In the final three chapters of the book – this one and the next two – I want tointroduce you to a topic with a different flavour from the rest of the book. Thisis a topic called mathematical analysis, or just analysis for short. This is a hugearea, which at an undergraduate level starts off with the study of the real andcomplex numbers, and functions defined on them. Of course we can’t coververy much of the subject – that would require several more books – but we willdo enough to prove several interesting results, such as putting the existence ofdecimal expressions for real numbers on a rigorous footing, and proving theexistence of nth roots of positive real numbers (stated in Proposition 5.1). Before we can start studying functions of the real numbers we need to gomore deeply into some basic properties of the real numbers themselves, andthat’s what we’ll do in this chapter.