Negative Numbers, Fractions and Irrational Numbers

1978 ◽  
pp. 91-113
Author(s):  
Felix Kaufmann
Keyword(s):  
Negative Numbers ◽  
Irrational Numbers

Les commentaires d'al-Māhānī et d'un anonyme du Livre X des Éléments d'Euclide

1999 ◽  
Vol 9 (1) ◽  
pp. 89-156 ◽  
Author(s):  
Marouane Ben Miled
Keyword(s):  
Negative Numbers ◽  
Irrational Numbers ◽  
Quadratic Equations ◽  
First Time

This paper presents the first edition, translation and analyse of al-Māhānī’s commentary of the Book X of Euclid’s Elements (9th century, the most ancient to have reached us) and of an anonymous’ one (prior to 968, among the first algebraic commentaries). For the first time, irrational numbers are defined and classified. The algebraisation of Elements’ X-91 to 102, on the basis of al-Khwārizmī’s Algebra, shows irrational numbers as solution to algebraic quadratic equations. The algebraic calculus makes here the first steps. On this occasion, negative numbers and their calculation rules appears. Simplifications imposed by the algebraic writings are sometimes in opposition with the conclusions of propositions conceived in a purely geometrical framework, revealing a contradiction between geometrical and algebraic goals. It will be resolved by the independant way algebra will take with mathematicians belonging to the tradition of al-Karajī and al-Samaw’al from the 11th-12th centuries on.

Exploring the Boundaries of the Number–Temporal Order Association

2015 ◽  
Vol 62 (3) ◽  
pp. 198-205 ◽  
Author(s):  
Dana Ganor-Stern
Keyword(s):  
Embodied Cognition ◽  
Mental Representation ◽  
Temporal Order ◽  
Past Research ◽  
Numerical Comparison ◽  
Negative Number ◽  
Comparison Task ◽  
Negative Numbers ◽  
The Past

Past research has shown that numbers are associated with order in time such that performance in a numerical comparison task is enhanced when number pairs appear in ascending order, when the larger number follows the smaller one. This was found in the past for the integers 1–9 ( Ben-Meir, Ganor-Stern, & Tzelgov, 2013 ; Müller & Schwarz, 2008 ). In the present study we explored whether the advantage for processing numbers in ascending order exists also for fractions and negative numbers. The results demonstrate this advantage for fraction pairs and for integer-fraction pairs. However, the opposite advantage for descending order was found for negative numbers and for positive-negative number pairs. These findings are interpreted in the context of embodied cognition approaches and current theories on the mental representation of fractions and negative numbers.

Positive Psychology’s A Priori Problem

2017 ◽  
Vol 58 (3) ◽  
pp. 313-342 ◽  
Author(s):  
Barbara S. Held
Keyword(s):  
Conceptual Framework ◽  
A Priori ◽  
A Posteriori ◽  
Negative Numbers ◽  
Psychological States ◽  
Psychological Reality ◽  
Negative Results ◽  
Optimal Functioning ◽  
Conceptual Muddles ◽  
Negative Phenomena

The positive/negative distinction works well in many fields—for example, in mathematics negative numbers hold their own, and in medical pathology negative results are usually celebrated. But in positive psychology negativity should be replaced with positivity for flourishing/optimal functioning to occur. That the designation of the psychological states and processes deemed positive (good/desirable) and negative (bad/undesirable) is made a priori, independent of circumstantial particularity, both intrapersonal and interpersonal, does not seem to bother positive psychologists. But it should, as it results in conceptual muddles and dead ends that cannot be solved within their conceptual framework of positivity and negativity. Especially problematic is an ambiguity I find in positive psychologists’ a priori and a posteriori understandings of positivity and negativity, an ambiguity about constitutive and causal relations that pervades their science and the conclusions drawn from it. By eliminating their a priori dichotomy of positivity and negativity, positive psychologists might well find themselves in a better position to put back together the psychological reality that they have fractured in their ontologically dubious move of carving up psychological reality a priori into positive and negative phenomena. They then might find themselves better placed to “broaden and build” their own science of flourishing.

Teaching Irrational Numbers Through Trigonometry

Resonance ◽  
2021 ◽  
Vol 26 (6) ◽  
pp. 813-827
Author(s):  
Sameen Ahmed Khan
Keyword(s):  
Irrational Numbers

scholarly journals A Study of Errors and Misconceptions in the Learning of Addition and Subtraction of Directed Numbers in Grade 8

SAGE Open ◽  
2016 ◽  
Vol 6 (4) ◽  
pp. 215824401667137 ◽  
Author(s):  
Judah Paul Makonye ◽  
Josiah Fakude
Keyword(s):  
Conceptual Understanding ◽  
Teaching And Learning ◽  
Study Data ◽  
Negative Numbers ◽  
Teaching And Learning Mathematics ◽  
Learning Mathematics ◽  
Number Lines ◽  
Addition And Subtraction ◽  
Logical Errors ◽  
Grade 8

The study focused on the errors and misconceptions that learners manifest in the addition and subtraction of directed numbers. Skemp’s notions of relational and instrumental understanding of mathematics and Sfard’s participation and acquisition metaphors of learning mathematics informed the study. Data were collected from 35 Grade 8 learners’ exercise book responses to directed numbers tasks as well as through interviews. Content analysis was based on Kilpatrick et al.’s strands of mathematical proficiency. The findings were as follows: 83.3% of learners have misconceptions, 16.7% have procedural errors, 67% have strategic errors, and 28.6% have logical errors on addition and subtraction of directed numbers. The sources of the errors seemed to be lack of reference to mediating artifacts such as number lines or other real contextual situations when learning to deal with directed numbers. Learners seemed obsessed with positive numbers and addition operation frames—the first number ideas they encountered in school. They could not easily accommodate negative numbers or the subtraction operation involving negative integers. Another stumbling block seemed to be poor proficiency in English, which is the language of teaching and learning mathematics. The study recommends that building conceptual understanding on directed numbers and operations on them must be encouraged through use of multirepresentations and other contexts meaningful to learners. For that reason, we urge delayed use of calculators.

scholarly journals On subrecursive representability of irrational numbers

Computability ◽  
2017 ◽  
Vol 6 (3) ◽  
pp. 249-276 ◽  
Author(s):  
Lars Kristiansen
Keyword(s):  
Irrational Numbers

On a Discrete Analogue of Inequalities of Opial and Yang

1968 ◽  
Vol 11 (1) ◽  
pp. 73-77 ◽  
Author(s):  
Cheng-Ming Lee
Keyword(s):  
Integral Inequality ◽  
Discrete Analogue ◽  
Absolutely Continuous ◽  
Negative Numbers ◽  
Image Position

Let be a non-decreasing sequence of non-negative numbers, and let U∘=0. Then we haveYang [3] proved the following integral inequality:If y(x) is absolutely continuous on a≤x≤X, with y(a) = 0, then

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