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 The Number Sense Represents (Rational) Numbers

2021 ◽  
pp. 1-57
Author(s):  
Sam Clarke ◽  
Jacob Beck
Keyword(s):  
Number Sense ◽  
Number System ◽  
Rational Numbers ◽  
Future Research ◽  
Approximate Number System ◽  
Real Numbers ◽  
Irrational Numbers ◽  
The Real ◽  
Turn On ◽  
Orthodox View

Abstract On a now orthodox view, humans and many other animals possess a “number sense,” or approximate number system (ANS), that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique—the arguments from congruency, confounds, and imprecision—and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical confounds or exotic substitutes for number, such as “numerosities” or “quanticals,” as critics propose. In so doing, we raise a neglected question: numbers of what kind? Proponents of the orthodox view have been remarkably coy on this issue. But this is unsatisfactory since the predictions of the orthodox view, including the situations in which the ANS is expected to succeed or fail, turn on the kind(s) of number being represented. In response, we propose that the ANS represents not only natural numbers (e.g. 7), but also non-natural rational numbers (e.g. 3.5). It does not represent irrational numbers (e.g. √2), however, and thereby fails to represent the real numbers more generally. This distances our proposal from existing conjectures, refines our understanding of the ANS, and paves the way for future research.

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.

Review: Arithmetic as Mathematics

1993 ◽  
Vol 24 (2) ◽  
pp. 172-176
Author(s):  
Vicky L. Kouba
Keyword(s):  
Complex Number ◽  
Mathematics Teaching ◽  
Number System ◽  
Whole Numbers ◽  
The Past ◽  
Children’S Learning ◽  
Flexible Knowledge ◽  
Children's Learning

Too often in the past “arithmetic” has been viewed as simple computing or simple acquisition of procedures. In contrast, the chapters in Analysis of Arithmetic for Mathematics Teaching portray arithmetic as a thoughtful building of a flexible knowledge of our complex number system, rich in concepts, connections, and patterns that are crucial for understanding most of the rest of mathematics. This book provides research foundations and conceptual analyses of the standard topics of arithmetic: understanding of and operations on whole numbers, fractions, and decimals. The authors provide updates on the results, theories, and thoughts about children's learning of arithmetic concepts and processes, a field that has coalesced over the past twenty years.

scholarly journals Advanced number system – The universe of all numbers where anything is possible

2020 ◽  
Author(s):  
Balram A Shah
Keyword(s):  
Multiple Solutions ◽  
Best Practice ◽  
Number System ◽  
Complex Numbers ◽  
Real Numbers ◽  
The Past ◽  
Indeterminate Form ◽  
The Universe ◽  
Imaginary Number ◽  
Mathematical Constant

This research introduces a new scope in mathematics with new numbers that already exist in everyday mathematics but very difficult to get noticed. These numbers are termed as advanced numbers where entire real numbers, including complex numbers are the subset of this number’s universe. Dividing by zero results in multiple solutions so it is the best practice to not divide by zero, but what if dividing by zero have a unique solution? These numbers carry additional details about every number that it produces unique results for every indeterminate form, it allows us to divide by zero and even allows us to deal with infinite values uniquely. So, related to this number, theories, framework, axioms, theorems and formulas are established and some problems are solved which had no confirmed solutions in the past. Problems solved in this article will help us to understand little more about imaginary number, calculus, infinite summation series, negative factorial, Euler’s number e and mathematical constant π in very new prospective. With these numbers, we also understand that zero and one are very sophisticated numbers than any numbers and can lead to form any number. Advance number system simply opens a new horizon for entire mathematics and holds so much detailed precision about every number that it may require computation intelligence and power in certain situations to evaluate it.

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.

Ratio-based perceptual foundations for rational numbers, and perhaps whole numbers, too?

2021 ◽  
Vol 44 ◽  
Author(s):  
Edward M. Hubbard ◽  
Percival G. Matthews
Keyword(s):  
Numerical Cognition ◽  
Processing System ◽  
Number System ◽  
Rational Numbers ◽  
Approximate Number System ◽  
Access Points ◽  
Whole Numbers ◽  
Two Systems ◽  
Systems View

Abstract Clarke and Beck suggest that the ratio processing system (RPS) may be a component of the approximate number system (ANS), which they suggest represents rational numbers. We argue that available evidence is inconsistent with their account and advocate for a two-systems view. This implies that there may be many access points for numerical cognition – and that privileging the ANS may be a mistake.

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.

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.

Numbers, numerosities, and new directions

2021 ◽  
Vol 44 ◽  
Author(s):  
Sam Clarke ◽  
Jacob Beck
Keyword(s):  
Number Sense ◽  
Number System ◽  
Rational Numbers ◽  
Theoretical Research ◽  
Approximate Number System ◽  
Irrational Numbers ◽  
Target Article ◽  
New Directions

Abstract In our target article, we argued that the number sense represents natural and rational numbers. Here, we respond to the 26 commentaries we received, highlighting new directions for empirical and theoretical research. We discuss two background assumptions, arguments against the number sense, whether the approximate number system (ANS) represents numbers or numerosities, and why the ANS represents rational (but not irrational) numbers.

On The Possibility of an Autonomous History of Modern Southeast Asia

1961 ◽  
Vol 2 (2) ◽  
pp. 73-105 ◽  
Author(s):  
John R. W. Small
Keyword(s):  
Southeast Asia ◽  
Contemporary History ◽  
The Past ◽  
History Of

It is generally accepted that history is an element of culture and the historian a member of society, thus, in Croce's aphorism, that the only true history is contemporary history. It follows from this that when there occur great changes in the contemporary scene, there must also be great changes in historiography, that the vision not merely of the present but also of the past must change.

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