by 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... 
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 20, 2021 | Functions, Number Theory
Introduction When we introduced the circular exponential, the trigonometric functions cosine and sine were defined as its real part and imaginary part. From this, we derived the analytical expressions: \(\cos x=\sum_{n=0}^{+\infty} (-1)^n\dfrac{x^{2n}}{(2n)!}\) and... 
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...