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 | 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... 
by Jean Barbet | Nov 10, 2020 | Number Theory, Set Theory
Integers are an extension of the natural numbers where the existence of subtraction provides a more appropriate framework for certain questions of arithmetic. They can be described axiomatically, but can also be constructed from the set of natural numbers and some... 
by Jean Barbet | Jun 22, 2020 | Number Theory, Set Theory
Mathematical science does not seek to define the notion of a natural number, but to understand the set of natural numbers. “Natural numbers have been made by God, everything else is the work of men”. Leopold Kronecker 1.We don’t define the natural... 
by Jean Barbet | Jun 20, 2020 | Number Theory, Set Theory
The real numbers are all the “quantities” that we can order, and we can “construct” them in various ways thanks to set theory. “Numbers govern the world.” Pythagoras The real numbers idealise all the “points” of the...