References for Teachers: The Theorem of Pythagoras

1951 ◽  
Vol 44 (8) ◽  
pp. 585-598
Author(s):  
William L. Schaaf
Keyword(s):  
Continued Fractions ◽  
Diophantine Equations ◽  
Irrational Number ◽  
Golden Section ◽  
Dynamic Symmetry ◽  
Squaring The Circle ◽  
Logarithmic Spirals ◽  
The Rich ◽  
The One ◽  
Star Polygons

Justly called the most celebrated theorem of geometry, the Pythagorean proposition is probably the one bit of mathematics which millions of laymen will remember long after they have forgotten whatever other mathematics they may once have known. The theorem is notable first because of the rich historical associations with which it is attended; secondly, because of the amazing variety of proofs which have been given; and thirdly, because further exploration quickly leads to interesting and unexpected byways, such as the Golden Section, dynamic symmetry, logarithmic spirals, angle trisection, duplication of the cube, squaring the circle, determining the value of π, the concept of irrational number, regular and star polygons and polyhedra, theory of numbers, constructibility of angles and polygons, continued fractions, phyllotaxy, musical scales, Diophantine equations, Heronian triangles, and Pythagorean number lore.

Optimal approximation by continued fractions

1991 ◽  
Vol 50 (3) ◽  
pp. 481-504 ◽  
Author(s):  
Wieb Bosma ◽  
Cor Kraaikamp
Keyword(s):  
Continued Fractions ◽  
Best Approximation ◽  
Continued Fraction ◽  
Irrational Number ◽  
Approximation Properties ◽  
Continued Fraction Expansion ◽  
The Best Approximation ◽  
Natural Sense ◽  
The One ◽  
Continued Fraction Expansions

AbstractAmong all possible semiregular continued fraction expansions of an irrational number the one with the best approximation properties, in a well-defined and natural sense, is determined. Some properties of this so called optimal continued fraction expansion are described.

Continued Fractions, Jacobi Symbols, and Quadratic Diophantine Equations

2000 ◽  
Vol 43 (2) ◽  
pp. 218-225 ◽  
Author(s):  
R. A. Mollin ◽  
A. J. van der Poorten
Keyword(s):  
Continued Fractions ◽  
Diophantine Equations ◽  
Norm Form

AbstractThe results herein continue observations on norm form equations and continued fractions begun and continued in the works [1]−[3], and [5]−[6].

Multidimensional regular C-fraction with independent variables corresponding to formal multiple power series

2019 ◽  
Vol 150 (4) ◽  
pp. 1853-1870 ◽  
Author(s):  
R. I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Continued Fractions ◽  
Numerical Experiments ◽  
The Other ◽  
Independent Variables ◽  
Classical Algorithm ◽  
Multiple Power ◽  
The One ◽  
The Given ◽  
Multiple Power Series

AbstractIn the paper the correspondence between a formal multiple power series and a special type of branched continued fractions, the so-called ‘multidimensional regular C-fractions with independent variables’ is analysed providing with an algorithm based upon the classical algorithm and that enables us to compute from the coefficients of the given formal multiple power series, the coefficients of the corresponding multidimensional regular C-fraction with independent variables. A few numerical experiments show, on the one hand, the efficiency of the proposed algorithm and, on the other, the power and feasibility of the method in order to numerically approximate certain multivariable functions from their formal multiple power series.

HISTORICAL HERITAGE AS A MEANS OF MORAL TRAINING OF SOCIAL TEACHERS

2021 ◽  
Vol 66 (2) ◽  
pp. 180-187
Author(s):  
І.R. Halitova ◽  
◽  
N.O. Atemkulova ◽  
G.K. Shirinbayeva ◽  
◽  
...  
Keyword(s):  
The Other ◽  
Human Society ◽  
The Past ◽  
Moral Training ◽  
Literary Heritage ◽  
The Rich ◽  
Historical Heritage ◽  
Social Side ◽  
The One ◽  
Social Pedagogue

