Decimal Representation Research Articles

Pascal's connection and fractions containing successive Padovan numbers in their decimal representation, reading left to right

In this paper, it is given a generalization of the relation of the repeating decimals which displays the successive terms of the Fibonacci sequence and Pascal’s rows. Additionally, two types of fractions that contain successive terms of the Padovan sequence in their decimal representation are given, by reading from left to right and giving numerical illustrations.

Applets in learning fractions: a look from instrumental genesis to emergency remote teaching

This article investigates contributions by the applets Cuisenaire environment, Geometry and Fraction Models for learning fractions from the measurement perspective in classes based on Exploratory Mathematics Teaching developed during the Emergency Remote Teaching. This qualitative-interpretative research was performed with students of 6th grade groups of Elementary School. Instrumental Genesis is the theoretical-methodological lens to analyze the intentional action of the subject, who using mental schemes of use, transforms the artifact into an instrument. Cuisenaire environment allowed students mobilize strategies to carry out multiplicative comparisons among the rods and establish fractions equivalence understanding symbolic representation and numerical magnitude signaling. The applet Geometry allowed students performing quadrilateral measurement (sides, perimeters, and areas), and using the applet Fraction Models they validate or not the same magnitude of fractional and decimal representations. Associating the three applets favored students to understand the product property, which when multiplying two fractions different from zero and one, the result may be less than one of the two factors.

Design of BCD Adder Using QCA Technology

Abstract: Quantum-dot Cellular Automata (QCA) technology is a promising alternative to traditional complementary metaloxide-semiconductor (CMOS) technology for implementing digital circuits due to its potential for high speed, low power consumption, and small size. In this paper, we present a design for a Binary Coded Decimal (BCD) adder using QCA technology. The BCD adder is a crucial building block for arithmetic operations in decimal representation. The proposed design uses QCA cells to perform the addition operation of two BCD digits, which are encoded as a 4-bit binary number. The design includes a full-adder circuit, a BCD-to-binary converter, and a carry-lookahead adder. Simulation results using QCA Designer software demonstrate the functionality and feasibility of the proposed BCD adder design. The proposed design provides a promising alternative to traditional CMOS-based BCD adders and can contribute to the development of QCA-based digital circuits

Developmental trajectories of numerical magnitude representations of fractions and decimals.

We examined the development of numerical magnitude representations of fractions and decimals from fourth to 12th grade. In Experiment 1, we assessed the rational number magnitude knowledge of 200 Chinese fourth, fifth, sixth, eighth, and 12th graders (92 girls and 108 boys) by presenting fraction and decimal magnitude comparison tasks as well as fraction and decimal 0-1 and 0-5 number line estimation tasks. Magnitude representations of decimals became accurate earlier, improved more rapidly, and reached a higher asymptotic accuracy than magnitude representations of fractions. Analyses of individual differences revealed positive relations between the accuracy of decimal and fraction magnitude representations at all ages. In Experiment 2, we presented an additional set of 24 fourth graders (14 girls and 10 boys) with the same tasks but with the decimals that were being compared varying in the number of decimal digits. The decimal advantage continued to be present for both magnitude comparison and estimation tasks, indicating that the greater accuracy with decimals was not limited to decimals with equal numbers of decimal digits, though unequal numbers of decimal digits did impact performance with decimals on both magnitude comparison and number line estimation tasks. Implications for understanding numerical development and education are discussed. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Verbal labels influence children's processing of decimal magnitudes

Verbal labels for math concepts influence multiple aspects of math learning. In this study, we examined the influence of point labels (e.g., 0.42 as “point four two”), decomposed labels (e.g., “four tenths and two hundredths”), and common-unit labels (e.g., “forty-two hundredths”) on children's processing and representation of decimal magnitudes. We randomly assigned 162 5th- and 6th-graders to briefly learn decomposed, common-unit, or point labels. Children then completed measures of decimal magnitude processing and representation. We found that the place-value labels (i.e., decomposed and common-unit labels) each showed unique advantages in reducing the whole-number bias, and common-unit labels also reduced componential processing. No difference was found in the ratio effect – which served as an index of the precision of decimal magnitude representation - among children from the three conditions. These findings add to our understanding of the role of verbal labels in math learning and have important implications for instructional practices.

