自然数与无限性思想的历史透视

无穷一直是诗人、艺术家、哲学家、神学家、科学家关注的焦点,它有着极为丰富的内涵,在不同的思想领域中有着不同的表现形式.自然数引出的无限多、无穷大等概念,打开了人类认识无限性的大门.对自然数序列"不可穷尽"的不同理解,产生了"实无限"与"潜无限"的数学哲学争论.     

三个不定方程式无自然数解的证明

采用分类法证明了三个不定方程式恒无自然数解。若此分类法的证明方法及步骤可行,则可能为比尔猜想的证明开创了新的思路。     

Complex fraction comparisons and the natural number bias: The role of benchmarks

People are often better at comparing fractions when the larger fraction has the larger rather than the smaller natural number components. However, there is conflicting evidence about whether this “natural number bias” occurs for complex fraction comparisons (e.g., 23/52 vs. 11/19). It is also unclear whether using benchmarks such as 1/2 or 1/4 enhances performance and reduces the bias (e.g., 11/19 > 1/2 and 23/52 < 1/2, hence 11/19 > 23/52). We asked 107 adults to solve complex fraction comparisons that did or did not afford using benchmarks, and we assessed response time and accuracy. We found a reverse bias (i.e., smaller components—larger fraction) that was greater among participants with lower mathematics experience. Fractions' proximity to 0 or 1 facilitated performance and decreased bias; effects of other benchmarks were nonsignificant. These results challenge the generality of the natural number bias in fraction comparison and highlight its variability.     

Variability in the natural number bias: Who,when, how,and why

When reasoning about rational numbers, people sometimes incorrectly apply principles or rules for natural numbers. Many factors affect whether participants display this natural number bias, including their age and experience, the affordances and constraints of the given task, and even the specific numbers in the given problem. In this paper, we argue that this variability can be conceptualized in terms of dynamic choices among problem-solving strategies. People's strategy choices vary as a function of their repertoire of available strategies and as a function of the specifics of the tasks, problems, and context. Further, we argue that the specific profiles of variability in strategy use that are observed in different participant groups can be conceptualized in terms of the strength and precision of the representations of numbers and operations that people in those groups possess. In our view, the natural number bias arises when people's representations of rational number magnitudes or rational number operations are not sufficiently strongly activated or sufficiently precise to guide performance on a specific task in a specific context. In these cases, participants' more highly activated or more precise representations for natural numbers may underlie and guide their performance. This account suggests that contexts and experiences (including instructional experiences) that help build, strengthen, and activate rational number representations should lead to improvements in performance.     

关于有限个自然数的幂和的组合恒等式计算法

自然数的幂和在幂指数较大的情况下计算是比较困难的。文章通过将自然数的幂表示为组合的线性结构,通过组合恒等式计算幂和,是一种行之有效的方法。     

《和为偶数的奇素数对的个数》之续篇

本续篇根据素数定理和有关无穷乘积,再度演化和为偶数的奇素数对的个数的求解公式,得出:和为偶数N的奇素数对的个数大于2N/πln2N,并且举几例比较结果.哥德巴赫猜想应该是和为偶数N的奇素数对的个数为1的一个特例。     

The role of inhibition in overcoming arithmetic natural number bias in the Chinese context: Evidence from behavioral and ERP experiments

The natural number bias (NNB) in arithmetic operations refers to the application of natural number properties to reasoning about rational numbers. Previous studies found the NNB interferes with students’ problem-solving. However, few studies have examined it in the Chinese context or the underlying mechanism by which it can be overcome. Addressing these gaps, in Experiments 1a (n = 31) and 1b (n = 30), we found that Chinese students demonstrate the NNB despite linguistic differences between Chinese and western languages. Experiment 2 (n = 38) adopted a negative priming paradigm and found that inhibitory control was necessary to overcome the NNB. Experiment 3 (n = 34) employed the event-related potential technique; we observed increased P2 amplitude when students solved congruent problems, and increased N2 and decreased P3 amplitude when they solved incongruent problems. These results indicated that the NNB is rooted in intuitive thinking, and overcoming this bias relies on inhibition.     

关于超实数域^＊R的基数的一些注记

证明了在扩大的分析的非标准模型中超实数域^*R，超有理数域^*Q，超自然数集^*N等集合的基数可以大于任何“标准基数”。     

和为偶数的奇素数对的个数

和为偶数N的奇数对可分为三种情况,第一种是奇合数对(这里把1看做奇合数);第二种是1个是奇合数、1个是奇素数的奇数对;第三种是奇素数对.小于N的奇合数的大约个数可以根据奇合数所含的因数情况来求出,和为N的奇合数对的大约个数也可以根据奇合数对所含的因数情况来求出,小于N的奇合数除两两组成和为N的奇合数对外,其余只能与小于N的奇素数组成和为N的奇数对.求出前两种和为N的奇数对的大约个数,就能求出和为N的奇素数对的大约个数.     

Introducing the empty number line

The empty number line (ENL) is an essential element of Dutch mathematics education, already internationally acknowledged as a successful model for developing mental strategies. The concept is a deceptively simple one but is based upon a complex structure of previous research findings. Several recent and influential UK publications have already misused the concept. This article is based upon the original work of Beishuisen from Leiden University and sets out a very simplified version of how the ENL is derived and built up for use in mental mathematics.     

