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The Neutrosophic Numeral System Algorithms are a set of techniques designed to handle uncertainty and ambiguity in numerical data. These algorithms use Neutrosophic Set Theory, a mathematical framework that deals with incomplete, indeterminate, and inconsistent information. In this paper, we provide an overview of different approaches used in Neutrosophic Numeral System Algorithms, including Neutrosophic Binary System, Neutrosophic Decimal System, and Neutrosophic Octal System. These systems use different bases and representations to account for degrees of truth, indeterminacy, and falsity in numerical data. We also explore the relationship between Neutrosophic Numeral System Algorithms and Number Neutrosophic Systems, which are another type of Neutrosophic System used for representing numerical data. Number Neutrosophic Systems use Neutrosophic Numbers to represent degrees of truth, indeterminacy, and falsity in numerical data, and they can be used in conjunction with Neutrosophic Numeral System Algorithms to handle uncertainty and ambiguity in decision-making and artificial intelligence applications. Moreover. We discuss the advantages and disadvantages of each algorithm and their potential applications in various fields. Finally, we highlight the importance of Neutrosophic cryptography in addressing uncertainty and ambiguity in decision making and artificial intelligence and discuss future research directions. Understanding Neutrosophic Numeral System Algorithms and their relationship with Number Neutrosophic Systems is crucial for developing effective techniques for handling uncertainty and ambiguity in numerical data in decision-making, pattern recognition, and artificial intelligence applications. [ABSTRACT FROM AUTHOR]

After the Hindu-Arabic decimal positional system was introduced in Europe, throughout many centuries textbooks on elementary arithmetic, intended for beginners, had a more or less fixed organization of content, usually starting with chapters on numeration. These chapters, as a rule, contained one or more examples of large integers the purpose of which was simply to be named (read out loud), sometimes also vice versa. This tradition apparently began with the two first texts that significantly contributed to the spread of the decimal system in Europe--the Latin translations of al-Khwarizmi's treatise on decimal arithmetic, and Leonardo's Liber Abaci, containing examples of reading a 16-digit and a 15-digit number respectively. Throughout the centuries, the order of magnitude of these introductory numbers increased, in general up to some 30 digits, but in some cases to over 60 digits. In this paper we examine the development and extent of this characteristic of introductory arithmetic textbooks from the period 13th-19th century, and the conditions which lead to this, now extinct, practice. [ABSTRACT FROM AUTHOR]

In schools, children are taught algorithm for extracting the square and cube roots of a number. However, they are not informed that Aryabhata (b. 476 CE) was the first Indian mathematician to present a systematic algorithm for the extraction of the square and cube root of a number. This algorithm has been succinctly presented by Aryabhata in two successive verses in the Ganitapada of his Aryabhatiyam composed in 499 CE. A careful study of the algorithm given by Aryabhata shows that it can be easily extended to find higher order roots. It also clearly reveals that the decimal place value system should have been in vogue in India for quite a long time before him, because this algorithm is heavily based on a thorough understanding of the decimal place value system. In our paper, we will discuss some of these aspects as well as outline how his algorithm can be extended to find the higher order roots. [ABSTRACT FROM AUTHOR]

Purpose: To report on the surgical maneuvers recommended for a successful unfolding of very young donors in order to accomplish an uneventful Descemet Membrane Endothelial Keratoplasty (DMEK) surgery. Methods: Five patients (three females and two males, mean age 71.2 ± 6.7 years) with Fuchs endothelial cell dystrophy who underwent DMEK with very young donors (between 20 and 30 years old) were included. The following demographic data were assessed: donor's age, donor's endothelial cell density (ECD), preservation time, recipient's age and sex and unfolding surgical time. Best-corrected visual acuity (BCVA; decimal system), ECD and corneal central thickness (CCT) were assessed preoperatively and at 6-month follow-up. Results: Donors' mean age was 23.6 ± 3.6 years (range 21 to 30) and the mean ECD was 2748.6 ± 162.6 cells/mm2. All of them underwent an uneventful DMEK as a single procedure performed by one experienced surgeon (MAG) with a mean unfolding time of 7.2 ± 4.9 min (range 4 to 15). The essential steps, including patient preparation as well as DMEK graft implantation, orientation, unrolling and centering are detailed. At 6 months, BCVA was 0.6 ± 0.2, ECD was 1945.0 ± 455.5 cells/mm2 and CCT was 497.0 ± 19.7 microns. Conclusions: We hereby present the keys to overcome tightly scrolled grafts of very young donors, which prove perfectly suitable for DMEK surgery. The graft shape tends towards a double-roll and specific maneuvers are strongly recommended. [ABSTRACT FROM AUTHOR]

