Showing papers on "Decimal published in 2014"

Mangarevan invention of binary steps for easier calculation

Abstract: When Leibniz demonstrated the advantages of the binary system for computations as early as 1703, he laid the foundation for computing machines. However, is a binary system also suitable for human cognition? One of two number systems traditionally used on Mangareva, a small island in French Polynesia, had three binary steps superposed onto a decimal structure. Here, we show how this system functions, how it facilitated arithmetic, and why it is unique. The Mangarevan invention of binary steps, centuries before their formal description by Leibniz, attests to the advancements possible in numeracy even in the absence of notation and thereby highlights the role of culture for the evolution of and diversity in numerical cognition.

Decimal fraction representations are not distinct from natural number representations – evidence from a combined eye-tracking and computational modeling approach

Abstract: Decimal fractions comply with the base-10 notational system of natural Arabic numbers. Nevertheless, recent research suggested that decimal fractions may be represented differently than natural numbers because two number processing effects (i.e., semantic interference and compatibility effects) differed in their size between decimal fractions and natural numbers. In the present study, we examined whether these differences indeed indicate that decimal fractions are represented differently from natural numbers. Therefore, we provided an alternative explanation for the semantic congruity effect, namely a string length congruity effect. Moreover, we suggest that the smaller compatibility effect for decimal fractions compared to natural numbers was driven by differences in processing strategy (sequential vs. parallel). To evaluate this claim, we manipulated the tenth and hundredth digits in a magnitude comparison task with participants’ eye movements recorded, while the unit digits remained identical. In addition, we evaluated whether our empirical findings could be simulated by an extended version of our computational model originally developed to simulate magnitude comparisons of two-digit natural numbers. In the eye-tracking study, we found evidence that participants processed decimal fractions more sequentially than natural numbers because of the identical leading digit. Importantly, our model was able to account for the smaller compatibility effect found for decimal fractions. Moreover, string length congruity was an alternative account for the prolonged reaction times for incongruent decimal pairs. Consequently, we suggest that representations of natural numbers and decimal fractions do not differ.

Decimals are not processed automatically, not even as being smaller than one.

Abstract: Common fractions have been found to be processed intentionally but not automatically, which led to the conclusion that they are not represented holistically in long-term memory. However, decimals are more similar to natural numbers in their form and thus might be better candidates to be holistically represented by educated adults. To test this hypothesis, we investigated the automatic processing of decimals by college students in 4 experiments. When decimals were presented in a familiar form (e.g., 0.3, 0.05) the length of the stimuli (i.e., the number of digits) dominated performance rather than the decimal value. When controlling for the number of digits and their location within the digit string, using the place-value task, decimals were not processed automatically in either a numerical comparison task or a physical comparison task. Under the same conditions, natural numbers were processed automatically. We conclude that decimals are not represented holistically. Results of mixed pairs of a decimal and a natural number suggest that, unlike common fractions, decimals are not automatically perceived as smaller than natural numbers. We conclude that decimal place-values (e.g., tenths, hundredths) are not represented well enough to be automatically activated, and we discuss possible explanations.

The Effects of Number-related Factors on Entry Performance

Abstract: Number entry is ubiquitous in user interface (UI) design, and in many applications --- such as finance, aviation, healthcare --- here, mitigating errors is critical. This paper examines the effects of factors such as the type of number (e.g., integer or decimal), number length (i.e., short or long) and display position (i.e., near or far) on entry errors. Until now, these factors have not been explored together. Using a touch-based numeric keypad, we found that number length influenced the probability of committing errors, while the position of presentation did not. Number type impacted user-corrected errors but not uncorrected errors. Number length, number type and display position affected input timings.The findings provide implications for the design of both number representations (e.g., decimal point appearance) and the sociotechnical systems that surround them (e.g., training practice).

