par Jean Barbet | Juin 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,... 
par Jean Barbet | Mai 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... 
par Jean Barbet | Mai 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 system »,... 
par 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 =... 
par Jean Barbet | Mar 20, 2021 | Algebra, Geometry, Non classé, Number Theory
The complex multiplication naturally extends to a multiplication in four dimensions, which defines on the space $ \mathbb{R}^4 $ the structure of the algebra $ \mathbb{H} $ of Hamilton’s quaternions. This multiplication can be interpreted geometrically using the... 
par Jean Barbet | Mar 12, 2021 | Algebra, Non classé, Number Theory
Gaussian integers are complex numbers with integer coordinates. Thanks to their norm, a kind of integer measure of their size, we can describe some of their arithmetic properties. In particular, we can determine which are the usual prime numbers that « remain » prime...