Thirty years ago, I published Signifying Nothing; The Semiotics of Zero a book about zero. It was framed as a semiotic investigation of zero as a signifying concept. My interest was the concept's role in Western culture, in the Hindu-Arabic notation for decimals, and within the contemporaneous elaboration of money and painting. In each of these three codes, a relation to 'nothing' -- to representing it -- is realized as a sign for absence of signs. In the process the notion that objects come before their names, that things are anterior to signs, is challenged. Instead, the action of zero is to be seen as constitutive of the whole numbers and not merely part of the notational apparatus of 'subsequently' naming them. Moreover, the same effect holds for the codes of painting and money no less than for numbers. The extract reproduced here outlines the nature of this phenomenon for the case of numbers. In the decades since my book was published other books and essays about zero have appeared likewise revealing its extraordinary and revolutionary nature. Not least the present initiative of the Zero Project to uncover zero's origin. [ABSTRACT FROM AUTHOR]

The study of the transition from the numerical system of writing to the decimal system of writing, including the quest to date the origin of zero as a place numeral, has actively engaged the attention of modern researchers and scholars alike. The subject is in fact so vast, gripping, as well as baffling that it will most likely continue to keep them engaged for quite a while to come. There is no need to reiterate the fact that India is very rich in epigraphical wealth. Hundreds and thousands of inscriptions have been reported from across the length and breadth of the country, written in various languages and in varied scripts, and following different dating systems. It is because of this abundance and the sheer complexity of these inscriptions, coming from different parts of India and belonging to different dynasties, that it has become an arduous task to pinpoint when exactly the numerical system of writing gave way to the decimal system in India, and when zero first made its appearance in Indian inscriptions. Further, the chronological sequencing of the changes in the system of writing -- from numerical to decimal -- that had taken place over time is also difficult to establish. In this chapter, while keeping the vastness of the subject in mind, an attempt has been made to re-examine some of the early dated inscriptions that have come to light in the last few years, both as part of the reports from archaeological excavations and from epigraphical explorations. Efforts have also been made to connect these new findings with the hypotheses of and the views expressed by earlier scholars. The goal of this exercise was to ascertain and confirm the presence and usage of decimal systems of writing, and also the use of zero as a placeholder, in South India. Thus, the primary task undertaken has been to review some of the already known dated inscriptions from Andhra Pradesh, Karnataka, Kerala, and Tamil Nadu. [ABSTRACT FROM AUTHOR]

Two of the characteristics of our modern concept of zero find equivalents in Ancient Mesopotamia. First, the Babylonian sexagesimal positional notation from its invention in the late third millennium BCE required a means of indicating a 'placeholder' in a number, a missing power of the base. Secondly, the needs of mathematical astronomy, developed in the first millennium BCE, had to deal with a numerical concept of 'nothing' as a full-fledged number, capable of entering into arithmetical operations. [ABSTRACT FROM AUTHOR]

If mathematics is the language the universe was written in and mathematics is discovered rather than invented, then Brahmagupta's 628 CE Sanskrit text first defines a zero as old as the universe. Brahmagupta's definition of zero as a sum of equal and opposite negative and positive quantities is the first scientific definition of zero found to be consistent with laws of motion and particle physics. However, having originated in the East, the full power of India's symmetric and scientific zero failed to migrate West via the medieval Arabic world to Renaissance Europe. Thus, only a trivial mathematical concept of zero emerged, representing either nothing on its own or an arithmetical placeholder used alongside other numbers. Zero as an arbitrary midpoint for measurement purposes within a single quantity is similarly trivial, whether it be centuries BCE or CE, or a measure above or below a surveyor's foundation line as used for construction in ancient Egypt. Thus, the oldest original extant definition of zero compatible with laws of physics describing our universe originated with Brahmagupta in India 628 CE. Yet long lost in transmission and translation, the sad aftermath in classrooms today is a zero in which negatives to its left are arbitrarily less than positives to its right. [ABSTRACT FROM AUTHOR]

Who invented zero? Interestingly, there is no one-line answer. And the acceptability of any attempted answer generally depends on the perception of the seeker, as the connotation of the word 'zero' can be perceived from several different but interrelated perspectives. Many ancient civilizations, including India, had their own version of zero or a zero-like concept or symbol as a representative of 'nothingness'; some as a philosophical conundrum, elsewhere some others even in a practical sense, such as a filler or a gap on their counting board; but except in India, none of these early and somewhat hesitant initial concepts did ultimately mature to its true mathematical potential. Today it is generally accepted worldwide that this peerless concept of a decimal place value system of enumeration in tandem with the true zero of our present-day mathematics, evolved in ancient India. Initially philosophically nurtured and analyzed during the early periods of oral tradition in India by several different schools of thought, eventually at some point of time the concept reflected in the Sanskrit word śūnya was transformed into a numeral for mathematical expression of 'nothing'. Perhaps at a later period, it bloomed into its full potential as a number, kha, an integer on which mathematical operations can be performed. Indeed, this unique feature makes kha, the Indian zero, the true progenitor of our modern mathematical zero. However, the exact time frame of this gradual evolution is still hotly debated, a recent controversy in this direction being the outcome of attempted radiocarbon dating of the famous Bakhshali manuscript by the Bodleian Library of Oxford. Going in the other direction, some scholars suggest that a trace of this concept, if not in total operational perspective, might have had a Greek origin that traveled to India during the Greek invasion of the northern part of the country in the pre-Mauryan period. A relatively recent third view professes the Chinese origin of the concept of zero as a placeholder, which might have traveled with the traders from China to the far eastern parts of Asia, to places like Cambodia, then under Buddhist influence that spread from mainland India, where it got the shape of a 'bold dot', the earliest known written form of zero. Scholars belonging to this school of thought want to credit mainland India only for 'garlanding' this concept of zero toward its modern shape, sometime around the eighth or ninth century CE. Embracing all these paradigms of claims and counterclaims about the origin of zero made till date, this article is a nutshell version of an odyssey, an unparalleled journey from a concept to a number in its own right, perhaps the most significant creation of the human mind ever. [ABSTRACT FROM AUTHOR]

