Chapter 2: The Real Numbers

The basis for everything we do in analysis is the real number system. In Calculus, most of our understanding of the real numbers is based on geometric intuition, together with simple algebra. In this chapter we shall give axioms for the real numbers (much like for groups or rings) which we can then use to prove useful facts about them. The ingredients that go into this are the usual arithmetic properties, together with a very subtle property related to the ordering of the reals. This means we have to look at how sets of numbers can be ordered, in careful detail.

Chapter Contents

Class 02
- Arithmetic: The axioms of a field, and some examples.
- Order: Linearly ordered sets
Class 03
- Ordered fields: Definition, examples, and basic properties
Class 04
- Properties of ordered sets
- Bounded sets
Class 05
Class 06
Class 07
- The absolute value function
- Open sets of real numbers

Analysis WebNotes by John Lindsay Orr.
Comments to the author: jorr@math.unl.edu

All contents copyright (C) 1996 John L. Orr
University of Nebraska--Lincoln
All rights reserved

Last modified: June 1996