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 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...
by Jean Barbet | Feb 13, 2021 | Algebra, Geometry, Non classé

Introduction In Vector angles: geometric intuition and algebraic definition, we defined and described the group of Euclidean plane vector angles algebraically, using an equivalence relation on unit vectors. Just as we can measure lengths, we learn at primary school...
by Jean Barbet | Feb 6, 2021 | Algebra, Geometry, Non classé

Vector angles are the usual oriented angles of Euclidean plane geometry. Thanks to the resources of naive set theory, they can be defined purely algebraically using an equivalence relation and the vectorial rotations of the plane. The operation of composing rotations...
by Jean Barbet | Jan 9, 2021 | Analysis, Functions, Non classé

From the complex exponential function, we can define a “circular exponential” function, which “wraps” the real line around the trigonometric circle, and makes it possible to rigorously define the cosine and sine trigonometric functions, which...
by Jean Barbet | Dec 29, 2020 | Analysis, Functions, Non classé

Some functions that can be differentiated indefinitely can be described ‘around each point’ as the sum of an power series. These are analytic functions, real or complex, the typical example being the exponential function, which can be extended to the whole complex...