Positional Number Systems Research Articles

The Archimedean Origin of Modern Positional Number Systems

A symbolic analysis of Archimedes’s periodical number system is developed, from which a natural link emerges with the modern positional number systems with zero. After the publication of Fibonacci’s Liber Abaci, the decimal Indo-Arabic positional system was the basis of the algorithmic and algebraic trend of modern mathematics, but even if zero plays a crucial role in algebra and mathematical analysis, zeroless positional systems show the same capability of producing efficient arithmetical algorithms based on operation tables over digits. The crucial role of digits is assessed, by considering a representation of numbers based on strings in lexicographic order. A new algorithm for the determination of decimal periods is presented by remarking on the cruciality of this topic in number theory. Periods of ordinal numbers and enumerations of recursive enumerability are shortly recalled. Concluding remarks are formulated about the deep relationship between numbers and information, which shed new light on a red line passing through the whole history of mathematics.

THE STRUCTURE OF THE COMPUTER SYSTEM IN THE RESIDUAL CLASSES

The subject of the article is the formulation and solution of the inverse problem of optimal redundancy in the system of residual classes (RNS) based on the use of the dynamic programming method. The solution of this problem makes it possible to improve the reliability of the operation of computer systems and components (CSC) in the RNS. The purpose of the article is to increase the reliability of the functioning of CSC, which are built on the basis of the use of RNS, without reducing the speed of calculations, as well as to calculate and compare the reliability, in terms of the probability of failure-free operation, of CSC in RNS and a tripled computing system that operates in a positional binary number system (PNS). Tasks: to analyze the influence of the number system used on the reliability of the CSC, taking into account the primary and secondary redundancy; to synthesize a computing system in RNS for a l-byte bit grid based on the use of the passive fault tolerance method (constant structural redundancy); formulate and solve the inverse problem of optimal redundancy in RNS based on the use of the dynamic programming method; to check the correctness of the results obtained, calculate the conditional amount of computer system equipment in the residual classes; evaluate the efficiency of using RNS to improve reliability when building a redundant CSC in relation to a redundant CSC in the PNS. Research methods: methods of analysis and synthesis of computer systems, number theory, coding theory in RNS, reliability theory. The following results are obtained. The paper shows that the use of PNS as a number system does not allow a radical increase in the performance and reliability of CSC. In this regard, the article developed the concept of using RNS as a number system for constructing a CSC. Based on this, the inverse problem of optimal redundancy in RNS is formulated and solved. Conclusions. As shown by the results of calculations and comparative analysis, the use of RNS provides a higher reliability of the CSC than the majority three-channel computing system in the PNS. The obtained research results can be used for the synthesis of fault-tolerant computer structures in RNS.

Web quests as a means of forming digital competence of schoolchildren

One of the main tasks of the modern school is the formation of a person with a sufficient level of digital competence. The formation of digital competence of schoolchildren will be facilitated by the inclusion of modern information and communication technologies in training as a subject of study, a means of interaction between participants in the educational process, a means of solving educational problems and planning educational activities. The use of a web quest at informatics lessons or for organizing independent work of students allows you to create conditions for the use of information and communication technologies in all the indicated aspects.The article describes an example of developing a web quest on the theme "Possibilities of the Python programming language in working with positional number systems" for students of 8th grade. This example made it possible to formulate general requirements for the structure of the web quest, for the information resource developed to accompany the web quest, for the forms of organizing the interaction of participants in the educational process, for ways of presenting the results of the work of students, for assessment procedures.

Investigating the rank of the number in a residue number system

The rank of a number in a residue number system indicates the count of transitions through a range when a number is converted to a positional number system and allows for more efficient non-modular operations and detection of values out of range. The main approach to calculate the rank is the use of the Chinese Remainder Theorem. In this article the approach which allows to compute the rank using a set of special numbers for which ranks are computed in advance is proposed. The simulation of the considered methods is done in the Python programming language. The results are analyzed and recommendations for further use of the proposed method are given.

