Triangular numbers

The triangular numbers, which are numbers associated with certain arrays of dots, were known to the ancient Greeks and viewed by them with reverence. Though possessing a simple definition, they are astonishingly rich in properties of various kinds, ranging from simple relationships between them and the square numbers to very complex relationships involving partitions, modular forms, etc. ?? topics which belong to advanced mathematics. They also possess many pretty combinatorial properties. In this expository article we survey a few of these properties.     

Afterword: embracing numbers?

ABSTRACT

While the use of numbers in governance has a long history, the kinds of numbers we now produce enable a range of new possibilities for monitoring, regulation and policy decision-making. Global policy actors are now calling for a steep increase in investment in education data. The growing trust in numbers has been critiqued by education policy scholars and social scientists, who have pointed to the reductionist nature of numbers and the dangerous decontextualisation of information which are leading to detrimental policies. The entry of big data poses even more complex epistemological and ontological challenges, many of which we do not fully understand as yet. This paper acknowledges these challenges, and at the same time speculates that big data might afford unique possibilities for new relational theories that may lead to better policy decision-making.     

Transfinite numbers

In a series of revolutionary articles written during the last quarter of the nineteenth century, the great German mathematician Georg Cantor removed the age-old mistrust of infinity and created an exceptionally beautiful and useful theory of transfinite numbers. This is an introductory article on this topic.     

Complex numbers and plane geometry

The representation of complex numbers as points of the Euclidean plane naturally leads to a two-way interaction between geometry and numbers. The geometry of the plane has a very deep influence in the study complex analytic functions. In this article, we illustrate the other way aspect by a few simple-minded application of complex numbers to give elegant solutions of problems in plane geometry, such as Ptolemy’s Theorem, Euler-line and Nine-point Circle Theorem. Anant R Shastri is a Professor at IIT, Bombay. His research inerests are in algebraic topology and algebraic geometry. He is also keen in math education and music. This article is based on a talk given to an audience consisting mainly students of class IX and X, at Nehru Science Centre under the aegis of Bombay Association for Science Education and Bombay Math. Colloq. on 25th Jan. 2003. An earlier version of this article was published in Bona Mathematica, Vol.14 Nos.1–2, 2003.     

中西方数字文化差异

世界上任何一种语言都是植根于特定的文化背景之中,反映着特定的文化内容。语言中数字是反映客观物质世界的规模、大小的数和量,而由于各自语言中所包涵的文化传统、思维观念以及对数字的感知取向不同,故而赋予数字的社会文化涵义也不尽相同。世界各民族都有自己的数字文化,数字蕴含的文化信息鲜明地体现了该民族文化的心理、传统习俗与特色。文章从中西方数字文化中有代表性的吉祥数字和禁忌数字入手,探讨其在汉、英两种语言中所体现出的中西方不同的文化内涵和民族文化习俗、宗教信仰、审美观等文化差异。     

关于费波那奇数与契贝谢夫多项式

建立了著名的费波那奇数与契贝谢夫多项式之间的关系,并由此推出一些包含费波那奇数与契贝谢夫多项式的一些恒等式。     

一类包含高阶Bernoulli数和高阶Euler数的积分计算公式

给出了一类包含高阶Bernoulli数和高阶Euler数的积分计算公式．     

利用函数思想 巧解数列问题

利用函数思想,给出了求递推数列通项及进行不等式证明的一种简便方法.     

从Fibonacci数的恒等变换到Lucas数的同余式

研究了Fibonacci数的和式∑a b=nUa^mUb^m/a!b!,得出了一些关于Fibonacci数与Lucas数的恒等变换和一些有趣的Lucas数的同余式。     

IEEE standard for floating point numbers

Floating point numbers are an important data type in computation which is used extensively. Yet, many users do not know the standard which is used in almost all computer hardware to store and process these. In this article, we explain the standards evolved by The Institute of Electrical and Electronic Engineers in 1985 and augmented in 2008 to represent floating point numbers and process them. This standard is now used by all computer manufacturers while designing floating point arithmetic units so that programs are portable among computers.     

