The Revenge of the Infinitesimals

Originating author is Michèle Artigue.

Infinitesimals played an essential role in the emergence and development of differential and integral calculus. The evident productivity of this calculus did not prevent recurrent and fierce debates about the nature of these objects and the legitimacy of their use. At the end of the 19th century, when the construction of real numbers from integers and the modern definition of the concept of limit provided a solid foundation for differential and integral calculus, infinitesimals and the associated metaphysics was rejected and their use perceived synonymous with bygone and poorly rigorous practices. However, the language of infinitesimals continued to be used, for example in physics and even in mathematics. It never completely disappeared from the informal discourse and heuristic thinking of a number of researchers.

Is this language thus really incompatible with mathematical rigour? What does it offer that is interesting and specific, which explains its permanence? Non-Standard Analysis developed in the 20th century and provided answers to these questions and enabled infinitesimals to take their revenge.

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Featured Book of the Month: Elementary Mathematics from an Advanced Standpoint

From now on we will feature a different book every month that is likely to be of interest to secondary teachers wanting to know more about mathematics. We have made the decision to use this feature to bring older books to the attention of a new generation of teachers (rather than to add to recent book promotions). All books must therefore be older than 10 years.
For our first featured book we turn to Felix Klein’s original works that stimulated the Klein Project, his three volume work Elementary Mathematics from an Advanced Standpoint. Only two volumes have been published in English, although all three are available in German, and have been published in Portuguese. It is exactly ten years since Dover reprinted the English versions.
These books are essentially Klein’s own notes for a series of lectures he gave to graduates of mathematics preparing to become teachers in the gymnasium’s of the time. Of course, Klein’s books discuss mathematics that is more than 100 years old (they were first published in German in 1908), but remain extraordinarily relevant for today’s world.

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Site of the Month: Il Giardino di 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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Site of the Month: 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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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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Site of the Month: Mathematics of Planet Earth

Mathematics of Planet Earth

Each month, we will feature a site that we feel is relevant to aims of the Klein Project.

The first site of the month is dedicated to: Mathematics of Planet Earth (MPE). MPE is an initiative of mathematical science organizations around the globe to demonstrate the ways in which mathematical sciences may help us to solve our world’s problems. Take a look at the MPE initiative. 

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Calculators, Power Series and Chebyshev Polynomials

Chebysev series

Originating author is Graeme Cohen.

Of all the familiar functions, such as trigonometric, exponential and logarithmic functions, surely the simplest to evaluate are polynomial functions. The purposes of this article are, first, to introduce the concept of a power series, which can be thought of as a polynomial function of infinite degree, and, second, to show their application to evaluating functions on a calculator. When a calculator gives values of trigonometric or exponential or logarithmic functions, the most straightforward way is to evaluate polynomial functions obtained by truncating power series that represent those functions and are sufficiently good approximations. But there are often better ways. We will, in particular, deduce a power series for \,\sin\,x and will see how to improve on the straightforward approach to approximating its values. That will involve Chebyshev polynomials, which are used in many ways for a similar purpose and in many other applications, as well. (For trigonometric functions, the Cordic algorithm is in fact often the preferred method of evaluation—the subject of another article here, perhaps.)

In the spirit of Felix Klein, there will be some reliance on a graphical approach. Other than that, we need only some basic trigonometry and calculus.

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The shocking behaviour of moving fluids

Figure 1: The shock wave caused by a supersonic jet.

Originating authors are David Mumford and Christiane Rousseau.

Foreword: This vignette is more difficult than others. However, in a few pages, it tells you how to explain in simple terms one of the most difficult open problems at the beginning of the 21st century. The vignette contains enrichment material, that you can choose to read or skip. The editors of the Klein blog hesitated for a while posting this vignette. After testing it with teachers during two Klein workshops, who expressed that they enjoy being challenged by more difficult vignettes, they decided to test it on the blog. They are eager to hear your comments, and if some of you were motivated by this topic.

You have probably heard some planes breaking the sound barrier. What does that mean? It means that a shock wave in the atmosphere is created as in Figure 1. But what is a shock wave? Imagine heavy traffic on highways as a wave. A shock wave corresponds to collisions. To explain this, we develop our intuition with a 1D model: traffic on a one lane road at different speeds. You know that collisions could occur if drivers do not adjust their speed. The atmosphere is a fluid, and traffic is a rough model of 1D fluid which is convenient to develop our intuition. Under which conditions do shock waves or other singularities occur in fluids? A million dollar prize is offered for answering this question. This is what we are going to explain you.

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Recurrence and induction

Originating authors are Michèle Artigue and Ferdinando Arzarello.
Given a square grid, it is easy to draw squares whose vertices are intersections of the grid lines. But is it possible to do so for other regular polygons, for instance an octagon ? The answer is : « No » and it can be proved, for the octagon, as follows (Payan, 1994) :

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