Topics in Elementary Mathematics

Code:

M181

Acronym:

M181

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2014/2015 - 1S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:M	88	Plano de estudos a partir de 2009	1	-	7,5	-

Teaching language

Portuguese

Objectives

To introduce the basic concepts of logic and elementary theory of sets on which mathematics and its language are based, and to deepen the study on elementary number theory.

Learning outcomes and competences

A student is expected to become familiar with deductive reasoning and the symbolic language of Mathematics, to deepen his knowledge on elementary number theory, and also to explore mathematical methods of proof.

Working method

Presencial

Program

1. Mathematical language and basic mathematical symbolism. Rudiments of logic. Examples of proofs.

2. Elementary theory of sets. Relations and functions.

3. Natural numbers. Mathematical induction and well-ordering principle. Notions of cardinality of infinite sets.

4. Integers. Divisibility and prime numbers. The division algorithm and the Euclidean algorithm. The Fundamental Theorem of Arithmetic. Congruences module a positive integer.

5. Representation of numbers in a given scale.  Finite and infinite, periodic and non-periodic decimal expansions. Characterization of the decimal expansions of the rational numbers through their representation as irreducible fractions.

 

Mandatory literature

; Apontamentos disponibilizados na página da UC
S. Kranz; Elements of advanced mathematics, Chapman & Hall/CRC, 2012
H. Stark; An introduction to number theory, MIT Press, 1991

Complementary Bibliography

K. Devlin; Sets, functions and logic, Chapman & Hall, 1992
C. Dodge; Sets, Logic and Numbers, Prindle, Weber and Schmidt, 1970
J.C. Santos; Números, U.Porto editorial, 2014
W. Sierpinski; Elementary Theory of Numbers, North-Holland, 1988

Teaching methods and learning activities

The content of the syllabus is presented at the lectures, and proposed exercises are solved by the students at the practical classes.

keywords

Physical sciences > Mathematics > Number theory
Physical sciences > Mathematics > Mathematical logic
Physical sciences > Mathematics > Algebra > Set theory

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	100,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	70,00
Frequência das aulas	70,00
Total:	140,00

Eligibility for exams

The students are not required to attend classes.

Calculation formula of final grade

The evaluation is done by an optional test and the final exam.

Course approval is obtained in the final exam, where a student may choose not to solve the first part of the exam that gets the grade obtained in the test or in the first part of the exam at the first exam season.