Aims

To develop concepts and techniques of mathematical proof, illustrated by results in algebra. To stimulate logical thinking and to develop students' skills at constructing mathematical arguments.

Module summary

This module introduces one of the two principal branches of pure mathematics: algebra via number systems. Integer arithmetic has always provoked fascination in mathematicians and has been a source of many famous theorems, for example Fermat's Last Theorem. The module will cover areas such as prime numbers and modular arithmetic, before concluding by presenting properties of the real numbers, including proofs of irrationality of numbers like the square root of 2 and e. However, the principal aim of the module is to develop students' understanding of proof and their ability to construct valid mathematical arguments, with examples from number systems providing the objects of study.

Outline Of Syllabus

Cardinality and countability.

Number theory including the Division Algorithm and the Euclidean algorithm.

Modular arithmetic.

Relations.

Rational and irrational numbers.

Teaching Rationale And Relationship

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking.

Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of programme accreditation.

Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.