Teoretyczne podstawy dzielenia z resztą liczb naturalnych wraz z uwagami o dzieleniu z resztą liczb całkowitych, wymiernych i rzeczywistych

In this article, I analyze the theoretical foundations of the division with remainder in the arithmetic of natural numbers. As a result of this analysis I justify that the notation a:b=c r s, where a, b, c, s are natural numbers and r denotes, is correct at school mathematics level and does not lead to a contrediction suggested by the author of the article (Semadeni, 1978). As a generalization of the division with remainder of natural numbers, I consider the division with remainder of integers, rational and real numbers.

Überbringen Störche Babys? Dinge, die wir Kindern nicht erzählen (können)

One of the main goals of the Complex Mathematics Education Experiment set by Tamás Varga was the following: “That is, the knowledge we provide fits the closest developmental zone and developmental level of the children; and yet is mathematically correct and forward-thinking. We do not tell stork tales.” (Varga, 1974, p. 1984.) We give some examples from the topic of number theory, where we cannot avoid telling “stork tales”, no matter how hard we try. In section 1.3 we describe some of the sources of disturbance. In section 2 we deal with the conflict of the different interpretations of some concepts occurring in primary/secondary school and university education, such as: “divisibility”, “divisor”, “common divisor”, “greatest common divisor”, “division with remainder”, the perceived or real special properties of zero, and “prime number”. We believe that it is important to make prospective teachers aware what facts they hide and why when teaching. In section 3 we present problems that can be discussed with children of different ages and different abstraction levels. Classification: D70, E40, F60, U60. Keywords: misconceptions and student errors, concept formation, treatment of mathematical concepts and definitions in mathematics education, number theory, educational games.

Algorithms and Data Structures for Sparse Polynomial Arithmetic

We provide a comprehensive presentation of algorithms, data structures, and implementation techniques for high-performance sparse multivariate polynomial arithmetic over the integers and rational numbers as implemented in the freely available Basic Polynomial Algebra Subprograms (BPAS) library. We report on an algorithm for sparse pseudo-division, based on the algorithms for division with remainder, multiplication, and addition, which are also examined herein. The pseudo-division and division with remainder operations are extended to multi-divisor pseudo-division and normal form algorithms, respectively, where the divisor set is assumed to form a triangular set. Our operations make use of two data structures for sparse distributed polynomials and sparse recursively viewed polynomials, with a keen focus on locality and memory usage for optimized performance on modern memory hierarchies. Experimentation shows that these new implementations compare favorably against competing implementations, performing between a factor of 3 better (for multiplication over the integers) to more than 4 orders of magnitude better (for pseudo-division with respect to a triangular set).