# Numbers, Proofs and Counting

## Overview

• Credit value: 30 credits at Level 4
• Convenor and tutor: Steve Noble
• Assessment: short, problem-based assignments (20%) and a three-hour examination (80%)

## Module description

This is a core Level 4 module on all BSc programmes involving mathematics. It covers key concepts of number systems, proof techniques, logic and mathematical reasoning, all of which are important preparation for more advanced modules.

### Indicative module syllabus

• The language of mathematics: statements; theorems; definitions; logical connectives (not, and, or, implies); truth tables; tautologies and equivalent statements; the universal and existential quantifiers; negating if/then statements and statements involving quantifiers; some elementary proofs; contrapositive proofs; converses and counterexamples; proof by contradiction; Euclid's proof of the infinity of primes; proof by induction; strong induction
• Integers, rationals and real numbers: the well ordering property; the division theorem; the fundamental theorem of arithmetic; the Euclidean division algorithm; algebraic properties of the natural numbers and integers; congruence and modular arithmetic; solution of linear congruences; the rational numbers, the real numbers; boundedness; the completeness axiom
• Complex numbers: definition and arithmetic with complex numbers; the argand diagram; loci
• Binary operations: definition and examples of binary operations; the properties of commutativity and associativity; identity elements and inverses; idempotents; operation multiplication tables; definition of a group; examples from number sets, matrices and permutations
• Counting and probability: basic counting; product rule; strings from finite alphabets; counting subsets and binomial coefficients; ordered subsets and permutations; addition rule; naïve idea of probability triples; conditional probabilities and independence; finding probabilities using counting results

## Learning objectives

By the end of this module, you will have:

• knowledge and understanding of, and the ability to use, mathematical and/or statistical techniques
• knowledge and understanding of a range of results in mathematics
• an appreciation of the need for proof in mathematics, and the ability to follow and construct mathematical arguments
• an understanding of the importance of assumptions and an awareness of where they are used and the possible consequences of their violation
• an ability to present, analyse and interpret data
• a deeper knowledge of some particular areas of mathematics, in particular, proof methods such as proof by induction and proof by contradiction; number sets and their properties, including integers, rational, real and complex numbers; modular congruences and their solutions, binary operations, counting and introductory probability.