A working model for rational numbers

1975 ◽  
Vol 22 (4) ◽  
pp. 328-332
Author(s):  
J. Paul Moulton
Keyword(s):  
Rational Number ◽  
Rational Numbers ◽  
Abstract Concept ◽  
Working Model ◽  
Cuisenaire Rods

There are many good models for the rational numbers-area models, Cuisenaire rods, lattices, for example. Each has its merits and its limitations, and none is so good that it should be used to the exclusion of all others. Indeed, it is partly through a variety of experiences that a person acquires the ability to relate the abstract concept of a rational number to the properties of the practical world. I should therefore like to suggest still another model, one that has certain advantages of its own.

scholarly journals On the integer part of the reciprocal of the Riemann zeta function tail at certain rational numbers in the critical strip

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
WonTae Hwang ◽  
Kyunghwan Song
Keyword(s):  
Zeta Function ◽  
Rational Number ◽  
Riemann Zeta Function ◽  
Rational Numbers ◽  
Critical Strip ◽  
Integer Part ◽  
Integral Points ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract We prove that the integer part of the reciprocal of the tail of $\zeta (s)$ ζ ( s ) at a rational number $s=\frac{1}{p}$ s = 1 p for any integer with $p \geq 5$ p ≥ 5 or $s=\frac{2}{p}$ s = 2 p for any odd integer with $p \geq 5$ p ≥ 5 can be described essentially as the integer part of an explicit quantity corresponding to it. To deal with the case when $s=\frac{2}{p}$ s = 2 p , we use a result on the finiteness of integral points of certain curves over $\mathbb{Q}$ Q .

Call for Manuscripts for the 2014 Focus Issue: Rational Number Sense – October 2012

2012 ◽  
Vol 18 (3) ◽  
pp. 189
Keyword(s):  
Rational Number ◽  
Number Sense ◽  
Rational Numbers

This call for manuscripts is requesting articles that address how to make sense of rational numbers in their myriad forms, including as fractions, ratios, rates, percentages, and decimals.

scholarly journals Gaussian Integers

2013 ◽  
Vol 21 (2) ◽  
pp. 115-125
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Daichi Mizushima ◽  
Yasunari Shidama
Keyword(s):  
Number Field ◽  
Rational Number ◽  
Rational Numbers ◽  
Gaussian Integer ◽  
Algebraic Integers ◽  
Quotient Field ◽  
Gaussian Integers ◽  
Integer Ring

Summary Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic.

scholarly journals Approximation of rational numbers by Dedekind sums

2014 ◽  
Vol 10 (05) ◽  
pp. 1241-1244 ◽  
Author(s):  
Kurt Girstmair
Keyword(s):  
Rational Number ◽  
Rational Numbers ◽  
Dedekind Sums ◽  
Dedekind Sum

Given a rational number x and a bound ε, we exhibit m, n such that |x - s(m, n)| < ε. Here s(m, n) is the classical Dedekind sum and the parameters m and n are completely explicit in terms of x and ε.

Have You Read …?

1969 ◽  
Vol 62 (3) ◽  
pp. 220-221
Author(s):  
Philip Peak
Keyword(s):  
Real Number ◽  
Rational Number ◽  
Geometric Approach ◽  
Rational Numbers ◽  
Polynomial Equations ◽  
Real Numbers ◽  
Basic Principles ◽  
New Ideas

One of the basic principles we follow in our teaching is to relate new ideas with old ideas. Dr. Forbes has done just this in his article about extending the concept of rational numbers to real numbers. He points out how this extension cannot follow the same pattern as that of extensions positive to negative integers or from integers to rationals. If we look to a definition for motivating the extension we at best can only say, “Some polynomial equations have no rational number solutions and do have some real number solutions.” We might use least-upperbound idea, or we might try motivating through nonperiodic infinite decimals. However, Dr. Forbes rejects all of these and makes the tie-in through a geometric approach.

scholarly journals The relational nature of rational numbers

Pythagoras ◽  
2015 ◽  
Vol 36 (1) ◽  
Author(s):  
Bruce Brown
Keyword(s):  
Conceptual Change ◽  
Number Field ◽  
Rational Number ◽  
Teaching And Learning ◽  
Field Research ◽  
Mathematical Structure ◽  
Rational Numbers ◽  
Number Line ◽  
Complex Nature ◽  
Research Projects

