Laurea magistrale in Matematica

The teaching aims to present the theory of continued fractions and its connections with Diophantine analysis and Diophantine approximation, with attention to epistemological and didactic aspects.

This course naturally fits into the Curriculum related to History and Didactics of Mathematics (Theoretical Curriculum) and its interdisicplinary pathways, but it can be taken by anyone who shows interest in the cultural and educational dimensions of mathematics.

Knowledge and understanding. It provides an understanding of the theory of continued fractions and its applications (to Diophantine analysis, Diophantine approximation,  cryptography, and more) and a framing of some mathematical topics within the field of number theory. By taking up basic themes and treating them from different points of view, it allows for strengthening knowledge. The use of different references and sources is intended to improve students' critical skills. The situations provided in the course, which are an application of the studied theory, are aimed at improving problem-solving skills, consolidating the mastery of scientific concepts and methods. The proposal of didactic activities and moments of reflection or group discussion are aimed at accustoming students to both autonomous scientific research and the use of theoretical and epistemological knowledge to construct teaching activity for secondary school and high school.

Ability to apply knowledge and understanding. The lectures, situations and activities within the course develop in the students the ability to solve and pose problems, establishing connections between various areas of mathematics; the ability to analyze mathematical texts from both scientific and didactic perspectives; the ability to use the skills acquired, both for research purposes and in order to develop didactic activities for secondary and high schools from the knowledge acquired; the ability to orient oneself in the bibliography and sitography.

Autonomy of judgement. The nature of this course induces students to improve their critical and argumentation skills and accustoms them to: recognizing errors or gaps in proofs, reflecting on the change of mathematical methodologies and tools over time, independently developing examples of activities for secondary school and high school, and asking themselves didactic and in-depth questions that give cultural value to the topics covered. 

Communications skills. The presentation of a didactic activity and the collective discussions accustom students to state their own point of view, argue, and understand the views of others, using various communicative tools. In addition, since many suggested sources for the course are in English, students become accustomed to using that language for scholarly communications.

Learning skills. The work required for this course helps to create in students a flexible mindset, useful both for further specialised studies and for teaching in secondary and high schools. Indeed, they acquire skills in setting up work in a rigorous manner and solving theoretical problems, critically approaching the study of a mathematical text, and communicating mathematics to a non-mathematical audience through the creation of didactic activities for secondary school and high school, including using appropriate software. They are also able to set up research independently.

The student acquires:

- Mastery from the theoretical point of view of the topics covered in this course;

- Knowledge of the epistemology underlying the main concepts and methods presented;

- Capacity to use the knowledge acquired to solve problems;

- Capacity to situate mathematics in a broader cultural context and to communicate it;

- Capacity to use acquired knowledge both for research purposes and to prepare didactic activities for secondary school and high school;

- Capacity to set up research independently.

The course focuses on the theory of continued fractions and its connections to Diophantine analysis and Diophantine approximation. Introduction to continued fractions. Euclid’s algorithm. Continued fraction development of rational numbers. Convergents and their properties. Linear Diophantine equations and continued fractions. Continued fraction development of irrational numbers. Convergents of an unlimited continued fraction. Geometric interpretation of continued fractions. Continued fractions and series. The equation x² = ax + 1, digressions on the golden ratio. Purely periodic continued fractions. Theorems. Reduced quadratic irrationals. Graphical representation of the periodic character of complete quotients. Euler-Lagrange Theorem. Continued fraction development of the square root of N (N>0, not perfect square). The Pell's equation x² - Ny² = ±1. Legendre's theorem on the equation x² - Ny² = -1. How to obtain the other solutions of the Pell’s equation from the minimal one. Diophantine approximation. Dirichlet's theorem, Hurwitz's theorem, Farey series and their properties.

In connection with mathematical developments, some insights into the content are illustrated from both epistemological and didactic perspectives. Continued fractions and the search for regularity; Archimedes, the root approximation of 3, the oxen problem; continued fractions and square roots; continued fractions of irrational numbers; continued fraction developments and second-degree equations; continued fractions and calendars; continued fractions, Ford's circles and developments.

Other applications of continued fractions. Proof of the infinity of prime numbers, differential equations solving, cryptography (Wiener attack).

For the proposal of a didactic activity for secondary and high schools, aspects related to the school Curriculum guidelines (2010) will be discussed, also using appropriate software.

The lessons (48 hours, 6 CFUs) are conducted in-person using slides, videos, other useful course materials, problems, etc., uploaded to the MOODLE platform. Group work and collective discussions led by the teachers are also part of the lessons.

Individual proposal of a didactic activity (weight 40%) developed by the students from the insights discussed during the course or from complementary aspects.
An oral examination (weight 60%) aimed at evaluating theoretical and epistemological competencies acquired in the course and the capacity of applying them to problem solving.

G. CARISTI, C. FIORI, S. INVERNIZZI, Dalle frazioni continue alla trascendenza di pi greco. Centocinquant'anni di matematica «dimenticata», Pitagora, 2012.

K. ROSEN, Elementary Number Theory and its Applications, Addison-Wesley, 1993.

A. WEIL, Number Theory. An Approach through History from Hammurabi to Legendre, Boston, Birkhäuser 1983.