# Numbers, Proofs and Counting

## Overview

• Credit value: 30 credits at Level 4
• Convenor and tutor: Steve Noble
• Assessment: short, problem-based assignments and online tests (25%), a one-hour in-class test (25%) and a two-hour examination (50%)

## 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
• 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

Counting and probability

• Basic counting
• Product rule
• Strings from finite alphabets
• Counting subsets and binomial coefficients
• Ordered subsets and permutations
• Naïve idea of probability triples
• Conditional probabilities
• 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.