It is commonly accepted that the knowledge and learning of rational numbers is more complex than that of the whole number field. This complexity includes the broader range of application of rational numbers, the increased level of technical complexity in the mathematical structure and symbol systems of this field and the more complex nature of many conceptual properties of the rational number field. Research on rational number learning is divided as to whether children’s difficulties in learning rational numbers arise only from the increased complexity or also include elements of conceptual change. This article argues for a fundamental conceptual difference between whole and rational numbers. It develops the position that rational numbers are fundamentally relational in nature and that the move from absolute counts to relative comparisons leads to a further level of abstraction in our understanding of number and quantity. The argument is based on a number of qualitative, in-depth research projects with children and adults. These research projects indicated the importance of such a relational understanding in both the learning and teaching of rational numbers, as well as in adult representations of rational numbers on the number line. Acknowledgement of such a conceptual change could have important consequences for the teaching and learning of rational numbers.

EXPRESS: People’s Preferences for Different Types of Rational Numbers in Linguistic Contexts

2022 ◽  
pp. 174702182210763
Author(s):  
Xiaoming Yang ◽  
Yunqi Wang
Keyword(s):  
Rational Number ◽  
General Pattern ◽  
Rational Numbers ◽  
The Other ◽  
Multiple Factors ◽  
Corpus Studies ◽  
Different Types ◽  
The One ◽  
General Linguistic ◽  
Preference Experiment

Rational numbers, like fractions, decimals, and percentages, differ in the concepts they prefer to express and the entities they prefer to describe as previously reported in display-rational number notation matching tasks and in math word problem compiling contexts. On the one hand, fractions and percentages are preferentially used to express a relation between two magnitudes, while decimals are preferentially used to represent a magnitude. On the other hand, fractions and decimals tend to be used to describe discrete and continuous entities, respectively. However, it remains unclear whether these reported distinctions can extend to more general linguistic contexts. It also remains unclear which factor, the concept to be expressed (magnitudes vs. relations between magnitudes) or the entity to be described (countable vs. continuous), is more predictive of people’s preferences for rational number notations. To explore these issues, two corpus studies and a number notation preference experiment were administered. The news and conversation corpus studies detected the general pattern of conceptual distinctions across rational number notations as observed in previous studies; the number notation preference experiment found that the concept to be expressed was more predictive of people’s preferences for number notations than the entity to be described. These findings indicate that people’s biased uses of rational numbers are constrained by multiple factors, especially by the type of concepts to be expressed, and more importantly, these biases are not specific to mathematical settings but are generalizable to broader linguistic contexts.

scholarly journals Hard Lessons: Why Rational Number Arithmetic Is So Difficult for So Many People

2017 ◽  
Vol 26 (4) ◽  
pp. 346-351 ◽  
Author(s):  
Robert S. Siegler ◽  
Hugues Lortie-Forgues
Keyword(s):  
Rational Number ◽  
Rational Numbers ◽  
Advanced Mathematics ◽  
Arithmetic Operations ◽  
Teacher Understanding ◽  
Decimal Arithmetic ◽  
Poor Understanding ◽  
Mathematics And Science

Fraction and decimal arithmetic pose large difficulties for many children and adults. This is a serious problem because proficiency with these skills is crucial for learning more advanced mathematics and science and for success in many occupations. This review identifies two main classes of difficulties that underlie poor understanding of rational number arithmetic: inherent and culturally contingent. Inherent sources of difficulty are ones that are imposed by the task of learning rational number arithmetic, such as complex relations among fraction arithmetic operations. They are present for all learners. Culturally contingent sources of difficulty are ones that vary among cultures, such as teacher understanding of rational numbers. They lead to poorer learning among students in some places rather than others. We conclude by discussing interventions that can improve learning of rational number arithmetic.

Teaching one of the differences between rational numbers and whole numbers

1971 ◽  
Vol 18 (5) ◽  
pp. 317-320
Author(s):  
Robert W. Prielipp
Keyword(s):  
Elementary School ◽  
School Children ◽  
Rational Number ◽  
Elementary School Children ◽  
Rational Numbers ◽  
Whole Numbers

What are some ways in which rational number (fractions) differ from whole numbers? How can we make these differences evident to elementary school children? We begin by looking at two ways in which rational numbers and whole numbers differ, and then we consider in depth a procedure that can be followed to teach one of these differences.

Teaching Rational Numbers—Intermediate Grades

1984 ◽  
Vol 31 (6) ◽  
pp. 40-42
Author(s):  
Marilyn Hall Jacobson
Keyword(s):  
Rational Number ◽  
Middle Grades ◽  
Rational Numbers ◽  
Intermediate Grades

The introduction of decimals is an important rational-number activity in the middle grades.

Sign in / Sign up

Export Citation Format

Share Document