Abstract
A precise axiomatic description of the real numbers is given, sufficient, as it turns out, to serve as a basis for all developments in analysis. The importance of the completeness axiom is emphasised, along with its manifestations of infimum and supremum.
As professor in the Polytechnic School in Zurich I found
myself for the first time obliged to lecture upon the
elements of the differential calculus and felt, more keenly
than ever before, the lack of a really scientific foundation
for arithmetic.
R. Dedekind. Essays on the theory of numbers
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This is the second group of exercises in Sect. 2.2. For this reason the numbering is continued from the previous set.
- 2.
Often the term “decimal fraction” is used to mean a rational number whose denominator is a power of 10. We use the term to mean a real number between 0 and 1 in its decimal representation.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2020 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Magnus, R. (2020). Real Numbers. In: Fundamental Mathematical Analysis. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-030-46321-2_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-46321-2_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-46320-5
Online ISBN: 978-3-030-46321-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)