The term 'Jaina mathematics' represents the mathematical concepts and ideas available in Jaina canonical and non-canonical literature. The literature related to the Karaṇānuyoga section of the Digambara Jaina tradition and the Gaṇītānuyoga section of the Śvetāmbara Jaina tradition have much content to hold mathematicians' interest. The classification of numbers in the three groups Saṃkhyāta (countable), Asaṃkhyāta (uncountable) and Ananta (infinite) can be found in ancient Jaina literature. The definition of Utkrṛsṣṭa saṃkhyāta (maximum countable) can be given only with the help of zero. Without the knowledge of numbers and zero it is not possible to define Uṭkrṛsṣta saṃkhyāta and other related quantities, such as Acalātma and Śīrsṣaprahelikā. But we find the use of zero in ancient mathematical texts such as the Bakhshali manuscript, texts of the Digambara Jaina tradition, such as SṢaṭakhaṇdāgama, Pancāstikāya, Mahābandha, Lokavibhāga, Tiloyapaṇṇattī, and the Dhavalā Commentary etc., and texts of the Śvetāmbara Jaina tradition such as Sthānāṃga sūtra, Jambūdvīpaprajñapti, Anuyogadvāra sūtra, Āvaśyaka Niryukti, Br1;ahatksṛetra Samāsa, etc., in the mathematical sense in many places. All these references are explored and analyzed in this chapter to give a comprehensive picture to the academic world. The study indicates that the use of zero and the decimal place value system was popular in the Jaina tradition during the first and second centuries CE. Keywords: SṢaṭkhaṇdāgama, Anuyogadvāra sūtra, Tiloyapaṇṇattī, Lokavībhāga, Bakhshālī manuscript,aṃkhyāta, Acalātma, Śūnya [ABSTRACT FROM AUTHOR]

In China, a decimal system of numeration is reflected in the number of oracle bones scripts between the fourteenth and eleventh centuries BCE. The place value system is clearly shown in the numbers on coinage from the eighth century BCE to the third century BCE. The decimal place value system was completely established in counting rods numeration, a description of the use of which is recorded in Mathematical Canon of Sun Zi of the late third century and the beginning of the fourth century; the earliest written counting rods numbers exist in a Dunhuang manuscript from around the tenth century. The procedures of multiplication, division or extraction of roots show that precise vacant positions in numbers are emphasized and strictly respected by the ancient practitioners. In thirteenth century mathematical works, the written symbol ... is systematically used as a placeholder in numbers. The 'procedure of the positive and the negative' is usually regarded as a general rule of addition and subtraction between positive, negative numbers and zero, but this procedure should be interpreted in the context of the mathematical method fangcheng. The general rules concerning the process of elimination numbers in which zero is involved are clearly expressed and adopted, even though there is no explicit declaration of the existence of the numeral zero. [ABSTRACT FROM AUTHOR]

As various articles in this volume demonstrate, zero has many forms and finds expression in a whole host of contexts, each as mystifying as the other. Each culture seems to have a very specific reaction to the number, and the reaction varies significantly from culture to culture. From the great hatred in Europe to the near obsession of their cousins in the Indian subcontinent, the entire spectrum of reactions can be found. Egyptians, Sumerians, and the Chinese managed to do some interesting algebra while circumventing zero completely. In this chapter we will investigate why cultures even with deeply common roots reacted so differently to zero. We take the example of the Indo-European cultures and their diametrically opposite reactions to zero. We suggest that this may be related to their experiences of nature post-separation, which were further amplified by their way of life. We then study the wider context of zero in its different manifestations and explore the relation between its different forms. We show that music seems to make a connecting link between these different aspects. [ABSTRACT FROM AUTHOR]

Currently, there are different claimants for the honor of inventing the zero as a digit, a part of the decimal system. These claims are made on the basis of evidence found in one particular country. None of the claims have, however, been able to convincingly rule out the rival claims. This chapter suggests that research along the terms of cross-cultural history and or connected history linking the threads of information coming from different civilizations regarding the knowledge of the zero may yield a different, border-crossing biography of the zero. [ABSTRACT FROM AUTHOR]

