par Jean Barbet | Déc 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... 
par 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 What is... 
par 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... 
par Jean Barbet | Juil 10, 2020 | Functions, Set Theory
A finite set is a set that can be counted using the natural numbers \(1,\ldots,n\) for a certain natural number \(n\). But what is counting ? And then, what is an infinite set? 1.Comparing sets : the notion of bijection The notions of finite set and infinite set, and... 
par Jean Barbet | Juin 24, 2020 | Set Theory
Naive set theory or « potato science » is the natural (and understandable!) foundation of mathematical science. « I know what time is. If you ask me, I don’t know anymore. » Augustine. This insightful quote from Augustine emphasises that there are concepts that... 
par Jean Barbet | Juin 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 numbers, but...