History and Culture
History of mathematics and cultural connections
Holiday geometry | Tangente Magazine
The holidays are also a chance to uncover the mathematics all around us
Raymond Smullyan (1919–2017): logician and magician
An eclectic mathematician, Raymond Smullyan, who died last June, was best known to the general public for his books of mathematical recreations centered on logic.
Tangente Awards
The Tangente Awards were launched to celebrate creativity inspired by mathematics and its links with many other fields, including literature, computing and the arts. Run by the Club Tangente association, these initiatives invite our readers to take part or vote at this time each year.
Literature and mathematics: Perec joins the Pléiade
Perec in the Pléiade, Nabokov in Big Data, and mathematics in a "digest"
Paul Erdős and the probabilistic view of arithmetic
With the development of probability theory in the early 20th century, a new field of inquiry opened up in arithmetic: the statistical study of integers. Paul Erdős was among the first mathematicians to grasp the significance of this new approach.
Why not base 12? The advantages of the duodecimal system
If you talk to people about the history of arithmetic, whatever your specific subject, you can expect someone to ask this question at the end: why is our numeral system base 10 when base 12 would be so much more practical?
Applying the Chinese remainder theorem to RSA cryptography
The Chinese remainder theorem finds, among other things, applications in cryptography
Chinese remainder theorem: history and applications | Tangente
The Chinese remainder theorem owes its name to ancient Chinese mathematicians’ interest in the arrangement problems it describes. Originally a source of puzzles, it has since found more practical applications, particularly in cryptography.
A logarithm beneath the hyperbola | Tangente Magazine
Although Napier introduced them purely as an aid to calculation, logarithms gradually permeated every branch of mathematics. Their fundamental nature emerged particularly clearly through their connections with geometry, trigonometry and the hyperbola.
Logarithms come to the table | Tangente Magazine
I'm speaking of a time that those under twenty cannot know...
Goldberg Variations and Exercices de style: maths and music
Bach delighted in weaving combinatorial and geometric structures into his compositions. His famous Goldberg Variations are a masterpiece of the genre. Vincent Rouquès, a Paris-based choirmaster, drew on this richness by pairing them with Raymond Queneau's Exercices de style...
Jean-Pierre Kahane: A committed mathematician
Jean-Pierre Kahane, who passed away last June at the age of 90, left his mark on his era in many ways: through his work in analysis, which earned him international renown and election to the Académie des sciences; and through his commitment, both political and in the popularization of mathematics.
The inverse of the exponential function
The introduction of logarithms can be traced back to the Renaissance, when they were used to solve computational problems. They have since found more theoretical applications and today even lie at the heart of some cryptographic systems—and hence of our computers.
John Napier's astonishing inventions
John Napier of Merchiston, the Scottish gentleman whose death four centuries ago is being commemorated this year, was no revolutionary. Yet his wondrous invention of logarithms made him one, opening up a new world for mathematics far beyond computation.
What names for the complex numbers? | Tangente
Imaginary numbers were already introduced by the Italian mathematician Girolamo Cardano in 1545, though they did not yet have a name at the time; their first formalization is due to his compatriot Rafaele Bombelli in 1572.
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.
The emergence of the complex plane | Tangente
The representation of the set of complex numbers by a plane appeared well after their invention.
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!