Abstract
Doing analysis in a rigorous way starts with understanding the properties of the real numbers. Readers will be familiar, in some sense, with the real numbers from studying calculus. A completely rigorous development of the real numbers requires checking many details. We attempt to justify one definition of the real numbers without carrying out the proofs.
Intuitively, we think of the real numbers as the points on a line stretching off to infinity in both directions. However, to make any sense of this, we must label all the points on this line and determine the relationship between them from different points of view. First, the real numbers form an algebraic object known as a field, meaning that one may add, subtract, and multiply real numbers and divide by nonzero real numbers. There is also an order on the real numbers compatible with these algebraic properties, and this leads to the notion of distance between two points.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2009 Springer-Verlag New York
About this chapter
Cite this chapter
Davidson, K.R., Donsig, A.P. (2009). The Real Numbers. In: Real Analysis and Applications. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-98098-0_2
Download citation
DOI: https://doi.org/10.1007/978-0-387-98098-0_2
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-98097-3
Online ISBN: 978-0-387-98098-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)