Drawing on a computer algorithm to advance future teachers’ knowledge of real numbers: A case study of task design

In our investigation of university students’ knowledge about real numbers in relation to computer algebra systems (CAS) and how it could be developed in view of their future activity as teachers, we used a computer algorithm as a case to explore the relationship between CAS and the knowledge of real numbers as decimal representations. Our work was carried out in the context of a course for university students who aim to become mathematics teachers in high schools. The main data consists of students’ written responses to an assignment of the course and interviews to clarify students’ perspectives in relation to the responses. The analysis of students’ work is based on the anthropological theory of the didactic (ATD). Our results indicate that simple CAS-routines have a potential to help university students (future teachers) to apply their university knowledge on certain problems related to the decimal representation of real number which are typically encountered but not well explained in high school.

AN APPRAISAL ALGORITHM FOR TESTS OF DIVISIBILITY USING MODULAR ARITHMETIC

Knowledge to execute wild conceptual mathematical computations helped immensely even out of the park. Knowing these quick calculations has been of great interest ever since. Divisibility tests were required to know whether a number (large enough) was divisible by a given integer or not? Let 0,m  and a be any integer. The symbol, mod , am was used to represent the residue when a was divided by m. In this piece of treatised work, modulo residue theory was employed to find tests of divisibilty for even numbers < 60 and elaborated the use of modular arithmetic from number theory in finding different tests of divisibility. Particularly, b adic expension of an integer N and its congruence modulo b was used to characterise a given integer regarding its divisibility rule. One of the characterisations was stated and proved that an integer N was divisible by 40 if and only if0 1 2 10( 2 ) a a a was divisible by 40, where 0 1 2 ,, a a awere the digits of N in its decimal representation. Finally, the framework proposed reduced the pitfalls by demonstating each established rule with the help of their recursive applications on large integers.

유리수의 순환마디에 대한 규칙성 분석

일반적으로, 유리수와 관련된 소수표현의 문제는 정수론에서 주로 다룬다. 이 논문에서는 잘 알려진 군론을 이 용하여 이러한 문제를 고찰해 보았다. 먼저, 유리수의 순환소수(repeating decimal)를 찾는 알고리즘을 고찰하여 이를 python으로 코딩을 해 보았다. 그리고 분모가 소수(prime)인 분수의 소수(decimal)표현을 군(group)의 관점에 서 고찰하였다. 구체적으로, 2와 5가아닌 소수 p에 대하여  의 순환마디 개수는 (Z , Xp)의 부분군 <10>의 위 수와 같다. 또한 (a=1,2,...,p-1)의 순환마디의 종류는 부분군 <10>의 잉여류의 개수만큼 나타난다.

Provably Secure and Fast Color Image Encryption Algorithm Based on S-Boxes and Hyperchaotic Map

The World Wide Web is experiencing a daily increase in data transmission because of developments in multimedia technologies. Consequently, each user should prioritize preventing illegal access of this data by encrypting it before moving it over the Internet. Numerous color image encryption schemes have been developed to protect data security and privacy, indifferent to the computation cost. However, most of these schemes have high computational complexities. This research proposes a fast color image scrambling and encryption algorithm depending on different chaotic map types and an S-box that relies on a hyperchaotic map principle. The first step involves converting color image values from decimal representation to binary representation in the scrambling stage by changing the location of the bits according to a proposed swapping algorithm. Next, in the second scrambling stage, the same process occurs after returning color image values from binary representation to decimal representation and generating an S-box with the assistance of two types of chaotic map, namely, a 2D Zaslavsky map and a 3D Hénon map. Thus, this S-box is relied upon to swap the locations of the pixels in the color image. The encryption procedure begins with the production of three key matrices using a hybrid technique that employs two low-complexity types of chaotic map, namely, a 1D Logistic map and a 3D Hénon map, followed by an XORed as a lightweight process between each key generated for the three matrices and the corresponding red, green, and blue image channels. According to the findings, the proposed scheme demonstrates the most efficiency in terms of lowering the computational cost and shows its effectiveness against a wide range of cryptographic attacks.

