The Real Numbers

Part of the Undergraduate Texts in Mathematics book series (UTM)


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.


Real Number Rational Number Positive Real Number Nonempty Subset Cauchy Sequence 
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Copyright information

© Springer-Verlag New York 2009

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of WaterlooOntarioCanada
  2. 2.Department of MathematicsUniversity of Nebraska-LincolnLincolnUSA

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