# COMPLEMENTARY MATHEMATICS I

## Learning outcomes of the course unit

Knowledge, understanding and links among the topics of the Course and the arguments of the other courses, with the aim to furnish an overview of basic Mathematics, also from an epistemological point of view. The course will prepare students to elaborate and apply their original ideas, also by means of problem solving activities.

## Course contents summary

The Babylonian and Egyptian mathematics.

The Greek mathematics: Thales, Pythagoras and his school, the crisis of incommensurables. Zeno’s paradoxes.

The three famous problems of Greek antiquity: quadrature of the circle, duplication of the cube, trisection of angle. Hippocrates and the quadrature of lunula.

Plato: arithmetic and geometry, the platonics polyedra.

Aristotle: the structure of deductive science, syllogisms. Induction principle.

Euclide: the “Elements”, commons notions, postulates and axioms, theory of Paralleles, proportions theory, magnitudes, prime numbers, equivalence in plane and space. Euclide’s work in modern epistemology.

Archimedes: from the measure of circle to the volume of sphere, the method of exhaustion.

Apollonius: conic sections.

Numerics systems: natural, integer, rational, real, complex numbers. The fundamental theorem of Algebra.

Numbers e,

Non-Euclidean geometries: hystorical and epistemological aspects, Poincaré’s and Klein’s models.

The Erlangen program and the transformations geometry: congruence, similarity, affinity, projectivity.

The geometrical transformations in Escher’s works.

The geometrical transformations in the space.

The problem of foundations of Geometry: the Hilbert’s axioms, indipendence, coherence, completeness.

## Recommended readings

F. Speranza, L. Ferrari (2008). Matematiche Complementari. Appunti delle lezioni. A.A. 1995/96. Marchini C., Pellegrino C., Vighi P. (Eds.). Parma: Servizio Editoriale Università di Parma.

F. Speranza, Scritti di Epistemologia della Matematica, Pitagora, Bologna, 1997.

E. Agazzi, D. Palladino, Le geometrie non euclidee e i fondamenti della geometria dal punto di vista elementare, La Scuola Editrice, Brescia, 1998.

C.B.Boyer, Storia della Matematica, Mondadori, Milano, 1980.

M. Dedò, Trasformazioni geometriche (con un’introduzione al modello di Poincaré), Decibel, Zanichelli, Bologna, 1996.

## Teaching methods

Lectures will be mainly in transmissive style, but with a steady dialogue with students which can be called to the blackboard for discussing problems, or for showing their understanding of and taking part to the course. Student will be asked to take part to seminar for studying in depth some course topics.

## Assessment methods and criteria

Oral examination.

Assessment will be made by a final oral, in which student must solve mathematical or interpretative problems.