Taschenrechner, Potenzreihen und Tschebyschow-Polynome

Chebysev series

Von Graeme Cohen.

Von allen bekannten Funktionen, wie zum Beispiel den Winkel-, Exponential- oder Logarithmusfunktionen, sind sicherlich die Funktionswerte von Polynomfunktionen am leichtesten zu berechnen. Dieser Artikel soll erstens den Begriff der Potenzreihe einführen, die auch als Polynomfunktion unendlichen Grades verstanden werden kann, und zweitens zeigen, wie mit ihrer Hilfe Funktionswerte von Funktionen mit einem Taschenrechner berechnet werden können. Wenn ein Taschenrechner Werte von trigonometrischen, exponentiellen oder logarithmischen Funktionen berechnen soll, so erreicht man dieses, indem die Funktionswerte von Polynomfunktionen berechnet werden, die man für solche Potenzreihen erhält, die für jene Funktionen repräsentativ sind und ausreichend gute Näherungen darstellen. Dies ist zwar der direkte Weg, aber es gibt oft bessere Möglichkeiten. Wir werden hier insbesondere eine Potenzreihe für die Funktion \sin x herleiten und darlegen, wie man den direkten Ansatz verändern kann, um ihre Werte besser zu approximieren. Dabei werden wir auf Tschebyschow-Polynome zurückgreifen, die in vielerlei Hinsicht für einen ähnlichen Zweck und in vielen weiteren Anwendungen verwendet werden. (Für trigonometrische Funktionen ist der CORDIC-Algorithmus oft die bevorzugte Auswertungsmethode – ein Thema, das sich vielleicht für einen weiteren Klein-Artikel anbieten würde.)

Im Sinne von Felix Klein greifen wir hier auf einen grafischen Ansatz zurück. Ansonsten verwenden wir nur grundlegende Kenntnisse und Techniken aus Trigonometrie und Analysis.

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Il Giardino di Archimede

Archimede.

This site of a museum in Florence not only encourages us to visit that beautiful city, but also contains materials for schools. It is easily navigated by those who do not speak Italian.
Il Giardino di Archimede

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Classifying objects

Originating author is Christiane Rousseau.
Mathematics offers tools for classifying objects. But is that of any practical use? More than we can imagine at first sight… It could allow us to conclude that a knot cannot be unknotted without cutting the rope, regardless how you move it in space. It could also tell you that you wrongly assembled your Rubik’s cube after dismantling it into pieces, and that there is no use trying to solve it.

In practice, classifying objects means grouping objects in classes of objects sharing some common properties. One very efficient way to do this is through an equivalence relation. Then, each object belongs to an equivalence class. But, this does not mean that we have an efficient way of describing an equivalence class! The mathematical notion of invariant offers an efficient way to do this. An invariant is some mathematical object (it could be simply a number) that is the same for all members of an equivalence class. Among the invariants we distinguish the complete invariants that characterize an equivalence class.

In this vignette, we will mostly work with examples and see how the notion of invariant is widely spread among mathematics, especially algebra and geometry. You will then be able to add your own examples.

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الأبعاد

الأبعاد

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حكاية مثلثين : مثلثات هيرون والمنحنيات الناقصية

حكاية مثلثين : مثلثات هيرون والمنحنيات الناقصية

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تلوين الخرائط وأساسات غروبنر Gröbner

تلوين الخرائط وأساسات غروبنر Gröbner

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NRICH

Starting in January, we began a “Site of the Month” feature.

eNRICHing mathematics

February’s site of the month is dedicated to: NRICH. NRICH is a team of mathematics teachers, who aim to enrich the mathematical experience for all. They provide engaging activities for students, as well as for teachers’ professional development. Take a look at NRICH’s page for upper-secondary students: NRICH

 

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O Teorema da Bola Cabeluda

Autor do original: João Pimentel Nunes.

O teorema da bola cabeluda é um resultado da Topologia, a disciplina matemática que estuda a forma dos espaços. Em grande parte, ele resulta do trabalho nos finais do século XIX do grande matemático francês Henri Poincaré [1], considerado um dos fundadores da Topologia.

Haverá poucos resultados matemáticos que nos sejam tão familiares dos gestos do quotidiano: muitos dos leitores confrontam-se todas as manhãs com o teorema da bola cabeluda, ao tentarem pentear o seu cabelo e verificando que há um remoínho persistente no topo das suas cabeças. De um modo simplificado, o teorema afirma que não é possível “pentear-se” uma superfície esférica coberta de “cabelo” sem se formarem “remoínhos” de algum tipo.

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The Hairy Ball Theorem

Originating author is João Pimentel Nunes

The Hairy Ball Theorem is from topology, that part of mathematics that is concerned with the form of spaces. For the most part, this result came from work at the end of the 19th century by Henri Poincaré[1], considered to be one of the founders of topology.

There are few mathematical results that are so familiar to us from everyday situations: many readers are faced every morning with the hairy ball theorem when they try to comb their hair and find a persistent whorl at the top of their heads. Stated simply, the Hairy Ball Theorem says that it is impossible to comb a spherical ball covered in hair so that there are no whorls.


Check out this cool video explanation of the theorem
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التناظر خطوةً خطوةً

التناظر خطوةً خطوةً

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