The line and real numbers
Representing real numbers as the points of a line is an idea that revolutionized geometry as much as analysis. Descartes, by locating the points of the plane with two numbers, played a major role in this. Since his contribution, we know how to express a line as an equation! The construction of real numbers in the 19th century is the culmination of the transition from an intuitive vision of the line to an axiomatic representation. This approach made it possible to study the notion of « proximity » of points, to define an interval and more generally to look at the topology of the real line.
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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?
Without straightedge, without compass: Mascheroni | Tangente
Every straightedge-and-compass construction can be carried out with compass alone. This rather extraordinary result comes as a surprise.
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!
Descartes and Cartesian coordinates | Tangente
To locate points in a plane, Cartesian coordinates are generally used. Descartes is credited with inventing this method, so much so that it bears his name. Wrongly?
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.
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.
These lines that bear a name (1)
Some lines carry a mathematician's name. They are generally lines associated with the triangle, linked to certain notable points.