2025
Arithmetics
Name: Arithmetics
Code: MAT14987L
6 ECTS
Duration: 15 weeks/156 hours
Scientific Area:
Mathematics
Teaching languages: Portuguese
Languages of tutoring support: Portuguese
Regime de Frequência: Presencial
Sustainable Development Goals
Learning Goals
1.Distinguish the different types of numbers through informal definitions and properties: natural, integers, rational, real, complex, as well as related operations.
2.Understand the distinction between number and numeral. Use the notions of numerical bases and decimals in representing numbers. Be able to relate aspects of arithmetic with geometric aspects and notions of ordering.
3. Master the concepts of prime factor, divisibility, partial order structure in the set of divisors, greatest common divisor and least common multiple.
4. Abstract the notions of arithmetic and understand modular arithmetic mod n. Understand that different arithmetic structures arise from different natural n.
5.Acquire skills to formulate and solve problems within the scope and/or with arithmetic techniques.
6.Relate concepts from other areas of mathematics and other disciplines with arithmetic concepts.
2.Understand the distinction between number and numeral. Use the notions of numerical bases and decimals in representing numbers. Be able to relate aspects of arithmetic with geometric aspects and notions of ordering.
3. Master the concepts of prime factor, divisibility, partial order structure in the set of divisors, greatest common divisor and least common multiple.
4. Abstract the notions of arithmetic and understand modular arithmetic mod n. Understand that different arithmetic structures arise from different natural n.
5.Acquire skills to formulate and solve problems within the scope and/or with arithmetic techniques.
6.Relate concepts from other areas of mathematics and other disciplines with arithmetic concepts.
Contents
A1 Natural numbers. Division algorithm. Numerical bases. Integers. Fractions.
A2 Rational numbers. Farey Tree. Ordering.
A3 Irrational numbers. Real numbers.
A4 Relationship of numbers with geometry on the line and in the plane.
B1 Natural. Prime numbers. Operations. Dividers. Hasse diagrams for dividers.
B2 Greatest common divisor, least common multiple. Euclid's algorithm.
B3 Examples of applications: gear wheels, others.
C1 Modular arithmetic. Multiplication tables. Examples of different properties depending on the structure of the factorization of n.
C2 Problems and applications: Divisibility criteria. Modular reduction.
C3 Application examples: Caesar cipher, affine cipher. Relate to written Portuguese. Frequency of occurrence of letters. Control digits in communications.
D1 Sequences of natural numbers. Arithmetic properties of some sequences. Sequences originating from counting problems Fibonacci and others. Discussion and presentation of the encyclopaedia of succes
A2 Rational numbers. Farey Tree. Ordering.
A3 Irrational numbers. Real numbers.
A4 Relationship of numbers with geometry on the line and in the plane.
B1 Natural. Prime numbers. Operations. Dividers. Hasse diagrams for dividers.
B2 Greatest common divisor, least common multiple. Euclid's algorithm.
B3 Examples of applications: gear wheels, others.
C1 Modular arithmetic. Multiplication tables. Examples of different properties depending on the structure of the factorization of n.
C2 Problems and applications: Divisibility criteria. Modular reduction.
C3 Application examples: Caesar cipher, affine cipher. Relate to written Portuguese. Frequency of occurrence of letters. Control digits in communications.
D1 Sequences of natural numbers. Arithmetic properties of some sequences. Sequences originating from counting problems Fibonacci and others. Discussion and presentation of the encyclopaedia of succes
Teaching Methods
Classes in theoretical-practical format, expository of content, articulated with the presentation of examples and exercises. The presentation moments alternate with the presentation of applications and discussion on solving associated problems. Applications can be proposed and presented by students and can be related to arithmetic or other subject areas. Moments to perform exercises in groups and individually. Calls to the board for students to present concepts and solve problems to their colleagues. The use of free software to solve certain problems but not all is assumed. Python, R, Excel or something else to be decided by the teacher. In this way, the recommendations and guidelines of the pedagogical model of the university of Évora are taken into account.
Assessment
Continuous assessment regime or a final exam assessment regime.
Continuous assessment will consist of several short moments - questions in class, with these assessment moments being seen simultaneously as assessment and practice/learning. They can be held every week or with a schedule to be agreed by the teacher at the beginning of the academic year. These moments must be more than 5 throughout the semester (1/3 of the total number of weeks).
In addition to the questions in class, a final written test will be carried out.
Each question in will have a fixed price (e.g. 2 points). The final grade for questions in is obtained by adding the partial grades for each question and normalized to 20 points. The weight of questions in will be 30% of the final grade. The final test will weigh 70% of the final grade.
Continuous assessment will consist of several short moments - questions in class, with these assessment moments being seen simultaneously as assessment and practice/learning. They can be held every week or with a schedule to be agreed by the teacher at the beginning of the academic year. These moments must be more than 5 throughout the semester (1/3 of the total number of weeks).
In addition to the questions in class, a final written test will be carried out.
Each question in will have a fixed price (e.g. 2 points). The final grade for questions in is obtained by adding the partial grades for each question and normalized to 20 points. The weight of questions in will be 30% of the final grade. The final test will weigh 70% of the final grade.
Teaching Staff
- Carlos Correia Ramos [responsible]
Certificações