The introduction of socio-pedagogical ideas into the historical and literary heritage enriches the content of training, makes it possible to enrich their practical skills through familiarity with historical experience, on the one hand, on the other hand, it enriches the inner world of social teachers as specialists, connecting the feeling and consciousness, thereby creating conditions for successful effective activities. In human society, various types of contradictions have always appeared at any time, but at the same time , methods and ways to eliminate them have been invented. Unfortunately, we have recently become interested in foreign technologies of training and education, their ideas, and have lost sight of the rich experience of the past, which includes methods and methods of social education of children and youth. The problem is that it is necessary to identify them and use them in practice. The activity of a social pedagogue , in particular, is associated with rehabilitation, socialization and other types of work among children, youth and adults. The history of social pedagogy spiritually enriches future specialists on the one hand, and on the other, helps to accumulate the experience of the past in order to use them in solving modern problems. Literary and historical materials concerning the social side of the life of the Kazakh people in this regard is important and essential.

Provincializing the International: Communist Print Worlds in Colonial India

2020 ◽  
Vol 89 ◽  
pp. 140-153
Author(s):  
Ali Raza
Keyword(s):  
Print Culture ◽  
British India ◽  
Colonial India ◽  
Western Province ◽  
The North ◽  
Colonial Archive ◽  
The Rich ◽  
The One ◽  
North Western ◽  
Global Project

Abstract This paper charts communist print worlds in colonial India during the interwar period. Beginning in the early 1920s, self-declared ‘Communist’ and ‘Bolshevik’ publications began surfacing across India. Through the example of the Kirti Kisan Sabha (Workers and Peasants Party: a communist group in the north-western province of Punjab), and its associated publications, this paper will provide a glimpse into the rich, diverse and imaginative print worlds of Indian communism. From 1926 onwards, Kirti publications became a part of a thriving print culture in which a dizzying variety of revolutionary, socialist and communist publications competed and conversed with the equally prolific and rich print worlds of their political and ideological rivals. Removed on the one hand from the ivory towers of party intellectuals, dense treatises and officious theses, and on the other hand from the framing of sedition, rebellion and fanaticism in the colonial archive, Kirti publications show how the global project of communist internationalism became distinctly provincialized and vernacularized in British India.

Achieving the Best Outcome

2002 ◽  
Vol 16 (1) ◽  
pp. 127-128 ◽  
Author(s):  
Peter Singer
Keyword(s):  
Central Point ◽  
The Rich ◽  
The One

The one central point in all my writing on this topic, from “Famine, Affluence and Morality” onward, has been that the failure of people in the rich nations to make any significant sacrifices in order to assist people who are dying from poverty related causes is ethically indefensible. It is not simply the absence of charity, let alone of moral saintliness: It is wrong, and one cannot claim to be a morally decent person unless one is doing far more than the typical comfortably-off person does.

scholarly journals CONTINUED FRACTIONS WITH BOUNDED PARTIAL QUOTIENTS

2002 ◽  
Vol 45 (3) ◽  
pp. 653-671 ◽  
Author(s):  
J. L. Davison
Keyword(s):  
Rate Of Convergence ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Irrational Number ◽  
Subject Classification ◽  
Specific Sequence ◽  
Secondary 11B37 ◽  
Quadratic Irrational ◽  
Infinite Word ◽  
Partial Quotients

AbstractPrecise bounds are given for the quantity$$ L(\alpha)=\frac{\limsup_{m\rightarrow\infty}(1/m)\ln q_m}{\liminf_{m\rightarrow\infty}(1/m)\ln q_m}, $$where $(q_m)$ is the classical sequence of denominators of convergents to the continued fraction $\alpha=[0,u_1,u_2,\dots]$ and $(u_m)$ is assumed bounded, with a distribution.If the infinite word $\bm{u}=u_1u_2\dots$ has arbitrarily large instances of segment repetition at or near the beginning of the word, then we quantify this property by means of a number $\gamma$, called the segment-repetition factor.If $\alpha$ is not a quadratic irrational, then we produce a specific sequence of quadratic irrational approximations to $\alpha$, the rate of convergence given in terms of $L$ and $\gamma$. As an application, we demonstrate the transcendence of some continued fractions, a typical one being of the form $[0,u_1,u_2,\dots]$ with $u_m=1+\lfloor m\theta\rfloor\Mod n$, $n\geq2$, and $\theta$ an irrational number which satisfies any of a given set of conditions.AMS 2000 Mathematics subject classification: Primary 11A55. Secondary 11B37

