Representation Of Natural Numbers Research Articles

Binary Numeration System with Alternating Signed Digits and Its Graph Theoretical Relationship

The binary number system is the basic number representation in computing. We can encode natural numbers with finite 0-1 sequences. The representation of natural numbers is based on this system. However, this poses problems and is technically not perfect. Several attempts have been made to handle integers (signed numbers). We mention only two: the balanced triple number system and the number system with base −2. Our paper introduces new possibilities. We also shed light on the graph theoretical background of the new number systems.

Efficient congruencing in ellipsephic sets: the quadratic case

In this paper, we bound the number of solutions to a quadratic Vinogradov system of equations in which the variables are required to satisfy digital restrictions in a given base. Certain sets of permitted digits, namely those giving rise to few representations of natural numbers as sums of elements of the digit set, allow us to obtain better bounds than would be possible using the size of the set alone. In particular, when the digits are required to be squares, we obtain diagonal behaviour with 12 variables.

Representation of natural numbers by sum of four squares of almost-prive having a special form

In this paper we consider the equation $x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} = N$, where $N$ is a sufficiently large integer and prove that if $\eta $ is quadratic irrational number and $0 < \lambda < \frac{1}{10}$, then it has a solution in almost-prime numbers $x_{1}, \ldots , x_{4}$, such that $\{ \eta x_{i} \} < N^{-\lambda }$ for $i = 1, \ldots , 4$.

Is thirty-two three tens and two ones? The embedded structure of cardinal numbers

The acquisition and representation of natural numbers have been a central topic in cognitive science. However, a key question in this topic about how humans acquire the capacity to understand that numbers make ‘infinite use of finite means’ (or that numbers are generative) has been left unanswered. While previous theories rely on the idea of the successor principle, we propose an alternative hypothesis that children's understanding of the syntactic rules for building complex numerals—or numerical syntax—is a crucial foundation for the acquisition of number concepts. In two independent studies, we assessed children's understanding of numerical syntax by probing their knowledge about the embedded structure of cardinal numbers using a novel task called Give-a-number Base-10 (Give-N10). In Give-N10, children were asked to give a large number of items (e.g., 32 items) from a pool that is organized in sets of ten items. Children's knowledge about the embedded structure of numbers (e.g., knowing that thirty-two items are composed of three tens and two ones) was assessed from their ability to use those sets. Study 1 tested English-speaking 4- to 10-year-olds and revealed that children's understanding of the embedded structure of numbers emerges relatively late in development (several months into kindergarten), beyond when they are capable of making a semantic induction over a local sequence of numbers. Moreover, performance in Give-N10 was predicted by other task measures that assessed children's knowledge about the syntactic rules that govern numerals (such as counting fluency), demonstrating the validity of the measure. In Study 2, this association was tested again in monolingual Korean kindergarteners (5–6 years), as we aimed to test the same effect in a language with a highly regular numeral system. It replicated the association between Give-N10 performance and counting fluency, and it also demonstrated that Korean-speaking children understand the embedded structure of cardinal numbers earlier in the acquisition path than English-speaking peers, suggesting that regularity in numerical syntax facilitates the acquisition of generative properties of numbers. Based on these observations and our theoretical analysis of the literature, we propose that the syntax for building complex numerals, not the successor principle, represents a structural platform for numerical thinking in young children.

Strategies Applied by Pupils During the Representation of Natural Numbers

Strategies Applied by Pupils During the Representation of Natural Numbers

Compaction of Church Numerals

In this study, we address the problem of compaction of Church numerals. Church numerals are unary representations of natural numbers on the scheme of lambda terms. We propose a novel decomposition scheme from a given natural number into an arithmetic expression using tetration, which enables us to obtain a compact representation of lambda terms that leads to the Church numeral of the natural number. For natural number n, we prove that the size of the lambda term obtained by the proposed method is O ( ( slog 2 n ) ( log n / log log n ) ) . Moreover, we experimentally confirmed that the proposed method outperforms binary representation of Church numerals on average, when n is less than approximately 10,000.

A New Set Theory for Analysis

We provide a canonical construction of the natural numbers in the universe of sets. Then, the power set of the natural numbers is given the structure of the real number system. For this, we prove the co-finite topology, C o f ( N ) , is isomorphic to the natural numbers. Then, we prove the power set of integers, 2 Z , contains a subset isomorphic to the non-negative real numbers, with all its defining structures of operations and order. We use these results to give the power set, 2 N , the structure of the real number system. We give simple rules for calculating addition, multiplication, subtraction, division, powers and rational powers of real numbers, and logarithms. Supremum and infimum functions are explicitly constructed, also. Section 6 contains the main results. We propose a new axiomatic basis for analysis, which represents real numbers as sets of natural numbers. We answer Benacerraf’s identification problem by giving a canonical representation of natural numbers, and then real numbers, in the universe of sets. In the last section, we provide a series of graphic representations and physical models of the real number system. We conclude that the system of real numbers is completely defined by the order structure of natural numbers and the operations in the universe of sets.

Extension of van der Corput algorithm to LS-sequences

The LS-sequences of points recently introduced by the author are a generalization of van der Corput sequences. They are constructed by reordering the points of the corresponding LS-sequences of partitions. Here we present another algorithm which is simpler to compute than the original construction and coincides with the classical one for van der Corput sequences. This algorithm is based on the representation of natural numbers in base L+S. Moreover, when S⩽L these sequences have low discrepancy and can be useful in Quasi Monte-Carlo methods.

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

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.