Reverse Mathematical Modelling

Background: Research in mathematics education shows that mathematical modelling is a practice that seeks to “translate” problem situations into possible mathematical models without, however, explaining the complexity involved in the reverse formulation, starting from the mathematical model to delimit the type of situation. Objective: To highlight the problem of reverse mathematical modelling, in the sense of a reverse formulation that goes from the mathematical model to the situation. Design: For this, a course of study and research was carried out guided by the investigative cycle of mathematical modelling that is based on theoretical-methodological resources of the anthropological theory of the didactic. Setting and Participants: Pre-service teachers of a teaching degree at a public institution were faced with a problem in an unusual context about the decimal number system and, more broadly, the positional number system. Data collection and analysis: We present an empirical approach based on research carried out by Ferreira (2020) with teachers in initial training. Results: The empirical results observed confirm the hypothesis of the existence of the problem of reverse mathematical modelling, even in the face of normative models, in the sense that they can faithfully describe a real situation. Conclusions: Ultimately, the study of a type of problem in an unusual context, in addition to highlighting the encounter of teachers with different objects of knowledge, revealed the remarkable difficulty in delimiting the type of quantification situation that can be associated with the mathematical model of the number and as stimulate future research on the teaching of reverse mathematical modelling.

MATHEMATICAL MODEL OF COMPUTER SYSTEM RELIABILITY IN RESIDUAL CLASSES

The subject of the article is the construction of a mathematical model of the reliability of a computer system (CS) that operates in a system of residual classes (RNS). This mathematical model is based on the use of structural sliding reservation. The purpose of the article is to increase the reliability of the CS, which operates in the system of residual classes, as well as to calculate and compare the reliability, in terms of the probability of failure-free operation, of the CS in the RNS and the ternary computing system, which operates in the positional binary number system (PBNS). Tasks: to analyze the influence of the number system used on the reliability of the CS; to investigate the properties of the RNS and determine their influence on the structure of the CS in the RNS; build a mathematical model of reliability and, on its basis, perform the calculation and comparative analysis of the reliability of the CS in the RNS and the tripled CS in the PBNS. Research methods: methods of analysis and synthesis of computer systems, number theory, coding theory in RNS, reliability theory. The following results are obtained. The paper shows that the provision of a given level of reliability in the design of the CS using the RNS is due to the presence of various types of redundancy at the same time: structural, informational, temporal, functional and load. In RNS, these types of redundancy can be effectively used to improve the reliability of the CS. The CS in the RNS represents a computational structure that is identical in structure and principle of operation to the sliding structural redundancy model in the PBNS in the case of a loaded (hot) mode of operation of the redundant elements. Proceeding from this, the paper presents a mathematical model of the reliability of the CS in the RNS. In this paper, the calculation and comparative analysis of the reliability of a tripled computing structure in a PBNS with an ideal majority element and a CS in an RNS with an ideal reliability automaton are carried out. Conclusions. As shown by the results of calculations and comparative analysis in some time intervals of operation, the probability of non-failure operation of the CS in the RNS is higher than the probability of non-failure operation of the aircraft in the PBNS with a tripled majority structure. This implies the effective use of the RNS to improve the reliability of the CS.

Performance Analysis of Hardware Implementations of Reverse Conversion from the Residue Number System

The Residue Number System (RNS) is a non-positional number system that allows parallel computations without transfers between digits. However, some operations in RNS require knowledge of the positional characteristic of a number. Among these operations is the conversion from RNS to the positional number system. The methods of reverse conversion for general form moduli based on the Chinese remainder theorem and the mixed-radix conversion are considered, as well as the optimized methods for special form moduli. In this paper, a method is proposed that develops the authors’ ideas based on the modified mixed-radix conversion and reference points. The modified method based on the mixed-radix conversion in this case makes it possible to replace the operation of finding the residue of division by a large modulo with the sequential calculation of the residue. The method of reference points allows to reduce the size of the stored information compared to the use of ROM to store all the residues of RNS. The application of this approach makes it possible to find a balance between the speed of the calculation and the hardware used, by varying the number of moduli of one method and the other.

NEW METHOD OF ON-LINE SUCCESSIVE-APPROXIMATION ADC CALIBRATION

A new method of successive approximation ADC calibration without interruption of the main conversion process is proposed. The method is based on the use of information redundancy in the form of redundant positional number systems. The method is based on the selection and analysis of unused combinations in the redundant ADC conversion characteristic.

Teacher Training and the use of the Abacus in the Practice of Quantifying Physical Quantities