scholarly journals Diophantine approximation by continued fractions

1991 ◽  
Vol 51 (2) ◽  
pp. 324-330 ◽  
Author(s):  
Jingcheng Tong
Keyword(s):  
Diophantine Approximation ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Irrational Number ◽  
Continued Fraction Expansion ◽  
Image Position

AbstractLet ξ be an irrational number with simple continued fraction expansion be its ith convergent. Let Mi = [ai+1,…, a1]+ [0; ai+2, ai+3,…]. In this paper we prove that Mn−1 < r and Mn R imply which generalizes a previous result of the author.

Catching Proteus: the collaborations of Wallis and Brounker. I. Squaring the circle

2000 ◽  
Vol 54 (3) ◽  
pp. 293-316 ◽  
Author(s):  
Jacqueline A. Stedall
Keyword(s):  
Continued Fractions ◽  
Royal Society ◽  
Paper Analyse ◽  
Main Subject ◽  
Joint Work ◽  
Squaring The Circle ◽  
John Wallis

William Brouncker ( ca. 1620–1684) was the inaugural President of The Royal Society, and John Wallis (1616–1703) one of its founder members. The two collaborated closely during the 1650s on some original and unusual mathematics, but while Wallis acquired a lasting reputation, Brouncker's work is no longer well known. The two parts of this paper analyse the joint work of Wallis and Brouncker and attempt to separate their respective contributions and very different mathematical styles. The main subject of Part I is Brouncker's discovery of continued fractions. He gave few clues as to how he obtained his results and I offer a possible reconstruction of his work. His quadrature of the hyperbola and rectification of the semicubical parabola are also discussed. Brouncker emerges as a skilled and intuitive mathematician.

The Vine and the Elm Tree: the Patristic Interpretation of Jesus’ Teachings on Wealth

1987 ◽  
Vol 24 ◽  
pp. 1-14 ◽  
Author(s):  
J. A. McGuckin
Keyword(s):  
New Testament ◽  
Biblical Criticism ◽  
Property Owner ◽  
Clement Of Alexandria ◽  
The Poor ◽  
The New Testament ◽  
The Subject ◽  
Original Message ◽  
The Rich ◽  
The One

If patristic tradition on the subject of wealth and possessions often appears ambivalent in its attitudes, then perhaps one of the reasons for this is that this tradition grows from an exegesis of Gospel teachings on the subject that themselves are far from being straightforward, even though they are immensely forthright. Clement of Alexandria, for example, has frequently been accused of twisting the simple and immediately obvious demand of Jesus: ‘Sell all you have and give to the poor’ (Mark 10.21) and subverting a radical vision of Jesus into a comfortable exhortation that any pious property-owner, bourgeois or aristocratic, could be happy to live with. If the rich young man had understood Christ’s real message, as Clement would have it (not so much to renounce his ownership of goods as to free his heart from attachment to them), then he might not have had such a crisis about following Jesus. Whether or not Clement’s case is, in the end, convincing as an exegesis, it none the less successfully raises all the implicit problems of interpreting the New Testament teachings on wealth in any kind of universalist sense—as teachings that are meant to apply to all, and for all time. And there are, consequently, many dangers in being too ready to dismiss Clement’s allegorism as an anachronistic exegesis, not least the danger of reverting to a different kind of biblical fundamentalism than the one Clement thought he was attacking; for contemporary biblical criticism, as it attempts to separate out the original message of Jesus and the insights of his later disciples, and to locate the original words in their correct historical and sociological milieu, has rightly warned us against over-confidence in our historical interpretations of Gospel material.

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