New Publications

1974 ◽  
Vol 67 (2) ◽  
pp. 152-155
Keyword(s):  
Word Problems ◽  
Rational Numbers ◽  
Natural Numbers ◽  
Irrational Numbers ◽  
Whole Numbers

The text provides a refresh on these topics: natural numbers, whole numbers, integers, rational numbers, decimals, and irrational numbers. There are two final chapters on geometry and selected applications. A good many “word” problems are included, along with lots of drill exercises. Exposition is rather brief. The treatment of topics is elementary throughout.— Skeen.

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.

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.

Math Roots: Ratios and Proportions: They Are Not All Greek to Me

2009 ◽  
Vol 14 (6) ◽  
pp. 370-378
Author(s):  
Joanne E. Snow ◽  
Mary K. Porter
Keyword(s):  
Number System ◽  
Rational Numbers ◽  
Real Numbers ◽  
Irrational Numbers ◽  
Line Segments ◽  
Whole Numbers ◽  
The Past ◽  
Geometric Objects ◽  
History Of

Today, the concept of number includes the sets of whole numbers, integers, rational numbers, and real numbers. This was not always so. At the time of Euclid (circa 330-270 BC), the only numbers used were whole numbers. To express quantitative relationships among geometric objects, such as line segments, triangles, circles, and spheres, the Greeks used ratios and proportions but not real numbers (fractions or irrational numbers). Although today we have full use of the number system, we still find ratios and proportions useful and effective when comparing quantities. In this article, we examine the history of ratios and proportions and their value to people from the past through the present.

scholarly journals Avoiding Fractional Powers over the Natural Numbers

10.37236/6678 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Lara Pudwell ◽  
Eric Rowland
Keyword(s):  
Fixed Point ◽  
Rational Numbers ◽  
Finite Alphabet ◽  
Its Sequence ◽  
Natural Numbers ◽  
Fractional Powers ◽  
Free Word

We study the lexicographically least infinite $a/b$-power-free word on the alphabet of non-negative integers. Frequently this word is a fixed point of a uniform morphism, or closely related to one. For example, the lexicographically least $7/4$-power-free word is a fixed point of a $50847$-uniform morphism. We identify the structure of the lexicographically least $a/b$-power-free word for three infinite families of rationals $a/b$ as well many "sporadic" rationals that do not seem to belong to general families. To accomplish this, we develop an automated procedure for proving $a/b$-power-freeness for morphisms of a certain form, both for explicit and symbolic rational numbers $a/b$. Finally, we establish a connection to words on a finite alphabet. Namely, the lexicographically least $27/23$-power-free word is in fact a word on the finite alphabet $\{0, 1, 2\}$, and its sequence of letters is $353$-automatic.

Order of operations in elementary arithmetic

1962 ◽  
Vol 9 (5) ◽  
pp. 263-267
Author(s):  
Marvin L. Bender
Keyword(s):  
Natural Numbers ◽  
Whole Numbers ◽  
Restricted Sense

We will assume acquaintance with the set of natural numbers N = {1, 2, 3, …} and the set of whole numbers W = {0, 1, 2, 3, …}. The fundamental operations of addition and multiplication, and the related operations of subtraction and division will also be assumed as well known to the reader. It is to be understood that subtraction and division are operations only in a restricted sense, since it is not always possible to subtract or divide in the set of whole numbers. A recognition of a few of the properties of these operations, at least in form if not in name, will be assumed: in particular, the commutative and associative properties of addition and multiplication, and the fact that they do not hold for subtraction and division. Finally, an understanding of the meaning of the relations indicated by =(equal) and ≠ (not equal) will be assumed.

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

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