Background: Several studies in mathematics education show the use of the physical abacus in the activities of teachers in initial graduation, emphasising, in their manipulation, practices with addition or subtraction operations, or yet, which express a direct translation of the numeral to the physical abacus and, vice versa, without, however, explaining part of the complexity involved the social practice of quantifying physical quantities, which even makes possible the structuring of the numeral. Objective: To show that the use of the abacus by teachers in initial training may not ensure the realisation or mastery of the practice of quantifying physical quantities with discrete units. Design: In this sense, a study and research path (SRP) was carried out based on the training of early years teachers. Environment and participants: 25 students in initial training of a degree course at a public educational institution participated in an activity on decimal number system (DNS), but largely positional number system (PNS), proposed based on a problem in an unusual context (for them), they mobilised to face and answer questions. Data collection and analysis: We present an excerpt from the empirical research forwarded by Ferreira (2020) with teachers in initial training. Results: The data observed in the empirical confirms the hypothesis of the existence of a problem regarding the use of the abacus as a possible facilitator in the structuring of non-decimal numbers through the practice of quantifying physical quantities. Conclusions: The results found with the teachers revealed, in addition to the problem of teacher training on what to teach and how to teach numerals, that using the abacus made it difficult, if not hindered, the practice of quantifying physical quantities with discrete units. We also found changes in the quality of the teachers’ relationships with structuring decimal and non-decimal numbers.

THE CONCEPT OF PERFORMING THE ADDITION OPERATION IN THE SYSTEM OF RESIDUAL CLASSES

The subject of the article is the development of a method for implementing the arithmetic operation of adding the residuals of numbers, which are represented in the system of residual classes (RNS). This method is based on the use of positional binary adders. The purpose of the article is to improve the performance of computer systems (CS) and their components by introducing new ways of organizing calculations based on the use of RNS. Tasks: to analyze and identify the shortcomings of the existing number systems that are used in the construction of computer systems and components; explore possible ways to eliminate the identified deficiencies; explore the structure of binary positional adders, taking into account the scheme for adding two residues of numbers modulo RNS; to develop a method for constructing adders modulo RNS, which is based on the use of a set of binary single-digit positional adders. Research methods: methods of analysis and synthesis of computer systems, number theory, coding theory in RNS. The following results are obtained. The paper shows that one of the promising ways to improve the performance of the CS is the use of RNS. The mathematical basis of RNS is the Chinese remainder theorem, which states that an integer operation on one large modulus can be replaced by a set of operations on coprime small modules. This opens up broad prospects for optimizing calculations. On the one hand, it is possible to significantly simplify the performance of complex and cumbersome calculations, including on low-resource computing platforms. On the other hand, calculations for different modules can be performed in parallel, which increases the performance of the CS. Conclusions. The article considers the operation of adding two numbers. This operation is the basis for both traditional positional number systems and RNS, i.e. forms the computational basis of all existing CS components. A new method for calculating the sum of the residuals of numbers modulo an arbitrary is proposed, and examples are given that clearly demonstrate the effectiveness of the proposed method. This method can be used in various computer applications, including for improving computing performance, ensuring fault tolerance, etc.

Algorithms of direct transformation of a position number to the system of residual classes and its inverse transformation

Increasing the speed of computer structures is associated with the involvement of new principles based on the use of the number system of residual classes. When performing a number of computational operations using a non-positional number system of residual classes, the problem of direct conversion of numbers from the positional number system to the number system of residual classes and the reverse conversion of numbers from the number system of residual classes to the positional one is solved. system. A number of works contain methods for implementing these operations of varying complexity, but their solution is far from perfect. The results of the study indicate the possibility of obtaining a more efficient solution that simplifies the practical implementation of operations for direct conversion of numbers from the positional number system to the number system of residual classes and inverse conversion of numbers. The aim of the study is to analyze the implementation of the operations of direct conversion of numbers from the positional number system to the number system of residual classes and the reverse conversion of numbers. The tools of the research methodology are system analysis, number theory and the Chinese remainder theorem. The research methodology consists in the analysis of special tables of residues by modules. An algorithm for performing complex operations based on the use of a polyadic code of a number presented in the number system of residual classes is proposed. A simple and efficient approach is proposed using iterative table operations with residuals. For each value of the given residuals, the constants are obtained by a simple selection from the corresponding table.The method underlying the implementation of these complex operations is to sum the chosen constants to the desired value, while the corresponding constants are subtracted from the values of the known remainders until the latter vanishes. The method is algorithmically simple and uncomplicated for circuit implementation, and it should be considered as one of the areas of research to improve the efficiency of computations in the system of residual classes.

An Efficient Method for Comparing Numbers and Determining the Sign of a Number in RNS for Even Ranges

Fully Homomorphic Encryption (FHE) permits processing information in the form of ciphertexts without decryption. It can ensure the security of information in common technologies used today, such as cloud computing, the Internet of Things, and machine learning, among others. A primary disadvantage for its practical application is the low efficiency of sign and comparison operations. Several FHE schemes use the Residue Number System (RNS) to decrease the time complexity of these operations. Converting from the RNS to the positional number system and calculating the positional characteristic of a number are standard approaches for both operations in the RNS domain. In this paper, we propose a new method for comparing numbers and determining the sign of a number in RNS. We focus on the even ranges that are computationally simple due to their peculiarities. We compare the performance of several state-of-art algorithms based on an implementation in C++ and relatively simple moduli with a bit depth from 24 to 64 bits. The experimental analysis shows a better performance of our approach for all the test cases; it improves the sign detection between 1.93 and 15.3 times and the number comparison within 1.55–11.35 times with respect to all the methods and configurations.