An Axiomatic System for a Physical or Digital but Continuous 3-Dimensional Euclidean Geometry, Without Infinite Many Points

This paper is concerned with finding an axiomatic system, so as to define the 3-dimensional Euclidean space, without utilizing the infinite ,that can imply all the known geometry for practical applied sciences and engineering applications through computers , and for more natural and perfect education of young people innbsp the Euclidean geometric thinking. In other words by utilizing only finite many visible and invisible points and only finite sets, and only real numbers with finite many digits, in the decimal representation. The inspiration comes from the physical matter , rigid, liquid and gaseous, which consists of only finite manynbsp particles in the physical reality. Or from the way that continuity is produced in a computer screen from only finite many invisible pixels . We present such a system of axioms and explain why it is chosen in such a way. The result is obviously not equivalent, in all the details, with the classical Euclidean geometry.nbsp Our main concern is consistency and adequacy but not independence of the axioms between them. It is obvious that within the space of a single paper, we do not attempt to produce all the main theorems of the Euclidean geometry, but present only the axioms.

Dielectric Polymer Property Prediction Using Recurrent Neural Networks with Optimizations.

Despite the growing success of machine learning for predicting structure-property relationships in molecules and materials, such as predicting the dielectric properties of polymers, it is still in its infancy. We report on the effectiveness of solving structure-property relationships for a computer-generated database of dielectric polymers using recurrent neural network (RNN) models. The implementation of a series of optimization strategies was crucial to achieving high learning speeds and sufficient accuracy: (1) binary and nonbinary representations of SMILES (Simplified Molecular Input Line System) fingerprints and (2) backpropagation with affine transformation of the input sequence (ATransformedBP) and resilient backpropagation with initial weight update parameter optimizations (iRPROP- optimized). For the investigated database of polymers, the binary SMILES representation was found to be superior to the decimal representation with respect to the training and prediction performance. All developed and optimized Elman-type RNN algorithms outperformed nonoptimized RNN models in the efficient prediction of nonlinear structure-activity relationships. The average relative standard deviation (RSD) remained well below 5%, and the maximum RSD did not exceed 30%. Moreover, we provide a C++ codebase as a testbed for a new generation of open programming languages that target increasingly diverse computer architectures.

Application of Monte-Carlo Simulations in Estimation of Pi

The term Pi can be defined as the share of the boundary of any circle to the diameter of that circle. Despite the circle’s dimension, this ratio is the same for all circles. Sometimes it is approximated as 22/7. In decimal form, the value of Pi is approximately 3.14. These approximations result in an error during precise calculations because actually, it is an irrational number. Its decimal representation is non-repeating & non-terminating. In this paper, an effort has been made to estimate the value of Pi using Monte Carlo Simulations.

Soma iterada de algarismos de um número racional

The digital roots S* (x), of a n positive integer is the digit 0 ≤ b ≤ 9 obtained through an iterative digit sum process, where each iteration is obtained from the previous result so that only the b digit remains. For example, the iterated sum of 999999 is 9 because 9 + 9 + 9 + 9 + 9 + 9 = 54 and 5 + 4 = 9. The sum of the digits of a positive integer, and even the digital roots, is a recurring subject in mathematical competitions and has been addressed in several papers, for example in Ghannam (2012), Ismirli (2014) or Lin (2016). Here we extend the application Sast to a positive rational number x with finite decimal representation. We highlight the following result: given a rational number x, with finite decimal representation, and the sum of its digits is 9, so when divided x by powers of 2, the number resulting also has the sum of its digits 9. Fact that also occurs when the x number is divided by powers of 5. Similar results were found when the x digit sum is 3 or 6.

НОВАЯ НЕОСОЗНАННАЯ ПАРАДИГМА В ОСНОВАНИИ МАТЕМАТИКИ И ФИЗИКИ

It is already the case that philosophical foundations of mathematics and physics, need a serious critical analysis and revision of a number of generally accepted assumptions. In the future, this work may lead to a shift in the paradigm related to mathematics and physics. The article deals with the problem of the ideas of actual and potential infinity being not distinguished in the «fragmented thinking» of many mathematicians. We consider it necessary to differentiate between the concept of «representation of a number by an infinite decimal fraction» and the concept of «writing a numeral». A real number can be written in different ways, but every number must be uniquely represented using an infinite decimal fraction. First of all, we overcome the ambiguity of the representation of number 1 by assuming a potentially infinite set of signs of a periodic fraction and an actually infinite set of signs of a non-periodic fraction. This leads to the following harmonious scientific and philosophical system required by broad-minded scientists. 1. Every real number, including 0, (9) is represented by a single point of a continuous number line. 2. Every irrational number in decimal representation, unlike a rational number, does not have the last digit. 3. Real space, as well as past and future time are not mathematically equal and are referents of potentially and actually infinite, countable and uncountable sets. 4. The motion of a quantum micro-object, as a fundamental particle, is mathematically imaginary because a quantum particle has a countable set of points in time that is insufficient to move temporally and an uncountable set of points in space that is excessive for moving along the trajectory. Therefore, the motion of a quantum particle can be described as the path of a point in the plane of a complex variable.

