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Dive into the fascinating world of mathematics with our captivating articles, written by passionate experts. Whether you are a curious amateur or a seasoned mathematician, our articles cover a wide range of topics, from fundamental concepts to the latest discoveries, including practical applications and philosophical reflections.
The zeta function and the Riemann hypothesis
The most important problem in contemporary mathematics can be stated in entirely elementary terms, requiring only a rudimentary knowledge of complex analysis. Despite mathematicians' titanic efforts, the Riemann hypothesis remains stubbornly out of reach.
The complex exponential
How can the classical exponential function be extended to complex numbers? Will its usual properties be preserved? Although the resulting extension is easy to study, the associated notion of a complex logarithm is more elusive. It was the subject of controversy in the 18th century.
Marden's theorem
Complex numbers, born of impossible calculations, found an unlikely geometric interpretation. This meeting of algebra and geometry is beautifully illustrated by theorems about the roots of a complex polynomial.
The geometry of complex numbers
René Descartes dreamed of turning a problem in pure geometry into an algebraic one. It took more than a century to realize that dream! Complex numbers opened up a new way to explore geometric figures and constructions.
The emergence of the complex plane | Tangente
The representation of the set of complex numbers by a plane appeared well after their invention.
Another transformation: inversion
Isometries (and more generally similarities) are not the only plane transformations that can be easily described using complex numbers. The same is true of inversion.
Some interesting similarities...
Isometries can be studied very neatly using complex numbers. But they are not the only transformations that can be described by simple formulas! Once we dispense with preserving lengths, the vast family of similarity transformations opens up before us.
Isometries of the plane
Studying isometries of the plane—reflections, translations, rotations, and so on—can sometimes be dizzying. What happens to a point or figure when several transformations are applied in succession? Complex numbers provide a representation that is as elegant as it is illuminating.
Complex numbers aren't so complicated
What do complex numbers really represent? How can we "picture" i² being equal to −1? A striking visual answer comes from interpreting multiplication geometrically. The icing on the cake is that the same model explains why "a negative times a negative makes a positive."
Conjugates, moduli and arguments
Once we accept the existence of a number i such that i² = −1, do we risk losing touch with physical reality? Quite the opposite: a profound and fruitful correspondence emerges, allowing questions of pure geometry to be solved through simple algebraic manipulations.
Complex numbers and trigonometry
The link between the exponential function and the familiar trigonometric functions is well known to anyone who has studied mathematics. Less well known is that this relationship extends to hyperbolic functions, which are useful in electrical engineering!
Online algorithmsMaking decisions without knowing the future
The gift of foreseeing the future would obviously be a considerable advantage. Who has never dreamed of it? Without a crystal ball, mathematics allows us to make decisions that cost us relatively little, even though we do not know what the future holds. Welcome to the world of online algorithms.
Poincaré, geometry and robotics | Tangente
Mathematics, and geometry in particular, is an essential component of that most technical of fields: robotics. The concept of space, which the mathematician Henri Poincaré (1854–1912) explored in depth, lies at the heart of the robotics engineer's work.
Logjam flaw in the HTTPS protocol | Tangente
A Franco-American team of researchers has just cheated… but for a good cause! Their goal? To verify the cryptographic mechanisms used to secure communications on the Internet.
The luthiers' secret code decrypted | Tangente
The luthiers of the Parisian house Gand & Bernardel, a famous dynasty of violin makers from 1795 to the Second World War, had a code that let them work on their slips with complete discretion.
Tangente turns 30: 30 ways to support it
The magazine Tangente, which is about to celebrate its 30th anniversary, has become an international benchmark and a fixture in the world of mathematics. Yet your support is more vital than ever to keep it going.
Betting on the presidential election: readers vs. the polls
Many of you took part in the challenge of coming as close as possible to the presidential election result. The winner is Élisabeth Pietravalle. Far behind, Ipsos proves to be the best polling institute.
Numbers in a triangle: math puzzles to explore
The arrangement of numbers in Pascal's triangle obviously lends itself to play, and many mathematical recreations refer to it.
Complex numbers of modulus 1
Initially mere formal symbols used in algebraic calculations, complex numbers came into widespread use from the 19th century onward thanks to their geometric interpretation... in almost every branch of mathematics! They therefore arise naturally in number theory.
Journey into the heart of a mathematical classic
If you know how to fill in Pascal's triangle, you may be less aware of the treasures it contains. Arithmetic, geometric, combinatorial: it has no shortage of astonishing mathematical properties! And there are undoubtedly many more still to be discovered…