FPGA-Implemented Fractal Decoder with Forward Error Correction in Short-Reach Optical Interconnects.

Forward error correction (FEC) codes combined with high-order modulator formats, i.e., coded modulation (CM), are essential in optical communication networks to achieve highly efficient and reliable communication. The task of providing additional error control in the design of CM systems with high-performance requirements remains urgent. As an additional control of CM systems, we propose to use indivisible error detection codes based on a positional number system. In this work, we evaluated the indivisible code using the average probability method (APM) for the binary symmetric channel (BSC), which has the simplicity, versatility and reliability of the estimate, which is close to reality. The APM allows for evaluation and compares indivisible codes according to parameters of correct transmission, and detectable and undetectable errors. Indivisible codes allow for the end-to-end (E2E) control of the transmission and processing of information in digital systems and design devices with a regular structure and high speed. This study researched a fractal decoder device for additional error control, implemented in field-programmable gate array (FPGA) software with FEC for short-reach optical interconnects with multilevel pulse amplitude (PAM-M) modulated with Gray code mapping. Indivisible codes with natural redundancy require far fewer hardware costs to develop and implement encoding and decoding devices with a sufficiently high error detection efficiency. We achieved a reduction in hardware costs for a fractal decoder by using the fractal property of the indivisible code from 10% to 30% for different n while receiving the reciprocal of the golden ratio.

Метод аппаратной реализации сверточной нейронной сети на основе системы остаточных классов

Convolutional Neural Networks (CNN) show high accuracy in pattern recognition solving problem but have high computational complexity, which leads to slow data processing. To increase the speed of CNN, we propose a hardware implementation method with calculations in the residue number system with moduli of a special type and . A hardware simulation of the proposed method on Field-Programmable Gate Array for LeNet-5 CNN is trained with the MNIST, FMNIST, and CIFAR-10 image databases. It has shown that the proposed approach can increase the clock frequency and performance of the device by 11%-12%, compared with the traditional approach based on the positional number system.

Residual classes based mathematical model of the computer system's reliability

It is also known, and it is practically shown that the use of a unposition number system in residual classes (RNS) allows you to drastically increase the speed of the computer system (CS). There are various approaches and difficulties regarding to the assessment of the reliability of the CS in RNS. So, the existing options of mathematical models do not always satisfy a comprehensive assessment of the reliability of the CS in RNS. The variant of mathematical model of reliability of CS, that functions in RNS, is considered in the article, based on the use of a mathematical model of structural sliding backup in the case of a loaded mode of operation of the backup elements, which is used in the positional binary number system (PBNS). A calculation and comparative security of the trooped computing structure (TCS) analysis was carried out in PBNS with an ideal majority element and the CS in RNS with an ideal reliability machine. The results of analysis showed the following. At the initial stage of the functioning of computational systems, reliability, by probability of trouble-free operation, the CS in RNS with two control bases above the reliability of the trooked positional computing structure, widely used in the PDS and with a smaller addition of an additional equipment insertion. It supposes the effective use of RNS for the increase of reliability of CS and computing devices on the initial stage of their functioning. For example, for the increase of reliability the practical use of unposition code structures is possible in RNS in the side digital calculable complexes of ballistic rockets and in the computing devices of pilotless aircrafts of brief action.

Istorijska geneza pojma broja i brojevnih sistema - mogućnosti primene u početnoj nastavi matematike (I)

The notion of natural number, as the basis of the entire further development of mathematics, appears as a result of gradual abstraction. The very notion of number has not been precisely defined and has changed throughout history, as has the way of recording. This paper aims to present the origin of natural numbers and their development to the present form. The essentially most important number systems that have evolved in parallel with the development of human consciousness, and which are guided by the need for man to record important events and facts, are discussed and presented. The importance of positional number systems, such as the Indo-Arabic number system we use today, was emphasized. Numerous studies show that the use of the history of mathematics in the teaching of mathematics can increase students 'motivation and students' interest in the subject. Based on a brief overview of existing research on the use of the history of mathematics in mathematics teaching, the paper presents examples of tasks for classroom teaching, which include elements of the history of natural numbers and number systems.

