
Author(s):  
Kayoko Shikishima-Tsuji ◽  
Masashi Katsura
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Positive Characteristic ◽  
Rational Numbers ◽  
Natural Numbers ◽  
Real Numbers ◽  
Power Series Ring ◽  
Series Ring ◽  
Formal Power Series Ring ◽  
Formal Power

Throughout this paper, we denote by N, Q and R the set of all natural numbers containing 0, the set of all rational numbers, and the set of all real numbers, respectively.

On natural numbers, integers, and rationals

1949 ◽  
Vol 14 (2) ◽  
pp. 81-84
Author(s):  
Frederic B. Fitch
Keyword(s):  
Natural Number ◽  
Rational Numbers ◽  
Simple Theory ◽  
Natural Numbers ◽  
Guiding Principle ◽  
Relative Product ◽  
Whole Number

A theory of natural numbers will be outlined in what follows. This theory will also be extended to give an account of positive and negative integers and positive and negative rational numbers. The system of logic used will be that of Whitehead and Russell's Principia mathematica with the simple theory of types. It will be assumed that the reader is familiar with the more elementary properties of relations and with such notions as the relative product of two relations, the square of a relation, the cube of a relation, and the various other whole-number powers of relations.The guiding principle of this theory is that the natural number zero is to be regarded as the relation of the zevoth power of a relation А to А itself, and the natural number 1 is to be regarded as the relation of the first power of a relation А to А itself, and the natural number 2 is to be regarded as the relation of the square of a relation А to А itself, and so on.

1. Numbers and algebra

2015 ◽  
pp. 1-10
Author(s):  
Peter M. Higgins
Keyword(s):  
Fundamental Theorem ◽  
Common Factor ◽  
Number System ◽  
Rational Numbers ◽  
Euclidean Algorithm ◽  
Greatest Common Divisor ◽  
Natural Numbers ◽  
Division Algorithm ◽  
Fundamental Theorem Of Arithmetic ◽  
And Algebra

‘Numbers and algebra’ introduces the number system and explains several terms used in algebra, including natural numbers, positive and negative integers, rational numbers, number factorization, the Fundamental Theorem of Arithmetic, Euclid’s Lemma, the Division Algorithm, and the Euclidean Algorithm. It proves that any common factor c of a and b is also a factor of any number of the form ax + by, and since the greatest common divisor (gcd) of a and b has this form, which may be found by reversing the steps of the Euclidean Algorithm, it follows that any common factor c of a and b divides their gcd d.

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