Numbers, Proofs and Counting

• 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%)

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

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.