Topics in Elementary Mathematics

Code:

M1024

Acronym:

M1024

Level:

100

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2024/2025 - 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	104	Official Study Plan	1	-	9	72	243
L:MA	105	Official Study Plan	1	-	9	72	243

Teaching language

Suitable for English-speaking students

Objectives

To introduce the basic concepts of mathematical logic, elementary set theory, combinatorics, and elementary number theory, that form the basis of much of what is taught in other courses of this degree. At the same time, one aims to familiarize the students with the logical-deductive reasoning of mathematics, and deepen their knowledge of the numbers, from the integers to the complex.

Learning outcomes and competences

It is intended that the students become familiar with the deductive reasoning and symbolic mathematical language, that they deepen their knowledge of some of the basic topics of mathematics, and explore basic techniques of mathematical proof.

Working method

Presencial

Program

• Rudiments of logic and identification of their use in mathematical proofs.


• Mathematical language and basic mathematical symbolism.


• Natural numbers and mathematical induction.


• Elementary set theory; binary relations, equivalence relations; notions about functions.


• Notions of cardinality of infinite sets.


• Arithmetic of integers: divisibility; division algorithm and the Euclidean algorithm; the fundamental theorem of Arithmetic; congruence module a positive integer; Fermat's Theorem and Euler's Theorem.


• Arithmetic of the polynomials with coefficients in Q, R or C: divisibility; division algorithm; roots of polynomials; rational roots of polynomials with integer coefficients; Eisenstein's criterion; reference to the Fundamental Theorem of Algebra.


• We can cover other topics, such as: representation of numbers in different bases; finite and infinite, periodic and non-periodic decimal expansions; algebraic and transcendent real numbers; complex numbers: their historical genesis and geometric interpretation.

Mandatory literature

António Machiavelo; Apontamentos disponibilizados na página da UC

Complementary Bibliography

Devlin Keith; Sets, functions and logic. ISBN: 0-412-45970-1
Halmos Paul Richard; Naive set theory. ISBN: 0-387-90092-6
Vinogradov I. M.; Elements of number theory. ISBN: 0-486-60259-1
Hahn Liang-Shin; Complex numbers and geometry. ISBN: 0-88385-510-0
Aigner Martin; Proofs from the book. ISBN: 3-540-63698-6

Teaching methods and learning activities

Lecturing on the various subjects in class, supported by notes made available at the course's page on Sigarra. Solving of exercises in practical classes, with the support of the respective teachers.

keywords

Physical sciences > Mathematics > Mathematical logic
Physical sciences > Mathematics > Algebra > Set theory
Physical sciences > Mathematics > Number 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	171,00
Frequência das aulas	72,00
Total:	243,00

Eligibility for exams

No requirement.

Calculation formula of final grade

During the semester there will be two tests, but distributed into mini-tests to be carried out at the practical classes, each corresponding to a question of one of those tests. Thus there will be a small test every other week, on average. Each of these two tests has a weight of 6 points towards the final grade.

The approval of the course is made at the final exam.

The final examination will have 3 parts corresponding to two of them to the two tests. The student may choose not to solve one or both of these parts of the exam, in which case (s)he will get the mark obtained in the corresponding test.