Mathematical Themes
Explore the major mathematical themes: geometry, algebra, analysis, arithmetic, logic, and many other fascinating fields.
A history of ordered means: AM, GM, HM | Tangente
“The” mean of two numbers is defined “naturally” according to the context. It is not always the familiar arithmetic mean! Several different notions coexist and are closely interconnected, as Liouville, Cauchy and Jensen clearly understood.
Solving inequalities: methods and examples | Tangente
Some of our readers no doubt have unpleasant memories of inequalities.
The usual inequality on real numbers | Tangente
Once their values are known, any two numbers can be compared—whether they are integers, fractions, or real numbers—and we can say which is smaller.
Ptolemy's little-known inequality explained | Tangente
The Greek mathematician, geographer and astronomer Claudius Ptolemy discovered a theorem about quadrilaterals inscribed in a circle in the second century CE. His result is certainly far less famous than that of his compatriot Pythagoras, but it is every bit as beautiful.
Dido's problem and isoperimetry | Tangente
For a given perimeter, the disk is the figure with the greatest area. This result, popularized by a trick attributed to Dido, the founder and first queen of Carthage, is the "isoperimetric theorem." It gave rise to many more general inequalities.
The riches of the triangle inequality | Tangente
The word "inequalities" often brings arithmetic or algebra to mind. But inequalities also have their place in geometry! The most famous of them, the celebrated triangle inequality, still has more than one trick up its sleeve.
Maryna Viazovska on CultureMATH | Tangente
The CultureMath website, produced by the École normale supérieure, features an interview well worth reading. Mathematics teacher Nathalie Braun interviews Maryna Viazovska.
2023 Abel Prize: Luis Caffarelli | Tangente
The 2023 Abel Prize was awarded to Luis Caffarelli
A tribute to Pierre Audin | Tangente
Pierre was rigorous, dependable, compassionate, attentive, determined and funny… and he seemed tireless. He slept very little, sending a great many emails at night—often at 3:14 a.m.
Charlotte Angas Scott, a pioneer | Tangente
A pioneer of algebraic geometry and a specialist in curves and surfaces, British mathematician Charlotte Angas Scott was forced to move to the United States to find an academic post. Her chief claim to fame is an elegant proof she devised for a difficult theorem by Max Noether.
Fermat's infinite descent explained | Tangente
Pierre de Fermat wrote that he had succeeded in proving that the area of a right triangle can never be an integer that is a perfect square. To do so, he introduced a method of reasoning that would go down in history: infinite descent.
Terence Tao: blogging and learning | Tangente
Terence Tao has created a professional blog featuring "updates on his research and expository papers, discussion of open problems, and other maths-related topics." What can his posts on learning mathematics teach us?
Terence Tao: the rise of a prodigy | Tangente
Given his life story and exceptional intellectual abilities, mathematician Terence Tao—born in Australia in 1975 and now a professor at the University of California, Los Angeles (UCLA), in the United States—has been dubbed the "Mozart of mathematics" by many.
Research
Singmaster's conjecture, the Polymath projects, the Green–Tao theorem…
Some questions about sums of integers…
You might think that everything has already been said and done about adding two integers. Think again! That mischievous carry, which can ripple from one step of the calculation to the next, gives rise to difficult questions. This elementary algorithm still raises formidable problems.
Mathematical games: 2023 final | Tangente
After two years without an international final because of the pandemic, the 36th Championship held its final outside France for the first time, in Lausanne, Switzerland, while the final round of the 37th Championship will take place on August 25 and 26, 2023, at Wroclaw University of Science and Technology in Poland
Annette and André Parent's legacy | Tangente
Annette Parent passed away on April 26, 2023. Her death was a source of immense sadness for everyone who knew her; Annette had a generosity and warmth that no one could forget.
Enter the Trophées Tangente | Tangente
Every summer, Club Tangente presents its awards. Alongside its literary prizes—the Tangente Book Prize and the Tangente High School Students' Prize—which recognize books devoted to mathematical culture, two competitions invite our readers to get creative. Don't hesitate to enter!
Generating the famous Pythagorean triples
Characterizing all right triangles in the plane with integer side lengths amounts to finding all Pythagorean triples. The geometric problem thus appears to become purely arithmetic! How did mathematicians go about completing this quest?
Pythagorean triangles
When the side lengths of a triangle form a Pythagorean triple, the triangle is called a Pythagorean triangle. Discover its properties...