Mathematical Content Knowledge for Teaching Elementary Mathematics: A Focus on Decimals

Abstract: IntroductionThe historical evolution of decimals as a representation of quantity rests largely on the development of place value and the use of zero in the numeration system. Far more difficult than using the notational system is understanding the quantities represented with the system (Irwin, 2001} in context. Of particular difficulty are decimal fractions (decimals}, rational numbers "which originate by subdivision of each unit interval into 10, then 100,1000, etc., equal segments" (Courant & Robbins, 1996, p. 61}. Research on children's conceptions of decimals illustrates a series of conceptual hurdles involved in interpreting and using the notational system (Resnick et al., 1989; Sackur-Grisvald & Leonard, 1985}. Because children build their understandings of decimals from their existing or coemergent understandings of multidigit whole numbers and fractions, they tend to over-apply concepts for these more familiar objects when the numerals being discussed are decimals. Findings from studies of children's understandings encouraged researchers to begin to explore prospective teachers' (PTs'} understandings of decimal notations (Putt, 1995; Thipkong & Davis, 1991}. Such studies unearthed parallels between categories of reasoning used by children and reasoning used by PTs, encouraging researchers to identify teachers' misconceptions as a source of children's faulty reasoning.Research on PTs' knowledge of decimal fractions has focused on exploring how decimals are interpreted and used in computation, and how mathematics educators might challenge existing beliefs about the use of decimal fractions. In this report, we focus primarily on terminating decimals that are included in primary school curriculum. A very small collection of reports focused on PTs' knowledge of decimals has been published over the last 25 years, but findings point to the importance of place value in PTs' understanding and application of decimals.Approaches and OrientationsIn the sections that follow, we have summarized historical influences in the study of PTs' knowledge of decimals, findings of published peer-reviewed papers from 1998 to 2011, and additional insights drawn from more recent work. Our approach to identification of articles was consistent with the method described in the introductory article of this Special Issue. In addition, our perspective on decimal understanding influenced our interpretations of the articles. We share this perspective to enable readers to gain insight into our interpretations.Our view of decimal is informed by explorations of PTs' understandings (D'Ambrosio & Kastberg, 2012; Kastberg & D'Ambrosio, 2011) of decimals using a framework including units, relationships between units, and additivity. As Courant and Robbins (1996) suggest, decimal units in the place value system involve repeatedly "subdividing" an individual unit into 10 parts. So if we begin with 1, then subdividing this unit into 10 parts produces 10 subunits 0.1. This action creates the opportunity for the development of relationships between 1 and 0.1, namely, that 1 is 10 times 0.1 and 0.1 is one tenth of 1. While this example involves adjacent units in the set of place value units {..., 10,1, 0.1, 0.01,...}, any two units in the set can be thought of as related multiplicatively. Finally, sums of multiples of the units can be used to create new decimals, an idea that is represented in expanded notation. For example, if we compare 0.606 and 0.66 using the additive structure, we can see that 0.606 = 0.6 + 0.006 and 0.66 = 0.6 + 0.06. This understanding and understanding of multiples of the units 0.001 and 0.01 allow us to quickly determine that 0.606 is less than 0.66. Understanding decimals as linear combinations of place value units allows us to compose and decompose decimals to quickly compare them. While there are certainly other views of decimals, it was this view that we held and used to make sense of the findings reported in the research. …

A Huffman Code Based Image Steganography Technique

Abstract: We present here a novel steganographic method based on Huffman coding and the least significant bit substitution in order to provide high embedding capacity, a strong security and imperceptible visual quality to secret message. Every eight bits of the secret image are first encoded by building a Huffman tree. After that those encoded bits of secret image are divided into 4 groups. Each part has a decimal value between 0 to 3. These decimal values determine the location where to embed the message in a particular pixel of cover image. To embed the message we just put a one in the corresponding location in a pixel of the cover image which identified by the decimal values of the secret image. Since Huffman Table reduces the size of the original image, an attacker cannot easily recover from the stego image those fine details of the original image that would enable him to mount a reliable attack. We have got comparable visual quality as the Peak Signal to Noise Ratio values lie between 30 dB to 31 dB.

The effect of citations on the significance of decimal places in the computation of journal impact factors

Abstract: Journal impact factors (JIF) are computed by Thomson Reuters to three decimal places. Some authors have cast doubt on the validity of the third decimal place in JIFs. In this paper I present a new approach to evaluate the significance of decimal places in JIFs. To do so, two modified JIFs were computed by adding or removing one citation to the number used by Thomson Reuters to compute the JIF for journals listed in the 2008 Journal Citation Report. The rationale is that one citation is the minimum amount of impact that can be observed and analyzed. Next, the modified JIFs were compared with the original JIF to identify the decimal place that changed as consequence of adding or removing one citation. The results suggest that for about two-third of journals, the number of places used by Thomson Reuters to compute JIFs can be considered appropriate for the most part.