F In what was probably his last writing on base-16 numeration, Leibniz [[12]] drew a table featuring the decimal numbers 0 to 40 represented in every base from 2 to 16. Hexadecimal, or base 16,[1] has been used as a computer language for decades. In his first writing on base 16, Leibniz [[9]] worked out how to convert the decimal number 1679, representing the year of his invention, into sedecimal numeration. 7 However, in 1867, W. B. Taylor, of the United States Patent Office, reviewed Nystrom's proposal and reached the conclusion that base 16 was not significantly more economical than base 10, as Nystrom had claimed. [Extracted from the article]

Simple Summary: This study aimed to teach the numerical sequence from 1 to 19 in the decimal system to six chimpanzees. The participants were three mother–child pairs. The original goal for chimpanzees was to touch the numerals on the display from 1 to 19, in this order. The baseline daily training was twofold: touching the adjacent numerals from 1 to X and X to 19 in ascending order. In two separate ways, chimpanzees succeeded at touching adjacent numerals in the range 1 to 19. Systematic tests assessed four factors: range (1 to 9 vs. 1 to 19), adjacency (adjacent vs. nonadjacent numerals), number of stimuli used (three, four, and five), and memory load (nonmemory vs. a memory task called the "masking task"). All four factors were important. A further test directly compared the performance of chimpanzees with that of human participants using the same apparatus and procedure. Both accuracy and response latency showed that processing two-digit numerals was more difficult than one-digit numerals in both species. A chimpanzee named Pal perfectly mastered the order of two-digit numerals just like humans. The difference between the two species was discussed in terms of species-specific global–local information processing. The sequence of Arabic numerals from 1 to 19 was taught to six chimpanzees, three pairs of mother and child. Each chimpanzee participant sat facing a touchscreen on which the numerals appeared in random positions within an imaginary 5-by-8 matrix. They had to touch the numerals in ascending order. Baseline training involved touching the adjacent numerals from 1 to X or from the numeral X to 19. Systematic tests revealed the following results: (1) The range 1 to 9 was easier than 1 to 19. (2) Adjacent numerals were easier than nonadjacent ones. (3) The "masking" (memory task) caused deterioration of performance. All these factors depended on the number of numerals simultaneously presented on the screen. A chimpanzee named Pal mastered the skill of ordering two-digit numerals with 100% accuracy. Human participants were tested in the same experiment with the same procedure. Both species showed relative difficulty in handling two-digit numerals. Global–local information processing is known to be different between humans and other primates. The assessment of chimpanzee performance and comparison with humans were discussed in terms of the possible difference in the global–local dual information processing of two-digit numerals. [ABSTRACT FROM AUTHOR]

The present article describes for the first time the book of Vaclav Josef Pelikan titled Arithmeticus Perfectus Qui tria numerare nescit. Seu Arithmetica dualis, In qua Numerando non proceditur, nisi ad duo, & tamen omnes quaestiones Arithmeticae negotio facili enodari possunt , published in Prague in 1712. The book is written in Latin on 86 pages and consists of a dedication, a message to the reader and four chapters. Operations in the binary system, including the extraction of square and cube roots, methods of converting numbers from the decimal system to the binary system and vice versa, etc., are given. In general, we may say that the book by Vaclav Josef Pelikan is the first fully fledged and methodologically sound textbook of arithmetic using the binary number system as well as containing original methods of solution. [ABSTRACT FROM AUTHOR]

Number transcoding is the cognitive task of converting between different numerical codes (i.e. visual "42", verbal "forty-two"). Visual symbolic to verbal transcoding and vice versa strongly relies on language proficiency. We evaluated transcoding of German-French bilinguals from Luxembourg in 5th, 8th, 11th graders and adults. In the Luxembourgish educational system, children acquire mathematics in German (LM1) until the 7th grade, and then the language of learning mathematic switches to French (LM2). French '70s '80s '90s are less transparent than '30s '40s '50s numbers, since they have a base-20 structure, which is not the case in German. Transcoding was evaluated with a reading aloud and a verbal-visual number matching task. Results of both tasks show a cognitive cost for transcoding numbers having a base-20 structure (i.e. '70s, '80s and '90s), such that response times were slower in all age groups. Furthermore, considering only base-10 numbers (i.e. '30s '40s '50s), it appeared that transcoding in LM2 (French) also entailed a cost. While participants across age groups tended to read numbers slower in LM2, this effect was limited to the youngest age group in the matching task. In addition, participants made more errors when reading LM2 numbers. In conclusion, we observed an age-independent language effect with numbers having a base-20 structure in French, reflecting their reduced transparency with respect to the decimal system. Moreover, we find an effect of language of math acquisition such that transcoding is less well mastered in LM2. This effect tended to persist until adulthood in the reading aloud task, while in the matching task performance both languages become similar in older adolescents and young adults. This study supports the link between numbers and language, especially highlighting the impact of language on reading numbers aloud from childhood to adulthood. [ABSTRACT FROM AUTHOR]

In the present article we explore a way to represent numbers with respect to the base − 3 2 using the set of digits {0, 1, 2}. Although this number system shares several properties with the classical decimal system, it shows some remarkable differences and reveals interesting new features. For instance, it is related to the field of 2-adic numbers, and to a self-affine "fractal" set that gives rise to a tiling of a non-Euclidean space. Moreover, it has relations to Mahler's 3 2 -problem and to the Josephus problem. [ABSTRACT FROM AUTHOR]

