Rational Numbers and Irrational Numbers

2014 ◽  
pp. 61-70
Author(s):  
Daniel Rosenthal ◽  
David Rosenthal ◽  
Peter Rosenthal
Keyword(s):  
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.

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

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.

Continued Fractions

2018 ◽  
pp. 35-42
Author(s):  
Christophe Reutenauer
Keyword(s):  
Continued Fractions ◽  
Error Term ◽  
Continued Fraction ◽  
Rational Numbers ◽  
Lexicographical Order ◽  
Quadratic Number ◽  
Real Numbers ◽  
Irrational Numbers ◽  
Upper Limit ◽  
Convergents Of Continued Fractions

Basic theory of continued fractions: finite continued fractions (for rational numbers) and infinite continued fractions (for irrational numbers). This also includes computation of the quadratic number with a given periodic continued fraction, conjugate quadratic numbers, and approximation of reals and convergents of continued fractions. The chapter then takes on quadratic bounds for the error term and Legendre’s theorem, and reals having the same expansion up to rank n. Next, it discusses Lagrange number and its characterization as an upper limit, and equivalence of real numbers (equivalent numbers have the same Lagrange number). Finally, it covers ordering real numbers by alternating lexicographical order on continued fractions.

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