Decimal numbers and safe interpretation of clinical pathology results

Abstract: Objective To determine the understanding of decimal numbers by medical laboratory scientists, doctors and nurses. Methods A Decimal Comparison Test determined the comprehension of decimals numbers. Additional questions sought the participants’ understanding of concentrations and reference ranges, and their preferences for the presentation of clinical pathology results. Results Of the 108 participants, 40% exhibited poor comprehension of decimal numbers. One-third of the medical laboratory scientists, a quarter of doctors, and half the nurses were characterised as lacking numeracy skills. The majority of participants (60%) thought it would be safer for results to be presented as whole numbers rather than as decimals with leading zeros. Conclusions The number of laboratory and clinical staff who show numeracy issues that could lead to misinterpretation of clinical pathology results and contribute to medical error strongly supports recommendations that pathology results should be presented as whole numbers.

Stable Laws and the Number of Ordinary

Abstract: Power total number of primes from the discharge of the decimal system is identified by the law of exponential growth with 14 fundamental physical constants. Model obtained on the parameters of the physical constants, proved less of the error and it gives more accurate predictions of the relative power of the set of prime numbers. The maximum absolute error of power (the number of primes), the traditional number is three times higher than suggested by us complete a number of prime numbers. Therefore, the traditional number 2, 3, 5, 7,. .. is only a special case. The transformation In10=2,30285… it was a rough rounded, leading to false identification of physico-mathematical regularities of different series of prime numbers. Model derived from physical constants, proved more accurate than the relative accuracy, and it gives more accurate predictions of the relative power of the set of prime numbers with increasing discharge the decimal number system.

Learning One-Digit Decimal Numbers by Measurement and Game Predicting Length.

Abstract: This paper aims to describe how students develop understanding of one-digit decimals. To achieve the aim, Local Instruction Theory (LIT) about the process of learning decimals and the means designed to support that learning are developed. Along with this idea, the framework of Realistic Mathematics Education (RME) is proposed. Based on the aim, design research methodology is used. This paper discusses learning activities of three meetings from teaching experiment of the focus group students of the fourth grade elementary school in Surabaya: SDIT Al Ghilmani. The data indicated that the learning activities promoted the students’ understanding of one-digit decimal numbers. Keyword: measurement, decimal numbers, number line  DOI : http://dx.doi.org/10.22342/jme.5.1.1447.35-46

Concept Development of Decimals in Chinese Elementary Students: A Conceptual Change Approach.

Abstract: The aim of this study was to examine the concept development of decimal numbers in 244 Chinese elementary students in grades 4–6. Three grades of students differed in their intuitive sense of decimals and conceptual understanding of decimals, with more strategic approaches used by older students. Misconceptions regarding the density nature of decimals indicated the progress in an ascending spiral trend (i.e., fourth graders performed the worst; fifth graders performed the best; and sixth graders regressed slightly), not in a linear trend. Misconceptions regarding decimal computation (i.e., multiplication makes bigger) generally decreased across grades. However, children's misconceptions regarding the density and infinity features of decimals appeared to be more persistent than misconceptions regarding decimal computation. Some students in higher grades continued to use the discreteness feature of whole numbers to explain the distance between two decimal numbers, indicating an intermediate level of understanding decimals. The findings revealed the effect of symbolic representation of interval end points and students' responses were contingent on the actual representations of interval end points. Students in all three grades demonstrated narrowed application of decimal values (e.g., merchandise), and their application of decimals was largely limited by their learning experiences.

Theoretical Foundations for the Analytical Computation of Coefficients of Basic Numbers of Krestenson's Transformation

Abstract: This paper presents theoretical foundations for the analytical transformation of coefficients of basic numbers of Krestenson's transformation, which significantly reduces the number of operations required to convert numbers from a residue number system to the decimal number system. An appropriate selection of modules makes it possible to efficiently use all processor registers.

What do Error Patterns tell us about Hong Kong Chinese and Australian Students' Understanding of Decimal Numbers?

Abstract: Mathematics educators have had a long standing interest in students’ understanding of decimal numbers. Most studies of students’ understanding of decimals have been conducted within Western cultural settings. The present study sought to gain insight into Chinese Hong Kong students’ and regional Australian students’ general performance on a variety of decimals tasks and to investigate students’ error patterns.