This paper presents an analysis of a hitherto barely known conceptual problem in the foundations of arithmetic. An unknown type of number, so-called complex numbers, first emerged in 18th century France and became connected with the claim of non-commutativity of their multiplication. These first French practitioners are here analysed, along with examination of the impact of the introduction of the metric decimal system. International dissemination of the notion is then analysed, in particular in the case of Brazil. The paper concludes by looking at the mathematical foundations of multiplying (physical) quantities. [ABSTRACT FROM AUTHOR]

Using the design method proposed by Kunii, this study mathematically explains the achievement of Kublai of merging two different societies to build a globalized nation after conquering China. The considered design method, called incrementally modular abstraction hierarchy, consists of seven levels, from the most abstract, homotopy, to the most concrete, computer graphics. Moreover, it allows a flexible design while moving up and down its hierarchy. Using this method, we attempt to restore the concepts of Kublai, aiming to build a globalized state by combining nomadic and agricultural cultures. Specifically, we examine the use of two capitals to resolve the problem of different lifestyles of the Mongolian Khan, who moved between the summer and winter camps of the nomads, and the Chinese Emperor, who lived in a fixed palace of the agricultural people. By incorporating the category theory into Kunii's method, we discuss the adjunction of the Mongolian 1000-household social and military system based on decimal numbers with the Chinese bureaucratic system. The inclusion of the category theory in his original method allows visualization by programming, making the design more versatile and flexible. [ABSTRACT FROM AUTHOR]

But the significance of tokens and notations exceeds this, as they enabled numbers to be separated from the process of counting.[140] Separating numbers from the items they counted by means of material proxies was at least as significant as separating the representation of quantity and commodity, for it helped influence numbers toward being conceived as objects more defined by their relations to one another than the things they enumerated, influencing numbers toward becoming a relational system. A particular number is not identical with any collection of terms having that number: the number 3 is not identical with the trio consisting of Brown, Jones, and Robinson. However, Sumerian number-words below I sixty i were quinary-decimal-vigesimal in structure.[137] As a result, sexagesimal factorability may have been used to express numbers higher than I sixty i , with numbers lower than I sixty i expressed decimally, analogous to way Hawaiian numbers were traditionally expressed (see Table 1). 79 Chrisomalis has compellingly argued against the idea that Mesopotamian peoples were particularly incompetent or concrete in their conceptualization of numbers: "If ... polyvalence and context-dependence imply an absence of abstract number concepts, then paradoxically, the quasi-literate Uruk accountants would be less numerate than the average Sumerian who did not use texts, only number words" (Chrisomalis, "Evaluating Ancient Numeracy" [[26]]: 4); "the accountants and scribes who used [tokens and numbers] were able to manage complex administrative tasks, and it is implausible that they did not recognize that '8 sheep' and '8 bushels of grain' had something in common" (Chrisomalis, "Cognitive and Cultural Foundations" [[25]]: 502); and "in order for these context-dependent numerals to represent a stage of "archaic arithmetic" in the evolution of numeration, ... we would expect similar systems to be present in other civilizations. [Extracted from the article]

The article reports that Fibonacci realized the value of the Hindu-Arabic system and his major contribution was to write a book in 1202, Liber Abaci, that took the system to Europe. Topics include mathematicians have worked on the Fibonacci sequence publishes articles related to Fibonacci's sequence; and Fibonacci's father was a Pisan merchant and customs official who was in charge of a trading post in Bugia.

E: Write the formula for the number Graph HT ht (5), where I n i represents any amount of ones in the number. Nothing can be done without it ... Let's see how it works out here: (The test subject then constructs a number series in the quinary system, while noting under each number its value in the decimal system.) In a preliminary series of experiments the test subjects, who - like all of us - knew how to express numbers in the decimal system, were asked to express numbers in the quinary system that they were given in the decimal system. Write the formula for the number Graph HT ht , i.e. a number expressed in ones, where I n i represents any amount of ones in a number. [Extracted from the article]

Stemming from an idea put forward by Loo-Keng Hua in 1962, this article describes an original way to perform arithmetic directly on infinite decimals. This approach leads to a new and elementary construction of the real number system via decimal representation. Based on the least upper bound property, a definition of trigonometric functions is also included, which settles an issue that Godfrey H. Hardy called "a fatal defect" in his Course of Pure Mathematics. [ABSTRACT FROM AUTHOR]

