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... 
by Jean Barbet | Dec 16, 2020 | Non classé, Number Theory, Set Theory
The prime natural numbers are those which have no divisors other than 1 and themselves. They exist in infinite number by Euclid’s theorem, which is not difficult to prove. 1.Prime numbers 1.1.Divisors and primes A prime number is a non-zero natural number (see... 
by Jean Barbet | Nov 20, 2020 | Non classé, Number Theory, Set Theory
1.The intuition of rational numbers Rational numbers, i.e. “fractional” numbers, such as \(-\frac 1 2, \frac{27}{4}, \frac{312}{-6783},\ldots\), form an intuitive set which we note \(\mathbb Q\). It is an extension of the set \(\mathbb Z\) of integers (see...