Knowledge
In-depth articles on mathematical knowledge and theories
Surfaces… made of straight lines!
A plane is generated by straight lines, and one might think it is the only surface that can be constructed in this way. But that is not so! Surfaces generated by straight lines even have a name: ruled surfaces.
Miraculous collinearity theorems!
A ruler, a compass, a sharp pencil—and the adventure has already begun! It can come as quite a surprise when three points turn out to lie on the same line. Plane geometry abounds in wondrous collinearity theorems.
Lines that have a name (2)
Some lines have a name, and some mathematicians have their own line.
Euclid: an introduction to the straight line
Written more than two thousand years ago, Euclid's Elements is a foundational text of classical geometry. The concept of the straight line is central to it. Let's meet this scholar of ancient Greece and take a closer look at his view of the straight line.
The topological line
The topology of the real line underpins the concept of a limit and hence, among other things, those of continuity and differentiability. It deals with notions of proximity between points that arise from the order relation.
Equations of a line
What defines a line, and in what geometric setting? Each possible answer leads to a representation from which the corresponding equations follow.
Geometry or numbers? The number line
It is easy to picture numbers as points on an oriented line with an origin. Yet this construction must be carried out with some rigor—especially if we want to represent infinitesimal or infinite numbers!
The number line: definitions versus intuition
In the 19th century, mathematicians sought rigorous definitions of the quantities that make up the continuum of the line. How did they rigorously characterize the real continuum?
The definition of the straight line through the ages
You thought you knew what a straight line is? Read on, and you may well start to doubt it…
Mathematical vs astrological semantics | Tangente
Horoscope advice can seem sensible. Why is that? While advice in objective areas consists solely of tautologies, subjective advice draws on a multiplicity of meanings, allowing clients always to interpret it as they wish.
Roots in the complex plane
The idea of a root extends readily to the field of complex numbers. In fact, it is even more at home there than among the real numbers, since this extension gives it a geometric interpretation. Welcome to the fascinating mathematical world of cyclotomy!
From Bézout's theorem for polynomials to the intersection of conics
Étienne Bézout is known for two theorems. One generalizes Bachet's theorem from integers to polynomials; the other concerns the intersection points of algebraic curves. The two are in fact related, but in a subtle way.
Bachet and Bézout: a winning mathematical duo
From Gauss's lemma to the Chinese remainder theorem, by way of numerous Diophantine equations, no problem seems able to resist the Bachet–Bézout theorem. Games, recreational puzzles, arithmetical tricks… Let's dive into mathematics!
A happy identity
The famous Bézout theorem—actually proved earlier by Bachet de Méziriac—may look simple, but it opens up many avenues in both arithmetic and algebra. This discovery makes it easier to solve a great many Diophantine equations, among other things...
Square roots
"The square root is my favorite," declared Boris Vian in Racine carrée (Square Root) in 1957. From antiquity to modern high-performance computing, many methods have been devised to calculate square roots in practice.
Mega primes: A new GIMPS record | Tangente
A quest that may seem far-fetched or pointless to some, but is of the utmost importance for our secret codes, is that of "mega primes" — in other words, prime numbers with more than a million digits.
Medical imaging: picturing disease
The spectacular advances in medical imaging in recent years have come from combining mathematical ingenuity with computing power, drawing on theories devised long before they found practical applications.
Connected health devices
Connected devices that monitor our health and allow doctors to intervene before an illness even develops seem to be the future of medicine. Yet they raise serious security concerns: via smartphones, they communicate over the Internet, a realm where hackers thrive…
Markov: chains of hope
For a disease transmitted by insects, a treatment campaign may not necessarily prove effective in the long term, as a particular case study using the mathematical concept of a Markov chain will show.
Differential equations in oncology
The effectiveness of chemotherapy in treating cancer depends on tailoring the dosage to the individual patient. To adjust it effectively, the physician must solve a first-order differential equation.