In contemporary Western society, a jubilee means celebration of an anniversary of some event. However, in the distant past, in the Judaic tradition, it meant a certain period of universal time: a jubilee was a period of 49 years (The Book of Jubilees). How did the transition from a calendar time-measuring unit, a period (49 years), to the festivity commemorating an event measured according to the decimal system (20, 50, 100-year jubilees) occur? In order to explain this phenomenon from the philosophical perspective, four concepts of time are distinguished: mythical, physical, historical and soul/consciousness time. When discussing the concept of natural (physical) time, which took shape in the period of classical antiquity (Physics by Aristotle) and later was taken over by natural sciences, it is explained how it differs from the mythical-religious concept of time, which is understood by religious anthropologists with the help of Bergson's idea of duration; what is the difference between the measurement of time in natural sciences and the measurement of mythical time, as well as between the mythical and mathematical understanding of a number. The religious calendar establishing ritual practices regulates human actions and relates them to cosmic cycles, but it has no correlation to their consciousness. The formation of the practice of commemorating a certain event requires a different relation with time: mythical time has to be replaced with historical time, which makes necessary self-consciousness and the concept of irreversible passing of time. The passage of time is retained through memory. Commemoration of an event is one of the forms of historical memory, when a reminiscence is externalised by ritualising it, by turning it into a ceremony. Thus, through ritual and by conferring a symbolic meaning to a number (a unit of the decimal system), elements of mythical time appear in historical time. A jubilee, as a periodical, ritual commemoration of a certain event, finds itself in the midway between mythical and historical time: it is a kind of degraded mythical time in historical time. [ABSTRACT FROM AUTHOR]

As a result, I y i years after 100 I c i , the day of the week for the given day will have increased by HT ht . Since a necessary and sufficient condition for I n i to be a leap year is that either HT ht and HT ht or HT ht and HT ht , we determine that the formula we seek should include the variables I c i and I y i . The day of the week for Doomsday in the year HT ht is HT ht Graph Proof. [Extracted from the article]

Reported herein is the novel achievement of uniform and tunable interference colors of nanocomposite films containing organized cellulose nanocrystals (CNCs) from dynamic hydrogel precursors. Homogeneous and amendable interference colors with broad range are obtained either by stacking the nanocomposite films to adjust the amount of CNCs in the propagation pathway of light or by regulating the rotation angles between the individual films to alter the relative organization of CNCs within the system. Moreover, the precise and controllable patterned CNC composite films with multicolors in one film are facilely fabricated for the first time from the patterned hydrogel precursors. Based on this stacking/rotation‐method, these patterned nanocomposite films with tunable interference colors can be further applied as fundamental elements for optical encryption by establishing a ternary‐coded decimal system with encoded decimal numerals, paving the way for the development of photonic functional materials based on CNCs. [ABSTRACT FROM AUTHOR]

Background: While it has been widely demonstrated that children's and adolescents' understanding of rational number plays an important role in their mathematics achievement, we have limited knowledge about the cognitive correlates of this understanding. Aims: The current study aimed at examining whether children's non‐symbolic ratio processing and their understanding of place‐value structure of whole numbers play a role in their understanding of fractions and decimals and whether their roles are different for fractions versus decimal understanding. Sample: A sample of 124 fourth graders was tested. Methods: Participants were tested on their symbolic rational number processing, non‐symbolic ratio processing, place‐value understanding of whole numbers, mathematics achievement, as well as a series of domain‐general and domain‐specific cognitive skills related to symbolic rational number processing. Results: The findings suggest that, while the understanding of place value of whole numbers significantly predicted the understanding of both fractions and decimals, non‐symbolic ratio processing specifically predicted the understanding of fractions, but not decimals. Conclusions: The findings highlight the roles of place‐value understanding and non‐symbolic ratio processing in the acquisition of symbolic rational numbers. [ABSTRACT FROM AUTHOR]

Peruvian author Ricardo Palma gained prominence as South America's preeminent literary figure with his tradiciones, but he also cultivated lesser-known genres such as poetry and theater, as well as the essay, the topic of the present study. This article investigates why Palma's "Sistema decimal entre los antiguos peruanos" is an essay and not a tradición, an apparent, yet erroneous category for it, since it appeared in Peruvian Traditions. In it, Palma utilizes a philological analysis of the anonymous play Ollantay and Inca Garcilaso de la Vega's Royal Commentaries to prove that the Inkas used the decimal system of numbers. The present article considers why this piece is an essay by comparing its constituent elements with the characteristics of the essay as José Luis Gómez Martínez outlines them, including discursivity, lack of a rigid structure, incitement to reflection, digression, and vagueness in quoting practices. If Palma completes a philological commentary on Ollantay and the Royal Commentaries, the present article accomplishes a philological analysis of Palma's "Sistema decimal," which reveals qualities properly belonging to the essay. [ABSTRACT FROM AUTHOR]

The important revision of the International System of Units (SI) that has taken place at the 26th General Conference on Weights and Measures in November 2018 is the culmination of some 250 years of development of ideas on how best to create a system of units of measurement closely linked to nature. In part 1 of this article, we outline the early history of units that led up to the creation of the decimal metric system in France at the end of the 18th century. In part 2, we describe the initiative of the French Government that led to the signing of the Metre Convention on 20 May 1875, which established the essential formal basis for the future international collaborations that have resulted in the revision of the SI adopted in November 2018. [ABSTRACT FROM AUTHOR]