伯努利数与集合的纯偶划分数

引入集合的纯偶划分数,给出了一些它的性质,用纯偶划分数得到了伯努利数的一种表示形式,得到正切数的一种递归表示,指出正切数与二进多项式的一个关系式.     

Fermion number fractionization

Solitons emerge as non-perturbative solutions of non-linear wave equations in classical and quantum theories. These are non-dispersive and localised packets of energy — remarkable properties for solutions of non-linear differential equations. In the presence of such objects, the solutions of Dirac equation lead to the curious phenomenon of ‘fractional fermion number’. Under normal conditions the fermion number takes strictly integral values. In this article, we describe this accidental discovery and its manifestation in polyacetylene chains, which has led to the development of organic conductors. (left) Kumar Rao is a Postdoctoral Fellow at PRL, Ahmedabad. He is interested in particle physics phenomenology as probed in particle colliders and formal aspects of quantum field theory. (right) Narendra Sahu is currently a postdoctoral fellow at Lancaster University, UK. His main research area includes Cosmology and Astroparticle physics. Currently he is working on dark matter and matter anti-matter asymmetry of the universe. (center) P K Panigrahi’s research interests are in the area of quantum computation, solitons in Bose Einstein condensates & nonlinear optical media. He is also deeply interested in science education and derives pleasure from long weekend walks.     

素数分布的一条规律

建立了平方位序分组相继数列素,以此为基础给出了素数分布的一个规律,运用这个规律证明了在相邻两个自然的平方之间至少存在一个素数,这一数论中的古典问题。     

人体细胞数目的探究

新编高中生物教材第一册第21页指出,人体细胞数目大约1014个.不少同学询问这个数目是怎样估算出来的,而且有的书籍中并不是这个数值,孰对孰错?带着这个问题,我们在高二(3)班成立了人体细胞数目探究课题组,课题组共有8名同学组成,组长王伟玲,指导教师李保兴.     

关于数论中的穷竭原则

The contents of the paper are to appear in book form shortly [4]and it has been given in the author’s lectures at Abdus Salam School of Math Sci.GC Univ.Lahore,Pakistan in Feb.,2011.We mean by the exhaustion principle that in a certain way,there is a rule which exhaust all the elements of a system,i.e.,a sort of classification.We shall illustrate this principle by significant examples.Since our examples are related to groups,the exhaustion is mostly of the form of a disjoint union of subsets Hd(dn) exhausting the whole group G:G=∪dnHd(disjoint).     

Directionality in the interrelations between approximate number,verbal number,and mathematics in preschool-aged children

A fundamental question in numerical development concerns the directional relation between an early-emerging non-verbal approximate number system (ANS) and culturally acquired verbal number and mathematics knowledge. Using path models on longitudinal data collected in preschool children (M_age = 3.86 years; N = 216; 99 males; 80.8% White; 10.8% Multiracial, 3.8% Latino; 1.9% Black; collected 2013–2017) over 1 year, this study showed that earlier verbal number knowledge was associated with later ANS precision (average β = .32), even after controlling for baseline differences in numerical, general cognitive, and language abilities. In contrast, earlier ANS precision was not associated with later verbal number knowledge (β = −.07) or mathematics abilities (average β = .10). These results suggest that learning about verbal numbers is associated with a sharpening of pre-existing non-verbal numerical abilities.     

Feeling number: grounding number sense in a sense of quantity

Drawing on results from psychology and from cultural and linguistic studies, we argue for an increased focus on developing quantity sense in school mathematics. We explore the notion of “feeling number”, a phrase that we offer in a twofold sense—resisting tendencies to feel numb-er (more numb) by developing a feeling for numbers and the quantities they represent. First, we distinguish between quantity sense and the relatively vague notion of number sense. Second, we consider the human capacity for quantity sense and place that in the context of related cultural issues, including verbal and symbolic representations of number. Third and more pragmatically, we offer teaching strategies that seem helpful in the development of quantity sense coupled with number sense. Finally, we argue that there is a moral imperative to connect number sense with such a quantity sense that allows students to feel the weight of numbers. It is important that learners develop a feeling for number, which includes a sense of what numbers are and what they can do.     

浅谈俄语中数字的用法

不同的民族，由于传统文化、地理环境、生活习俗、宗教信仰、语言习惯等方面的不同，有关数字的用法和内涵也有差别，从而形成了各具特色的民族数字文化。本文主要介绍俄罗斯的数字文化。     

关于孪生素数猜想证明的探讨

孪生素数猜想 ,即孪生素数是否无穷多 [1] ,是数论三大问题之一 .“所谓数论三大问题就是费尔马问题、孪生素数问题和哥德巴赫猜想 [1]” .我们在前人研究的基础上 ,先找出了勾股数组的排列顺序表[2 ] ,从中发现了大于 2的素数表达式 [3]和孪生素数的表达式 [4 ] ,在 [2 ]、[3]、[4 ]研究的基础上本文对孪生素数猜想证明做了进一步的探讨