# Numbers, Proofs and Counting

## Overview

• Credit value: 30 credits at Level 4
• Convenor: Professor Steven Noble
• Assessment: two problem sets (10% and 15%), a one-hour test (25%) and a two-hour examination (50%)

## Module description

In this module we cover key concepts of number systems, proof techniques, logic and mathematical reasoning, all of which are important preparation for more advanced modules.

### Indicative syllabus

#### The language of mathematics

• Statements, theorems, definitions and logical connectives
• Quantifiers and negating statements
• Elementary proof techniques
• Proof by induction

#### Integers, rationals, real and complex numbers

• The division algorithm and the Euclidean algorithm
• Congruence and modular arithmetic
• Boundedness and the least upper bound property
• Arithmetic involving complex numbers and the Argand diagram
• Binary operations

#### Counting and probability

• Product rule, counting strings and subsets
• Probability triples, conditional probabilities

## Learning objectives

By the end of this module, you will be able to:

• use mathematical and/or statistical techniques
• understand a range of results in mathematics
• appreciate the need for proof in mathematics, and follow and construct mathematical arguments
• understand the importance of assumptions and where they are used and the possible consequences of their violation
• show a deeper knowledge of particular areas of mathematics, in particular, rigorous mathematical statements, number sets, counting and probability.