Highlights • The whole number bias is at the root of errors in decimal magnitude comparison tasks. • The whole number bias must be inhibited to compare the magnitude of decimal numbers. • The inhibition of the whole number bias is largely automatized by 12 years of age. Abstract A major source of errors in decimal magnitude comparison tasks is the inappropriate application of whole number rules. Specifically, when comparing the magnitude of decimal numbers and the smallest number has the greatest number of digits after the decimal point (e.g., 0.9 vs. 0.476), using a property of whole numbers such as "the greater the number of digits, the greater its magnitude" may lead to erroneous answers. By using a negative priming paradigm, the current study aimed to determine whether the ability of seventh graders and adults to compare decimals where the smallest number has the greatest number of digits after the decimal point was partly rooted in the ability to inhibit the "the greater the number of digits, the greater its magnitude" misconception. We found that after participants needed to compare decimal numbers in which the smallest number has the greatest number of digits after the decimal point (e.g., 0.9 vs. 0.476), they were less efficient at comparing decimal numbers in which the largest number has the greatest number of digits after the decimal point (e.g., 0.826 vs. 0.3) than they were after comparing decimal numbers with the same number of digits after the decimal point (e.g., 0.981 vs. 0.444). The negative priming effects reported in seventh graders and adults suggest that inhibitory control is needed at all ages to avoid errors when comparing decimals where the smallest number has the greatest number of digits after the decimal point. [ABSTRACT FROM AUTHOR]

For integer g ≥ 5, we prove that any positive integer can be written as a sum of three palindromes in base g. [ABSTRACT FROM AUTHOR]

Financial and commercial applications depend on decimal arithmetic because they must produce results that match exactly those obtained by human calculations. Decimal multiplication is a frequently used operation in these applications and also in the design of decimal floating-point units. In this paper we propose a new architecture for parallel decimal multiplication that improves the area of previous decimal multipliers while keeping the best performances. A decimal adder [1] based on a mixed BCD/excess-6 representation of the operands is utilized. A new partial product generation unit is proposed based on a 5221 recoding of the multiplier digits. With the proposed multiplier, we are able to improve on state-of-the-art parallel decimal multipliers targeting LUT-6 FPGAs. Compared to previous decimal multipliers, implementation results for 2, 4, 8, 16, 32 and 34-digits show that the proposed multiplier achieves over 20% better area without performance degradation. [ABSTRACT FROM AUTHOR]

Even and odd numbers appear early in history of mathematics [9], as they serve to describe the property of objects easily noticeable by human eye [7]. Although the use of parity allowed to discover irrational numbers [6], there is a common opinion that this property is "not rich enough to become the main content focus of any particular research" [9]. On the other hand, due to the use of decimal system, divisibility by 2 is often regarded as the property of the last digit of a number (similarly to divisibility by 5, but not to divisibility by any other primes), which probably restricts its use for any advanced purposes. The article aims to extend the definition of parity towards its notion in binary representation of integers, thus making an alternative to the articles grouped in [5], [4], and [3] branches, formalized in Mizar [1], [2].Even and odd numbers appear early in history of mathematics [9], as they serve to describe the property of objects easily noticeable by human eye [7]. Although the use of parity allowed to discover irrational numbers [6], there is a common opinion that this property is "not rich enough to become the main content focus of any particular research" [9]. On the other hand, due to the use of decimal system, divisibility by 2 is often regarded as the property of the last digit of a number (similarly to divisibility by 5, but not to divisibility by any other primes), which probably restricts its use for any advanced purposes. The article aims to extend the definition of parity towards its notion in binary representation of integers, thus making an alternative to the articles grouped in [5], [4], and [3] branches, formalized in Mizar [1], [2]. [ABSTRACT FROM AUTHOR]

This article reframes previously identified misconceptions about repeating decimals by describing these misconceptions as limited understandings of how mathematics concepts are referenced. In particular, misconceptions about repeating decimals and their quotient of integer representations are recast as limited understandings of mathematics as a discipline that derives its content from representational systems and the denotations they provided. Under this framework, arguments (e.g., proofs) that convert repeating decimals to their quotient of integer representations provide content for “rational number,” which is represented in multiple ways, each offering distinct opportunities for mathematical activity. The notion of an argument as content is illustrated as arguments providing access to a concept. One Grade 8 student’s struggle with understanding rational number is used to illustrate this framework and its implications for teaching and learning. [ABSTRACT FROM AUTHOR]

Many children and adults have difficulty gaining a comprehensive understanding of rational numbers. Although fractions are taught before decimals and percentages in many countries, including the USA, a number of researchers have argued that decimals are easier to learn than fractions and therefore teaching them first might mitigate children’s difficulty with rational numbers in general. We evaluate this proposal by discussing evidence regarding whether decimals are in fact easier to understand than fractions and whether teaching decimals before fractions leads to superior learning. Our review indicates that decimals are not generally easier to understand than fractions, though they are easier on some tasks. Learners have similar difficulty in understanding fraction and decimal magnitudes, arithmetic, and density, as well as with converting from either notation to the other. There was too little research on knowledge of percentages to include them in the comparisons or to establish the ideal order of instruction of the three types of rational numbers. Although existing research is insufficient to determine the best sequence for teaching the three rational number formats, we recommend several types of research that could help in addressing the issue in the future. [ABSTRACT FROM AUTHOR]

