Rational and Irrational Numbers. Approximation of Numbers by Rational Numbers (Diophantine Approximation)

2003 ◽  
pp. 85-107
Author(s):  
Paul Erdős ◽  
János Surányi
Keyword(s):  
Diophantine Approximation ◽  
Rational Numbers ◽  
Irrational Numbers

Rational Numbers and Irrational Numbers

2018 ◽  
pp. 63-72
Author(s):  
Daniel Rosenthal ◽  
David Rosenthal ◽  
Peter Rosenthal
Keyword(s):  
Rational Numbers ◽  
Irrational Numbers

A number pencil

1967 ◽  
Vol 14 (7) ◽  
pp. 557-559
Author(s):  
David M. Clarkson
Keyword(s):  
Elementary Teachers ◽  
Sixth Grade ◽  
Rational Numbers ◽  
Class Discussion ◽  
Irrational Numbers ◽  
Whole Numbers ◽  
Number Lines

So much use is being made of number lines these days that it may not occur to elementary teachers to represent numbers in other ways. There are, in fact, many ways to picture whole numbers geometrically as arrays of squares or triangles or other shapes. Often, important insights into, for example, oddness and evenness can be gained by such representations. The following account of a sixth-grade class discussion of fractions shows how a “number pencil” can be constructed to represent all the positive rational numbers, and, by a similar method, also the negative rationals. An extension of this could even be made to obtain a number pencil picturing certain irrational numbers.

The simultaneous diophantine approximation of certain kth roots

1970 ◽  
Vol 67 (1) ◽  
pp. 75-86 ◽  
Author(s):  
Charles F. Osgood
Keyword(s):  
Diophantine Approximation ◽  
Simultaneous Approximation ◽  
Rational Numbers ◽  
Simultaneous Diophantine Approximation ◽  
Explicit Bounds ◽  
Positive Integers

In a recent paper in this journal(1) Baker obtained a result giving effectively computable bounds on the simultaneous approximation of certain numbers of the form (ab-1)nj/n by rational numbers of the formpιq-1 where a, b, n, the nι and q are positive integers while the pι are non-negative integers. In this paper we shall obtain explicit bounds on the simultaneous approximation of certain numbers of the form rβ1, …, rβn where β and each rι, 1 ≤ j ≤ n, are positive rational numbers.

Exceptional algebraic properties of the three quadratic irrationalities observed in quasicrystals

2001 ◽  
Vol 79 (2-3) ◽  
pp. 687-696 ◽  
Author(s):  
Z Masáková ◽  
J Patera ◽  
E Pelantová
Keyword(s):  
Binary Operation ◽  
Rational Numbers ◽  
Point Sets ◽  
Irrational Numbers ◽  
Algebraic Properties ◽  
Equivalent Characterization

There are only three irrationalities directly related to experimentally observed quasicrystals, namely, those which appear in extensions of rational numbers by Ö5, Ö2, Ö3. In this article, we demonstrate that the algebraically defined aperiodic point sets with precisely these three irrational numbers play an exceptional role. The exceptional role stems from the possibility of equivalent characterization of these point sets using one binary operation. PACS Nos.: 61.90+d, 61.50-f

On a Hardy-Littlewood problem of diophantine approximation

1939 ◽  
Vol 35 (4) ◽  
pp. 527-547 ◽  
Author(s):  
D. C. Spencer
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Diophantine Approximation ◽  
Irrational Numbers ◽  
Image Position

1. Let .When r is a positive integer, various writers have considered sums of the formwhere ω1 and ω2 are two positive numbers whose ratio θ = ω1/ω2 is irrational and ξ is a real number satisfying 0 ≤ ξ < ω1. In particular, Hardy and Littlewood (2,3,4), Ostrowski(9), Hecke(6), Behnke(1), and Khintchine(7) have given best possible approximations for sums of this type for various classes of irrational numbers. Most writers have confined themselves to the case r = 1, in which

Irrational Numbers Can In-Spiral You

2007 ◽  
Vol 12 (8) ◽  
pp. 442-446
Author(s):  
Leslie D. Lewis
Keyword(s):  
Rational Numbers ◽  
Pythagorean Theorem ◽  
Irrational Numbers ◽  
Natural Context

Introducing students to the pythagorean theorem presents a natural context for investigating what irrational numbers are and how they differ from rational numbers. This artistic project allows students to visualize, discuss, and create a product that displays irrational and rational numbers.

scholarly journals Do we have a sense for irrational numbers?

2017 ◽  
Vol 2 (3) ◽  
pp. 170-189 ◽  
Author(s):  
Andreas Obersteiner ◽  
Veronika Hofreiter
Keyword(s):  
Correct Response ◽  
Response Times ◽  
Number Sense ◽  
Irrational Number ◽  
Rational Numbers ◽  
Cube Root ◽  
Irrational Numbers ◽  
Whole Numbers ◽  
Symbolic Notation ◽  
Whole Number

Number sense requires, at least, an ability to assess magnitude information represented by number symbols. Most educated adults are able to assess magnitude information of rational numbers fairly quickly, including whole numbers and fractions. It is to date unclear whether educated adults without training are able to assess magnitudes of irrational numbers, such as the cube root of 41. In a computerized experiment, we asked mathematically skilled adults to repeatedly choose the larger of two irrational numbers as quickly as possible. Participants were highly accurate on problems in which reasoning about the exact or approximate value of the irrational numbers’ whole number components (e.g., 3 and 41 in the cube root of 41) yielded the correct response. However, they performed at random chance level when these strategies were invalid and the problem required reasoning about the irrational number magnitudes as a whole. Response times suggested that participants hardly even tried to assess magnitudes of the irrational numbers as a whole, and if they did, were largely unsuccessful. We conclude that even mathematically skilled adults struggle with quickly assessing magnitudes of irrational numbers in their symbolic notation. Without practice, number sense seems to be restricted to rational numbers.

scholarly journals Farey Sequences for Thin Groups

2021 ◽  
Author(s):  
Christopher Lutsko
Keyword(s):  
Explicit Formula ◽  
Diophantine Approximation ◽  
Continued Fraction ◽  
Rational Numbers ◽  
Gauss Map ◽  
Gauss Measure ◽  
Farey Sequence ◽  
Farey Sequences ◽  
Partial Quotients

Abstract The Farey sequence is the set of rational numbers with bounded denominator. We introduce the concept of a generalized Farey sequence. While these sequences arise naturally in the study of discrete and thin subgroups, they can be used to study interesting number theoretic sequences—for example rationals whose continued fraction partial quotients are subject to congruence conditions. We show that these sequences equidistribute and the gap distribution converges and answer an associated problem in Diophantine approximation. Moreover, for one example, we derive an explicit formula for the gap distribution. For this example, we construct the analogue of the Gauss measure, which is ergodic for the Gauss map. This allows us to prove a theorem about the associated Gauss–Kuzmin statistics.

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