A versatile tool
In many mathematical fields, inequalities play an important role. This is the case for example in probability, with the Bienaymé‒Tchebychev inequality, or in all that relates to optimization. Many physical phenomena are also modeled by inequalities; thermodynamics gives us a striking example. In economic and social sciences, the study of inequalities is a central topic, whether they concern wealth, labor, housing, education…
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Social inequality: the Gini coefficient | Tangente
Statisticians often publish averages, reducing a snapshot of a country's socioeconomic situation to a single figure. But these averages often (always?) conceal stark disparities. One way to bring them into sharper focus is to quantify inequality as well.
The birth of thermodynamics and Clausius | Tangente
The study of heat transformed our view of the world by prompting us to consider whether a physical "arrow of time" exists. It all comes down to one remarkably simple relation: the Clausius inequality.
Cauchy–Schwarz in graph theory | Tangente
Take ten points and join some of them with line segments, but never form a triangle (that is, a triangle whose vertices are among the original ten points). What is the maximum number of segments you can draw?
Bienaymé–Chebyshev inequality in probability | Tangente
Summarizing a set of observations with a few key numbers is essential. But how much do these numbers tell us about the frequencies within particular intervals? The Bienaymé–Chebyshev inequality provides a first quantitative answer.
A little order…: order relations in mathematics | Tangente
Dictionary words, like points in the plane, can be put in order. Each choice of method—alphabetical order, lexicographic order, and so on—corresponds to an ordering: it is up to each of us to choose the order relation that suits our needs!
And order was established
The notion of order is part of our daily lives, whether we are putting objects away, comparing quantities or deciding what order to tackle a series of tasks in. Yet it took a very long time for these natural activities to be conceptualized mathematically.
Sorting algorithms: complexity and methods | Tangente
When a collection of data contains different values, one common task is to sort them—that is, to arrange them in a given order: the usual numerical order, lexicographic order as in a dictionary, or any other order defined by a total ordering relation.