by Jean Barbet | Jun 22, 2021 | Algebra, Geometry, Non classé
The visual intuition through which we represent the Euclidean plane suggests that we can orient it according to a direction of rotation. This intuition reflects a rigorous mathematical definition of the orientation of the plane, which involves choosing a basis and,... 
by Jean Barbet | May 29, 2021 | Logic, Set Theory
Russell’s paradox or antinomy is a very simple paradox in naive set theory, which arises when one tries to define a “set of all sets”. Its resolution relies on the introduction of the notion of class and the distinction of sets among classes. Thanks... 
by Jean Barbet | May 23, 2021 | Algebra, Geometry
The linear transformations of the Euclidean plane are the invertible linear applications, i.e. of non-zero determinant. They allow us to move from one basis of the plane to another, and the orthogonal transformations, i.e. the vectorial isometries, exchange the... 
by Jean Barbet | May 8, 2021 | Algebra, Geometry, Non classé
The representation of the Euclidean plane as the Cartesian product \(\mathbb R^2\) allows us to decompose any vector of the plane into two coordinates, its abscissa and its ordinate. This decomposition is linked to a particular and natural “representation... 
by Jean Barbet | Mar 25, 2021 | Algebra, Geometry, Non classé, Number Theory
Descartes’ analytical method, which allows the Euclidean plane to be represented as the Cartesian product $ \mathbb{R}^2 $ through the theory of real numbers, also makes it possible to represent Euclidean space as the Cartesian product $ \mathbb{R}^3 =...