The fifth lecture in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers:
• The algebra of rational numbers: sums and products of rational numbers are rational (etc.)
• Comparison with irrational numbers: investigation of combinations of rational numbers and irrational numbers.
• Density of the rationals and the irrationals in the real line.
Note the factor 2 error in the last sketch proof, as noted in the call-out.

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