What is a Real Number? A 5 Minute Introduction

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  • Thread starter Thread starter Greg Bernhardt
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    Introduction
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SUMMARY

Real numbers are defined through two primary methods: Cauchy sequences and Dedekind cuts. The discussion emphasizes the necessity of advanced mathematical concepts, such as equivalence classes, to accurately define real numbers. These definitions highlight the complexity underlying what may seem like a straightforward concept. Understanding these methods is crucial for grasping the foundational aspects of real analysis.

PREREQUISITES
  • Understanding of Cauchy sequences
  • Familiarity with Dedekind cuts
  • Basic knowledge of equivalence classes
  • Foundational concepts in real analysis
NEXT STEPS
  • Research the formal definition of Cauchy sequences
  • Explore Dedekind cuts in detail
  • Study equivalence classes and their applications
  • Investigate the field axioms related to real numbers
USEFUL FOR

Students of mathematics, educators in real analysis, and anyone interested in the foundational concepts of real numbers will benefit from this discussion.

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There are two main ways to introduce real numbers. Cauchy sequences are one possibility, Dedekind cuts are another. It is surprising how such a for us common thing like real numbers need some elaborated mathematics in order to properly define them. Even the approach by Cauchy sequences needs the concept of equivalence classes.
 

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