The fourth lecture in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers Highest Common Factor (HCF, also known as Greatest Common Divisor, GCD) for integers, and lowest terms for rational numbers.How and why indirect proof (proof by contradiction) works. Examples of indirect proofs, including the proof that the square root of two is irrational.. These videos are also available for download on iTunes U at: itunes.apple.com/us/itunes-u/foundations-pure-mathematics/id950755120
Dr Feinstein's blog may be viewed at: explainingmaths.wordpress.com Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham.

1. Introduction to Pure Mathematics
2. Sets and Numbers
3. Workshop 1: About this module
4. Definitions and Direct Proofs
5. Rational and Irrational Numbers
6. Workshop 2
7. More on Rational and Irrational Numbers
8. Bezout's Lemma and Prime Factorization
9. Workshop 3
10. Sets and Subsets
11. Cartesian Products and Relations
12. Workshop 4
13. Equivalence Relations and Equivalence Classes
14. Unions and Partitions
15. Workshop 5
16. Equivalence Classes and Modular Arithmetic
17. Decimal expansions and rational numbers
18. Workshop 6
19. Functions and their graphs
20. Functions and sets
21. Workshop 7
22. Properties of functions
23. Finite sets and cardinality
24. Workshop 8
25. Permutations of finite sets
26. Permutations continued
27. Workshop 9
28. Cardinality for infinite sets
29. Conclusion of Cardinality for infinite sets