Ternary Mathematics and 3D Placement of Logical Elements Justification

Aims/ Objectives: The research presented in the following application aims to prove use of Ternary Maths for calculating machines and to simplify the process of calculating In it we will try to justify the use of triplets and describe how it works. An earlier research presented in “Logical Principles in Ternary Mathematics” [1,2,3] shows that we can transit from one expression of a number such as a "component form" to another, e.g a decimal, or still another, that is it’s vector form [4]. The aim of our further research is to explain why we associate Triplets of numbers in such choice {-1,0,1} and not the numbers 1,2,3 for example, or a set {1,2,3} The explanation seems obvious as a set of decimal numbers consists of 10 entries not 3 At the same time we have to prove that the mentioned set of triplets is a unique and the only one to be used as a Ternary Set or a base, as we might call it, for our calculating machines.

Changing the Order of Factors Does Not Change the Product but Does Affect Students’ Answers, Especially Girls’ Answers

This study is aimed at exploring how different formulations of the same mathematical item may influence students’ answers, and whether or not boys and girls are equally affected by differences in presentation. An experimental design was employed: the same stem-items (i.e., items with the same mathematical content and question intent) were formulated differently and administered to a probability sample of 1647 students (grade 8). All the achievement tests were anchored via a set of common items. Students’ answers, equated and then analysed using the Rasch model, confirmed that different formulations affect students’ performances and thus the psychometric functionality of items, with discernible differences according to gender. In particular, we explored students’ sensitivity to the effect of a typical misconception about multiplication with decimal numbers (often called “multiplication makes bigger”) and tested the hypothesis that girls are more prone than boys to be negatively affected by misconception.

Method for extracting square and cube roots in the binary number system

The article describes in detail the methods of extracting square and cube roots in the binary number system. The method for extracting the square root of a binary number is similar to the corresponding method for decimal numbers, which is called the "column method". As for decimal numbers, when choosing the next digit of the root, twice the current value of the root, represented in the binary system, is used. When extracting the cube root (also "column"), there are two differences from the decimal system. The first is that instead of 300 (the product of 3 and 100), the binary number 1100 is used (that is, the product of the binary equivalents of the numbers 3 and 4). The second difference is that instead of the number 30 (the product of 3 and 10), the binary number 110 is used (that is, the product of binary analogs numbers 3 and 2). To facilitate the selection of the next root digit (0 or 1), a number of standard values have been calculated, depending on the current root value. Assignments for independent work of students are offered.

An Analysis of Students Error on Inversely Proportional Question at Grade VIII F SMP Negeri 9 Palu

The purpose of this study was to describe the students error on inversely proportional question at grade VIII F SMP Negeri 9 Palu. This type of research is qualitative research. Data were collected using data collection techniques, namely tests and interviews. The research subjects consisted of 3 people, namely C students, HD students and YYA students. The results of the study based on the students completion showed the mistakes made by the students in solving the comparison questions against the scores that were done. Based on the solution, conceptual and procedural errors were obtained. Conceptual errors include: a) factual errors consisting of: 1) students have not been able to write down the information contained in the questions, 2) students have not been able to understand the use of the equal sign correctly, 3) students have not been able to change the information contained in the questions into a model mathematics and 4) students have not been able to write conclusions correctly. b) Misconceptions consist of: 1) students do not understand the concept of reversing value comparisons, namely students working on comparison questions of turning values ​​using the principle of value comparisons, 2) students have not been able to operate a reversed value comparison and 3) students do not understand the concept of multiplication and division of integers and decimal numbers. Procedural errors consist of: 1) students have not been able to work on questions using the principle of reversing value comparisons, 2) students make mistakes operating multiplication and division and 3) students have not been able to determine the final result correctly.

An Improved Generalized Quantum-Inspired Evolutionary Algorithm for Multiple Knapsack Problem

This article describes how the 0/1 Multiple Knapsack Problem (MKP), a generalization of popular 0/1 Knapsack Problem, is NP-hard and harder than simple Knapsack Problem. Solution of MKP involves two levels of choice – one for selecting an item to be placed and the other for selecting the knapsack in which it is to be placed. Quantum Inspired Evolutionary Algorithms (QIEAs), a subclass of Evolutionary algorithms, have been shown to be effective in solving difficult problems particularly NP-hard combinatorial optimization problems. QIEAs provide a general framework which needs to be customized according to the requirements of a given problem to obtain good solutions in reasonable time. An existing QIEA for MKP (QIEA-MKP) is based on the representation where a Q-bit collapse into a binary number. But decimal numbers are required to identify the knapsack where an item is placed. The implementation based on such representation suffers from overhead of frequent conversion from binary numbers to decimal numbers and vice versa. The generalized QIEA (GQIEA) is based on a representation where a Q-bit can collapse into an integer and thus no inter conversion between binary and decimal is required. A set of carefully selected features have been incorporated in proposed GQIEA-MKP to obtain better solutions in lesser time. Comparison with QIEA-MKP shows that GQIEA-MKP outperforms it in providing better solutions in lesser time for large sized MKPs. The generalization proposed can be used with advantage in other Combinatorial Optimization problems with integer strings as solutions.

ANALISA KOMPUTASI PARALEL MENGURUTKAN DATA DENGAN METODE RADIX DAN SELECTION

Increasing computing power is now achieved by replacing the programming paradigm with parallel programming. Parallel computing is a method of solving problems by dividing the computational load into small parts of the computation sub-process. This study describes the comparative analysis of parallel computations in the Selection Sort and Radix Sort algorithms. The data used are in the form of whole numbers and decimal numbers totaling 100 to 2 million data. The test was carried out with three scenarios, namely using two processors, four processors, and 3 computers connected to each other via a LAN network. The results showed that the parallel Selection Sort algorithm for small data was better than the parallel Radix Sort. On the other hand, parallel Radix Sort is better for millions of data than Selection Sort.

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.

An Efficient Algorithm for 2-Dimensional Pattern Matching Problem

Pattern matching is the area of computer science which deals with security and analysis of data. This work proposes two 2D pattern matching algorithms based on two different input domains. The first algorithm is for the case when the given pattern contains only two symbols, that is, binary symbols 0 and 1. The second algorithm is in the case when the given pattern contains decimal numbers, that is, the collection of symbols between 0 and 9. The algorithms proposed in this manuscript convert the given pattern into an equivalent binary or decimal number, correspondingly find the cofactors of the same dimension and convert these cofactors into numbers if a particular cofactor number matches indicate the matching of the pattern. Furthermore, the algorithm is enhanced for decimal numbers. In the case of decimal numbers, each row of the pattern is changed to its decimal equivalent, and then, modulo with a suitable prime number changes the decimal equivalent into a number less than the prime number. If the number mismatched pattern does not exist, the complexity of the proposed algorithm is very low as compared to other traditional algorithms.