Complements of Mathematics 1

The course will focus on the concept of number.

In the first part, with a historical-epistemological approach, some fundamental historical developments related to the concept of number will be presented, starting from the Pythagorean conception of numbers and proportions, passing through the theory of relationships and of commensurable and incommensurable quantities elaborated by Eudoxus and systematized in the Elements of Euclid, to arrive at further evolutions of the concept of number linked to borderline problems between geometry, arithmetic, algebra and analysis (constructions with ruler and compass and analogy between geometric and algebraic problems, negative and algebraic numbers, analytic geometry, algebraic structures , arithmeticization of the continuum, transcendental numbers, Diophantine equations).

In the second part of the course the main properties of fields will be presented, with a contemporary approach. In particular, the following topics will be covered: split fields and fields, real and complex numbers, formally real, Archimedean and non-Archimedean fields, algebraic closures, real and Archimedean, complete fields and uniqueness of real numbers.