Longest arithmetic progressions of palindromes

A positive integer n is called a palindrome if the decimal representation of n reads the same backward as forward. Domotorp asked and Tao answered that the sequence of palindromes does not contain arbitrarily long arithmetic progressions. From Tao's comments, it is expected that the length of such the progressions should be less than 108. However, his method only leads to an upper bound such as 108 and does not give the exact value of the length. In this article, we determine the length of longest arithmetic progressions of palindromes.

Comments on “Improved designs of digit-by-digit decimal multiplier”

Decimal multiplication is one of the complex operations in the applications that still radix-10 representations are preferred. To improve the performance, Ahmed et al. proposed some fast and area efficient binary-to-decimal converters to be used for both partial product generation and partial product reduction stages of digit-by-digit decimal multiplication. These converters that transform each binary result to binary coded decimal (BCD) representation utilize two new types of binary-to-decimal converter cells in a cascading structure. These cells are called fast binary-to-decimal (FBD) and low-area binary-to-decimal (LABD) converters. In this paper, we show that despite the fact that their architectural framework for digit-by-digit decimal multiplication and the proposed hybrid multi-operand binary-to-decimal converter are correct, some logical equations presented for FBD converter cell are incorrect. Moreover, there exist ambiguities in presenting the logical equations of both FBD and LABD converter cells. Thus, we represent the correct and optimized equations for FBD and LABD converter cells using truth tables.

The Anomaly of Anomaly of Anomaly

The Anomaly of Anomaly of Anomaly

Exploring Real Numbers as Rational Number Sequences With Prospective Mathematics Teachers

The understandings prospective mathematics teachers develop by focusing on quantities and quantitative relationships within real numbers have the potential for enhancing their future students’ understanding of real numbers. In this article, we propose an instructional sequence that addresses quantitative relationships for the construction of real numbers as rational number sequences. We found that the instructional sequence enhanced prospective teachers’ understanding of real numbers by considering them as quantities and explaining them by using rational number sequences. In particular, results showed that prospective teachers reasoned about fractions and decimal representations of rational numbers using long division, the division algorithm, and diagrams. This further prompted their reasoning with decimal representations of rational and irrational numbers as rational number sequences, which leads to authentic construction of real numbers. Enacting the instructional sequence provides lenses for mathematics teacher educators to notice and eliminate difficulties of their students while developing relationships among multiple representations of real numbers.

스프레드시트 환경에서 근사 분수와 직선의 기울기를 이용한 무리수의 대안적 지도법

The purpose of this study was to explore the possibility of interconnection between non-fractional representation and non-circular decimal representation in irrational teaching and learning of school mathematics and to examine the possibility of field application. First, through the analysis of previous studies, it was found that the core of understanding irrational concepts is the intuitive perception of convergence and the inference of acyclicity. And through mathematical analysis, we confirmed that finite decimal sequences and approximate fractional sequences should be learning contents. In this regard, we applied engineering tools in teaching and learning activity of irrational numbers and analyzed the process by which engineering tools become learners’ mathematical tools and then learning occurs, that is, instrument genesis. As a result, in the activity of finding finite fractions of square roots, students gained a meaningful understanding by focusing on the meaning of square root symbols and square roots as objects from inputting functions and operations and using drag. And in the activity to solve the task ‘Can a graph cross a grid point?’, Students could infer the acyclicity of the decimal representation of irrational numbers by observing the approximate fraction as the slope. Thus, this study is expected to be a basic reference of teaching and learning of irrational numbers to construct a mental image of real numbers completeness axiom.

The Real Numbers: Not All Decimals Are Fractions

Legend has it that the first person in ancient Greece who discovered that there are numbers that cannot be written as fractions was thrown overboard from a ship. Centuries later, while we regularly use numbers that cannot be written as fractions, those numbers that can be written as fractions remain powerful tools. What makes fractions so special? We explore how we can recognize the decimal representation of fractions and how fractions can be used to approximate any real number as closely as we wish.