About this book
Sorting, comparing, ordering are everyday activities since time immemorial that were formalized and perfected by integrating into the field of mathematics in the form of inequalities or order relations. It was in geometry that the first explicit inequalities appeared in various problems dealt with by the Greeks of Antiquity. Among them, we find the most famous one: the triangle inequality. A turning point in the use of inequalities was undoubtedly their introduction in analysis. Thus, the notions of limit or continuity call upon inequalities. We encounter them in differential and integral calculus. In particular, some of them, the best known being those of Cauchy-Schwarz or Minkowski, facilitated the creation of new spaces, allowing a theoretical framework to be given to physics. We find them in probability, in chemistry, in quantum mechanics or... in social sciences to describe and analyze a world of inequalities.
Contents
File 1: To bring some order
To be able to talk about inequalities, one must first be able to compare: for this, it is necessary to define what we mean by order.
Topics covered
Order in disorder Solving inequalities A story of well-arranged averages The different averages Averages without words The power of sets And order was established
File 2: A geometric origin
Comparing quantities is one of the most natural activities. Before knowing all the mathematical tricks, it is useful to return to the geometric origins of the question, in order to visualize the quantities involved.
Topics covered
The riches of the triangle The problem of Dido A little-known inequality, that of Ptolemy Approximately π Areas of the plane The Hornich-Hlawka generalization
File 3: Analysis to prove
If geometric intuition helps us to compare, it is mathematical analysis that will allow us to quantify variations, to find upper or lower bounds.
Topics covered
A small anthology Convexity inequalities The Cauchy-Schwarz inequality Cauchy-Schwarz and its variants Even in graphs! Chebyshev and monotonic sequences Symmetry, an effective tool
File 4: A versatile tool
In many scientific fields, inequalities play an important role. This is the case, for example, in probability or for processes related to optimization. Many physical phenomena are also modeled by inequalities.
Topics covered
In probability, the Bienaymé-Chebyshev inequality Results for prime numbers The birth of thermodynamics Sorting an array Heisenberg's indeterminacy The surprises of quantum mechanics
File 5: A world of inequalities
In economic and social sciences, the study of inequalities is a central topic, whether they concern wealth, work, housing, education, consumption, sports... All fields are a source of inequalities.
Topics covered
Measuring social inequalities From Lorenz to Gini Inequalities for a consumer Political (in)equality under the scrutiny of mathematics Measuring gaps between the sexes
Contents
File 1: To bring some order
To be able to talk about inequalities, one must first be able to compare: for this, it is necessary to define what we mean by order.
Topics covered
Order in disorder Solving inequalities A story of well-arranged averages The different averages Averages without words The power of sets And order was established
File 2: A geometric origin
Comparing quantities is one of the most natural activities. Before knowing all the mathematical tricks, it is useful to return to the geometric origins of the question, in order to visualize the quantities involved.
Topics covered
The riches of the triangle The problem of Dido A little-known inequality, that of Ptolemy Approximately π Areas of the plane The Hornich-Hlawka generalization
File 3: Analysis to prove
If geometric intuition helps us to compare, it is mathematical analysis that will allow us to quantify variations, to find upper or lower bounds.
Topics covered
A small anthology Convexity inequalities The Cauchy-Schwarz inequality Cauchy-Schwarz and its variants Even in graphs! Chebyshev and monotonic sequences Symmetry, an effective tool
File 4: A versatile tool
In many scientific fields, inequalities play an important role. This is the case, for example, in probability or for processes related to optimization. Many physical phenomena are also modeled by inequalities.
Topics covered
In probability, the Bienaymé-Chebyshev inequality Results for prime numbers The birth of thermodynamics Sorting an array Heisenberg's indeterminacy The surprises of quantum mechanics
File 5: A world of inequalities
In economic and social sciences, the study of inequalities is a central topic, whether they concern wealth, work, housing, education, consumption, sports... All fields are a source of inequalities.
Topics covered
Measuring social inequalities From Lorenz to Gini Inequalities for a consumer Political (in)equality under the scrutiny of mathematics Measuring gaps between the sexes