Representation of natural numbers by sums of four squares of integers having a special form

This paper obtains an asymptotic formula for the number of solutions to the equation \( l_1^2 + { }l_2^2 + l_3^2 + l_4^2 = N \) in integers l 1, l 2, l 3, l 4 such that a < {ηl j } < b, where η is a quadratic irrational number, 0 ≤ a < b ≤ 1, j = 1, 2, 3, 4.

The decimal representation of real numbers

The representation of natural numbers in decimal form is an unequivocal procedure while for the representation of real numbers some ambiguities concerning the existence of infinitely many digits equal to 9 still emerge. One of the most frequently confronted misunderstandings is whether 0.999 … equals 1 or not, and if not what number does this sequence of digits represent. In this review article, the decimal representation of any real number is explicitly presented. In particular, it is investigated whether this representation is unique or not. A condition is given that guarantees the uniqueness of decimal representation for a subset of real numbers, while for the remaining numbers two decimal representations exist. It is also investigated which sequences of digits cannot be accepted as decimal representations of real numbers. Moreover, analogous results are presented in the case where a real number is represented in systems with base n, n being a natural number greater than 1.

Representation of Numbers by Quadratic Forms. Main Results of the Research Done in Georgia

Abstract This is a survey of the main results obtained by the mathematicians of Georgia on the representation of natural numbers by integral quadratic forms before 2000.

Complexity, decidability and completeness

AbstractWe give resource bounded versions of the Completeness Theorem for propositional and predicate logic. For example, it is well known that every computable consistent propositional theory has a computable complete consistent extension. We show that, when length is measured relative to the binary representation of natural numbers and formulas, every polynomial time decidable propositional theory has an exponential time (EXPTIME) complete consistent extension whereas there is a nondeterministic polynomial time (NP) decidable theory which has no polynomial time complete consistent extension when length is measured relative to the binary representation of natural numbers and formulas. It is well known that a propositional theory is axiomatizable (respectively decidable) if and only if it may be represented as the set of infinite paths through a computable tree (respectively a computable tree with no dead ends). We show that any polynomial time decidable theory may be represented as the set of paths through a polynomial time decidable tree. On the other hand, the statement that every polynomial time decidable tree represents the set of complete consistent extensions of some theory which is polynomial time decidable, relative to the tally representation of natural numbers and formulas, is equivalent to P = NP. For predicate logic, we develop a complexity theoretic version of the Henkin construction to prove a complexity theoretic version of the Completeness Theorem. Our results imply that that any polynomial space decidable theory Δ possesses a polynomial space computable model which is exponential space decidable and thus Δ has an exponential space complete consistent extension. Similar results are obtained for other notions of complexity.

On the Representation of Natural Numbers in Positional Numeral Systems1

On the Representation of Natural Numbers in Positional Numeral Systems1 In this paper we show that every natural number can be uniquely represented as a base-b numeral. The formalization is based on the proof presented in [11]. We also prove selected divisibility criteria in the base-10 numeral system.

On the Representation of Natural Numbers as Sums of Squares

(2005). On the Representation of Natural Numbers as Sums of Squares. The American Mathematical Monthly: Vol. 112, No. 2, pp. 168-171.

On the Representation of Natural Numbers as Sums of Squares

On the Representation of Natural Numbers as Sums of Squares

Digital Sum Problems for the Gray Code Representation of Natural Numbers

We give simple explicit formulas of the power sum and the exponential sum of digital sums of the Gray code representation of natural numbers by use of the distribution function of the singular measure.

Representation of natural numbers in quantum mechanics

This paper represents one approach to making explicit some of the assumptions and conditions implied in the widespread representation of numbers by composite quantum systems. Any nonempty set and associated operations is a set of natural numbers or a model of arithmetic if the set and operations satisfy the axioms of number theory or arithmetic. This paper is limited to $k\ensuremath{-}\mathrm{ary}$ representations of length L and to the axioms for arithmetic modulo ${k}^{L}.$ A model of the axioms is described based on an abstract L-fold tensor product Hilbert space ${\mathcal{H}}^{\mathrm{arith}}.$ Unitary maps of this space onto a physical parameter based product space ${\mathcal{H}}^{\mathrm{phy}}$ are then described. Each of these maps makes states in ${\mathcal{H}}^{\mathrm{phy}},$ and the induced operators, a model of the axioms. Consequences of the existence of many of these maps are discussed along with the dependence of Grover's and Shor's algorithms on these maps. The importance of the main physical requirement, that the basic arithmetic operations are efficiently implementable, is discussed. This condition states that there exist physically realizable Hamiltonians that can implement the basic arithmetic operations and that the space-time and thermodynamic resources required are polynomial in L.

Representation of natural numbers as the sum of values of polynomials of primes

Representation of natural numbers as the sum of values of polynomials of primes

On sets of integers with prescribed gaps

For a fixed setI of positive integers we consider the set of paths (po,...,p k ) of arbitrary length satisfyingp l −pl−1∈I forl=2,...,k andp0=1,p k =n. Equipping it with the uniform distribution, the random path lengthT n is studied. Asymptotic expansions of the moments ofT n are derived and its asymptotic normality is proved. The step lengthsp l −pl−1 are seen to follow asymptotically a restricted geometrical distribution. Analogous results are given for the free boundary case in which the values ofp0 andp k are not specified. In the special caseI={m+1,m+2,...} (for some fixed m∈ℕ) we derive the exact distribution of a random “m-gap” subset of {1,...,n} and exhibit some connections to the theory of representations of natural numbers. A simple mechanism for generating a randomm-gap subset is also presented.