Efficient ASIC and FPGA Implementation of Binary-Coded Decimal Digit Multipliers

Abstract: Partial product generation (PPG), in radix-10 multiplication hardware, is often done through selection of pre-computed decimal multiples of the multiplicand. However, ASIC and FPGA realization of classical PPG via digit-by-digit multiplication has recently attracted some researchers. For example, a sequential multiplier, squarer, divider, FPGA parallel multiplier, and array multiplier are all based on a specific binary-coded decimal (BCD) digit multiplier (BDM). Most BDMs, as we have encountered, compute the binary product of two 1-digit BCD operands, and convert it to 2-digit BCD product. We provide our own version of two of these works with some adjustments and improvements, and offer two new low-cost BDMs in this category. However, a recent FPGA BDM uses straightforward truth table approach from scratch and skips binary product generation. We redesign the latter via low-level FPGA programming, and also provide its ASIC realization. We synthesize all the studied and new designs on ASIC and FPGA platforms, exhaustively check them for correctness, and compare their performance, to show that our two new designs, and the ASIC and new FPGA realizations of the aforementioned fully truth table-based design, outperform the previous ones in terms of one or more figures of merit.

Delay-based processing-in-wire for design of QCA serial decimal arithmetic units

Abstract: Quantum-dot cellular automata (QCA) technology is now considered to be one of the prospective technologies for a nanocomputer creation. The physical properties of QCA and its expanding range of computer applications make it expedient to use the novel paradigm of nanocomputer architecture: serial decimal storage-transfer-processing. The delay-based encoding of decimal digits allows the use a delay element as a main element of QCA serial arithmetic units. The simple implementation of the delay element by a short length of QCA wire results in reduction of complexity and of the area required for a QCA circuit. The theoretical basics of delay-based processing-in-wire and design examples of QCA serial decimal arithmetic units are presented.

Series Primes in Binary

Abstract: To prove the famous Riemann hypothesis, that the real part of the root is always exactly equal to 1/2, a series of 500 and the other prime numbers has been converted from decimal to binary number system. At the same time was a clear non-trivial zeros. Any prime number can be represented as quantized into binary digital signal. Quantization step to not dilute a number of prime numbers is 1. Number of levels (binary digits) depends on the power of the quantized number of primes. As a result, we get two types of zeros - the trivial and nontrivial. Capacity of a finite number of primes must be taken based on the completeness of block incidence matrix. Average statistical indicator is a binary number, and influencing variable - itself a prime number. The binary representation allows to visualize and geometric patterns in the full range of prime numbers.

A unified flagged prefix constant addition-subtraction scheme for design of area and power efficient binary floating-point and constant integer arithmetic circuits

Abstract: This paper presents a unified logic for flagged prefix addition-subtraction that eliminates the need to perform constant addition and subtraction in two separate blocks. The logic is based on a modified algorithm for constant subtraction that allows us to achieve the unification which is not possible with traditional algorithms. Thus we are able to eliminate the most crucial challenge that practical implementation of constant flagged structures faces. We present the applications of the proposed logic in the exponent biasing circuits of a binary floating-point unit and in a signed-digit decimal adder. Synthesis results show that close to 42% reduction in area and 24% in power is achieved when the unified logic is used with numerically large values like exponent biases. Even for numerically smaller constants like those used in signed-digit decimal adders, we get substantial benefit, with the area reducing by 12.3% and power by 12.4% in this case, thereby demonstrating the effectiveness of the proposed scheme for both small and large constants. Additionally, the propagation delay does not vary by more than 5–6% and power-delay product comes down by almost 20% in both the cases. On account of their power and area efficiency, proposed designs incorporating the unified logic can serve as good frameworks for Embedded DSP and financial applications.

Method for optimizing binary code in language having access to binary coded decimal variable, and computer and computer program

Abstract: A method for optimizing binary code in a language having access to binary coded decimal variable. The method includes: generating a first compiler expression of the binary code; analyzing a use-definition and/or a definition-use for the first compiler expression; generating a second compiler expression by identifying logical binary coded decimal (BCD) variables in the first compiler expression; assigning temporary variables to the logical BCD variables, wherein the second compiler expression includes packed decimal operations and the assigned temporary variables; and converting a packed decimal operation in the second compiler expression and an assigned temporary variable to a decimal floating point (DFP) if sign information and precision information are not lost during conversion from BCD to DFP, wherein identifying logical BCD variables includes: in the use-definition and/or definition-use of operands, regarding an operand of definition and an operand of use as the same logical BCD variables.