Istorijska geneza pojma broja i brojevnih sistema - mogućnosti primene u početnoj nastavi matematike (II)

The history of mathematics and its discoveries have a significant role in the modern teaching of mathematics because they increase the motivation and interest of students in the subject. This paper aims to consider the possibilities of its application in practice based on a brief overview of existing research on the use of the history of mathematics in mathematics teaching. Examples of tasks related to the history of natural numbers and number systems that can be applied in the initial teaching of mathematics are presented. In the conclusion of the paper, an overview of number systems through history is summarized (Part I, NORMA XXVII, 1/2022), with special reference to the importance of positional number system, i.e. the advantages of the Indo-Arabic system, with recommendations for introducing these contents in elementary mathematics

ВИСОКОЛІНІЙНА СИСТЕМА АНАЛОГ-КОД-АНАЛОГ З ВАГОВОЮ НАДЛИШКОВІСТЮ НА БАЗІ ПЕРЕТВОРЮВАЧІВ СТРУМІВ

At present, analog-to-digital and digital-to-analog converters are widely used for operations with signals and allow recording and storing signals, outputting a signal in analog form, performing operations with signals, and generating arbitrary waveforms. In a number of applications of analog-to-digital and digital-to-analog converters for signal processing, high resolution, low noise and high linearity of the analog-to-digital-to-analog transfer characteristic are required to provide low non-linear, frequency, phase distortion of signals and a high signal-to-noise ratio. A method for constructing a high-linear analog-code-analog system with weight redundancy based on current converters is considered. The features of the use of redundant positional number systems in analog-to-digital and digital-to-analog conversion are considered. The ability of redundant positional number systems to ensure the absence of «gaps» in the conversion characteristic caused by deviations of the real weights of digits from their calculated values, as well as the ability to significantly correct the initial errors of digital-to-analog converters, namely the linearity of the conversion characteristic, is indicated. The ability of redundant positional number systems to provide, despite the lengthening of the bit grid, a significant increase in performance is indicated. The scheme of analog-code-analogue converter based on redundant positional number systems with accumulation of buffer samples of signals in memory is presented. Calculation of errors of the analog-code-analogue signal reproduced by the converter is given. The relevance and practical expediency of using redundant number systems to obtain high bit capacity and speed of analog-code-analog systems using inaccurate components is substantiated.

Binomial Number System

This paper presents and first scientifically substantiates the generalized theory of binomial number systems (BNS) and the method of their formation for reliable digital signal processing (DSP), transmission, and data storage. The method is obtained based on the general theory of positional number systems (PNS) with conditions and number functions for converting BNS with a binary alphabet, also allowing to generate matrix BNS, linear-cyclic, and multivalued number systems. Generated by BNS, binomial numbers possess the error detection property. A characteristic property of binomial numbers is the ability, on their basis, to form various combinatorial configurations based on the binomial coefficients, e.g., compositions or constant-weight (CW) codes. The theory of positional binary BNS construction and generation of binary binomial numbers are proposed. The basic properties and possible areas of application of BNS researched, particularly for the formation and numbering of combinatorial objects, are indicated. The CW binomial code is designed based on binary binomial numbers with variable code lengths. BNS is efficiently used to develop error detection digital devices and has the property of compressing information.

Сравнение чисел и анализ переполнения в модулярной арифметике

The method of modular arithmetic consists in operating not with a number, but with its remainders after division by some integers. In the modular number system or the number system in the residual classes, a multi-bit integer in the positional number system is represented as a sequence of several positional numbers. These numbers are the remainders (residues) of dividing the original number into some modules that are mutually prime integers. The advantage of the modular representation is that it is very simple to perform addition, subtraction and multiplication operations. In parallel execution of operations, the use of modular arithmetic can significantly reduce the computation time. However, there are drawbacks to modular representation that limit its use. These include a slow conversion of numbers from modular to positional representation; the complexity of comparing numbers in modular representation; the difficulty in performing the division operation; and the difficulty of determining the presence of an overflow. The use of modular arithmetic is justified if there are fast algorithms for calculating a number from a set of remainders. This article describes a fast algorithm for converting numbers from modular representation to positional representation based on a geometric approach. The review is carried out for the case of a comparison system with two modules. It is also shown that as a result of increasing numbers in positional calculus, they successively change in a spiral on the surface of a two-dimensional torus. Based on this approach, a fast algorithm for comparing numbers and an algorithm for detecting an overflow during addition and multiplication of numbers in modular representation were developed. Consideration for the multidimensional case is possible when analyzing a multidimensional torus and studying the behavior of the turns on its surface.