QR分解和Cholesky分解的Rice条件数

条件数是在计算过程中由于误差引起的放大系数, 所以条件数理论在误差分析中占有非常重要的地位. 本文运用Rice关于条件数的一般理论, 采取一种统一的方式, 在单参数扰动的情况下, 定义了与正定对称矩阵的Cholesky 分解和一般矩阵的QR分解有关的一些矩阵因子的条件数. 利用解析展开和矩阵向量方程的方法, 求出了用Frobenius 范数所定义的Rice条件数的具体表达式. 所得结果与常小文的结果类似. 在Cholesky分解情况下, 与因子矩阵L 相对应的条件数 KL是 Stewart条件数K的一个下界.     

几个有关Euler数的恒等式

利用生成函数法研究了正割函数与欧拉数的关系,得到了几个关于欧拉数的有趣的恒等式.     

Numbers defy the law of large numbers

As the number of independent tosses of a fair coin grows, the rates of heads and tails tend to equality. This is misinterpreted by many students as being true also for the absolute numbers of the two outcomes, which, conversely, depart unboundedly from each other in the process. Eradicating that misconception, as by coin‐tossing experiments, should be incorporated early on into learning the law of large numbers.     

概数的性质及语义解释

本文在对重要语法著作关于概数的论述的基础上，对概数的性质、表示概数的主要方法及其语义解释和概数表示指定区间中不同数的概率进行了探讨。     

What young children know about numbers

Two sources of evidence that young children know some things about number without the benefit of school instruction are reviewed: (a) Preschoolers follow counting principles in order to represent the numerical value of a display and can reason arithmetically about numbers they represent accurately; (b) children in the early primary grades achieve an understanding that there is no largest number and that numbers never end. It is suggested that early numerical abilities, like early language abilities, are universal cognitive abilities. The notion of number readiness in terms of Piagetian theory is discussed. Consideration is given to the kind of further research required to bridge the research findings and educational practice.     

质量数为什么会大于原子质量

质量数和原子质量是中学化学中常见的两个基本概念,前者表示某核素原子核中质子数与中子数之和,后者表示某核素一个原子的质量;搞清两者的相对大小,对学习化学知识有很大的意义。     

课堂教学质量的区间值模糊评判

通过一元实函数和可信度的引入，给出了fuzzy区间数的一种排序方法，并就区间数fuzzy综合评判建立了一种课堂教学质量的综合评判模型．     

Mayer-Jensen shell model and magic numbers

Many nuclear models have been put forward since 1932. Among them the collective model proposed by Aage Niels Bohr and Ben Roy Mottelson and the nuclear shell model proposed by Maria Goeppert Mayer and Johannes Hans D Jensen are the two most successful models. A number of experimental facts like the existence of magic numbers compiled by Maria Mayer led to the discovery of the nuclear shell model. The addition of a nuclear spin-orbit coupling force to the mean field of the nucleons successfully predicted the nuclear magic numbers and many other properties of nuclei. R Velusamy is the Head of the PG Department of Physics, Ayya Nadar Janaki Ammal College, Sivakasi, Tamilnadu. His current interests are in quantum dots and wave packet dynamics.     

数字中有爱国主义教育

本文从数字的直观性、形象性、可信度高及可比性强等特点出发,说明它对中国历史课程中进行爱国主义教育有着重要作用.     

Policy as numbers: ac/counting for educational research

This paper provides an account and a critique of the rise of the contemporary policy as numbers phenomenon and considers its effects on policy and for educational research. Policy as numbers is located within the literatures on numbers in politics and the statistics/state relationship and, while recognising the longevity of the latter relationship, it is argued that the governance turn and neo-liberalism have strengthened the role of numbers in contemporary education policy. This phenomenon is situated in the contemporary ‘structure of feeling’, which sees politics reduced to managing the everyday and the evisceration of a progressive imaginary. The paper then documents the impact within education, focusing both on the emergent global education policy field and on the national agenda in Australian schooling and the related rise of ‘gap talk’, both globally and nationally. The paper concludes by drawing out some implications for educational research, suggesting that we as educational researchers are also being positioned by policy as numbers.