Numeral systems in Alor-Pantar languages

Abstract: This chapter presents an in-depth analysis of numeral forms and systems in the Alor-Pantar (AP) languages. The AP family reflects a typologically rare combination of mono-morphemic ‘six’ with quinary forms for numerals ‘seven’ to ‘nine’, a pattern which we reconstruct to go back to proto-AP. We focus on the structure of cardinal numerals, highlighting the diversity of the numeral systems involved. We reconstruct numeral forms to different levels of the AP family, and argue that AP numeral systems have been complicated at different stages by reorganisations of patterns of numeral formation and by borrowings. This has led to patchwork numeral systems in the modern languages, incorporating to different extents: (i) quaternary, quinary and decimal bases; (ii) additive, subtractive and multiplicative procedures, and; (iii) non-numeral lexemes such as ‘single’ and ‘take away’. Complementing the historical reconstruction with an areal perspective, we compare the numerals in the AP family with those of the Austronesian languages in their immediate vicinity and show that contact-induced borrowing of forms and structures has affected numeral paradigms in both AP languages and their Austronesian neighbors.

Decimal engine for energy-efficient multicore processors

Abstract: Prior work demonstrated the use of specialized pro-cessors, or accelerators, be energy-efficient for binary floating-point (BFP) division and square root, and for decimal floating-point (DFP) operations In the dark silicon era, where not all the circuits on the die can be powered simultaneously, we propose a hybrid BFP/DFP engine to perform BFP division and DFP addition, multiplication and division The main purpose of this engine is to offload the binary floating-point units for this type of operations and reduce the latency for decimal operations, and power and temperature for the whole die

Improved design of high-frequency sequential decimal multipliers

Abstract: Hardware implementation of decimal arithmetic operations has become a hot topic for research during the last decade. Among various operations, decimal multiplication is considered as one of the most complicated dyadic operations, which requires high-cost hardware implementation. Therefore, the processor industry has opted to use the sequential decimal multipliers to reduce the high cost of parallel architectures. However, the main drawback of iterative multipliers is their high latency. In this reported work, the focus has been on reducing the latency of decimal sequential multipliers while maintaining a low cost of area. Consequently, a high-frequency sequential decimal multiplier is proposed whose cycle time is reduced to the latency of a binary half-adder plus that of a decimal multiply-by-two operation, which overall is less than that of a decimal carry-save adder. The synthesis results reveal that the proposed sequential multiplier works with a higher clock frequency than the fastest previous decimal multiplier which in turn leads to overall latency advantage.

Exploring Rounding Errors in Matlab Using Extended Precision

Abstract: We describe a simple package of Matlab programs which implements an extended-precision class in Matlab. We give some examples of how this class can be used to demonstrate the effects of rounding errors and truncation errors in scientific computing. The package is based on a representation called Double-Double, which represents each floating-point real as an unevalu- ated sum of IEEE double-precision floating point numbers. This allows Matlab computations that are accurate to 30 decimal digits. The data structure, basic arithmetic and elementary functions are implemented as a Matlab class, entirely using the Matlab programming language.

Realization of Bi-Quinary Coded Decimal Adder in Quantum Dot Cellular Automata

Abstract: Bi-quinary coded parallel adder design with Quantum Cellular Automata is presented in this brief contribution. The nano-electronic computer architecture using QCA technology is in infancy stage. It requires more advancement with new approaches. In this paper the design of parallel decimal adder is proposed using bi-quinary encoding techniques with algorithm. The circuits are implemented using QCA designer tool and analyzed using simulation result. The signal propagation delay, complexity, required area, hardware cost are calculated and compare with previously proposed decimal QCA adders.

Flow supervision method and device

Abstract: The invention discloses a flow supervision method and device. Decimal parts generated in a process of operation by adopting a flow supervision algorithm is hidden in a system counter, and in a process of counting of the system counter, the decimal parts are accumulated and carry compensation is performed in time. By adoption of the flow supervision method and device, storage space of a decimal part of an intermediate variable of each queue can be removed, and errors brought by the decimal parts are eliminated, thereby enabling concrete realization to be accurate and error-free, and saving storage space at the same time.