Fraction and decimal concepts are notoriously difficult for children to learn yet are a major component of elementary and middle school math curriculum and an important prerequisite for higher order mathematics (i.e., algebra). Thus, recently there has been a push to understand how children think about rational number magnitudes in order to understand how to promote rational number understanding. However, prior work investigating these questions has focused almost exclusively on fraction notation, overlooking the open questions of how children integrate rational number magnitudes presented in distinct notations (i.e., fractions, decimals, and whole numbers) and whether understanding of these distinct notations may independently contribute to pre-algebra ability. In the current study, we investigated rational number magnitude and arithmetic performance in both fraction and decimal notation in fourth- to seventh-grade children. We then explored how these measures of rational number ability predicted pre-algebra ability. Results reveal that children do represent the magnitudes of fractions and decimals as falling within a single numerical continuum and that, despite greater experience with fraction notation, children are more accurate when processing decimal notation than when processing fraction notation. Regression analyses revealed that both magnitude and arithmetic performance predicted pre-algebra ability, but magnitude understanding may be particularly unique and depend on notation. The educational implications of differences between children in the current study and previous work with adults are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we prove several congruences for Appell–Lerch sums which generalize some conjectures given by Chan. [ABSTRACT FROM AUTHOR]

Since the days of Perrin (1908) [ 1 ], microscopy methods have played an important role in the study of colloidal suspensions. Along with the continued development of new imaging techniques, colloid scientists have also implemented a sophisticated range of computational analyses. These analysis techniques are often the unsung heroes that hold the promise of unlocking scientific mysteries at the next decimal place of colloid science. They now enable precision measurements of particle location and size (Bierbaum et al., 2017; Kurita et al., 2012) as well as measurements of local stresses and forces (Lin et al., 2016). Here, we spotlight these exciting advances focusing on the analysis of simple brightfield and confocal microscope images of dense colloidal suspensions as well as the scientific mysteries they may unravel. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Multiplicative understanding is essential for mathematics learning and is supported by models for multiplication, such as equal groups and rectangular area, different calculations and arithmetical properties, such as distributivity. We investigated two students’ multiplicative understanding through their connections between models for multiplication, calculations and arithmetical properties and how their connections changed during the school years when multiplication is extended to multi-digits and decimal numbers. The case studies were conducted by individual interviews over five semesters. The students did not connect calculations to models for multiplication, but showed a robust conceptualisation of multiplication as repeated addition or equal groups. This supported their utilisation of distributivity to multi-digits, but constrained their utilisation of commutativity and for one student to make sense of decimal multiplication [ABSTRACT FROM AUTHOR]

Decimal arithmetic has recovered the attention in the field of computer arithmetic due to decimal precision requirements of application domains like financial, commercial and internet. In this paper, we propose a new decimal adder on FPGA based on a mixed BCD/excess-6 representation that improves the state-of-the-art decimal adders targeting high-end FPGAs. Using the proposed decimal adder, a multioperand adder and a mixed binary/decimal adder are also proposed. The results show that the new decimal adder is very efficient improving the area and delay of previous state of the art decimal adders, multioperand decimal addition and binary/decimal addition. [ABSTRACT FROM AUTHOR]

Hardware and software implementations of decimal arithmetic have resurfaced in recent years to overcome the limitations of binary arithmetic. Traditionally, decimal arithmetic units have been designed as application-specific hardware modules. But there is an emerging trend towards the design and implementation of decimal arithmetic operations on reconfigurable structures. This paper contributes to this trend by proposing a reconfigurable architecture, namely DARA, for high performance implementation of decimal arithmetic operations. Some basic decimal arithmetic operations were implemented on DARA and synthesized subsequently. The results show that DARA has a delay overhead of 26% and area overhead of 54% on average compared to an ASIC implementation of the same operations. At the same time, if those basic operations had been implemented on a modern commercial FPGA, DARA would have outperformed the commercial device in terms of delay and area by a factor of almost 4 and 9, respectively. [ABSTRACT FROM AUTHOR]

The aim of this study was to investigate Hong Kong Grade 4 students’ understanding of the decimal notation system including their knowledge of decimal quantities. This is a unique study because most previous studies were conducted in Western cultural settings; therefore we were interested to see whether Chinese students have the same kinds of misconceptions as Western students given the Chinese number naming system is relatively transparent and explicit. Three hundred and forty-one students participated in a written test on decimal numbers. Thirty-two students were interviewed to further explore their mathematical reasoning. In summary, the results indicated that many students had mastered reasonable knowledge of decimal notation and quantities, which may be attributed to the Chinese linguistic clarity of decimal numbers. More importantly, the results showed that some students’ construction of decimal concepts have been adversely affected by persistent misconceptions arising from whole number bias. Two kinds of whole number misconceptions, namely “-ths suffix error” and “reversed place value progression error”, were revealed in this study. This paper suggests that a framework theory approach to conceptual change may be an alternative approach to addressing students’ learning difficulties in decimals. [ABSTRACT FROM AUTHOR]