# Difference between revisions of "Real analysis"

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The first serious consideration of the real numbers was by [[Archimedes]] and followed by other [[Greek]]s such as [[Euclid]], [[Pappus]], and [[Zeno]]. To honor Archimedes' contribution, real analysts have named a property of the real numbers the [[Archimedean|Archimedean property]]. Real analysis remained in [[geometry]]'s shadow until the development of the subfield of [[calculus]]. This subject [[coordinatization|coordinatized]] all geometry known at the time, subsuming it into its scope. | The first serious consideration of the real numbers was by [[Archimedes]] and followed by other [[Greek]]s such as [[Euclid]], [[Pappus]], and [[Zeno]]. To honor Archimedes' contribution, real analysts have named a property of the real numbers the [[Archimedean|Archimedean property]]. Real analysis remained in [[geometry]]'s shadow until the development of the subfield of [[calculus]]. This subject [[coordinatization|coordinatized]] all geometry known at the time, subsuming it into its scope. | ||

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+ | The foundations of real analysis were shaken at the end of the 19th century with the work of [[Richard Dedekind]], who relaid the Archimedean-style groundwork with his own radical concepts. His [[Dedekind cut]]s undercut the assumption of the continuity of the real line, by ''cutting'' at ''gaps'' between points. Mathematicians were worried that his techniques used the dubious [[Axiom of Choice]] and seemingly [[elementary_proof|non-elementary]] methods. However, the unifying ideas of [[Cauchy]], specifically that of the [[Cauchy sequence]] and [[complete metric space|completeness]], helped eliminate doubts and gain acceptance for Dedekind's ideas among real analysts. Dedekind cuts are now viewed as a solid foundation for real analysis, more than Archimedes' ideas ever were.<ref>http://plato.stanford.edu/entries/dedekind-foundations/#FouAna</ref> | ||

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+ | ====References==== | ||

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+ | {{reflist}} | ||

[[category:mathematics]] | [[category:mathematics]] |

## Revision as of 16:48, 11 August 2008

**Real analysis** is a field in mathematics that focuses on the set of real numbers, their properties, sequences and functions. Included in this branch of mathematics is concepts of limits and convergence, calculus, and properties of real-valued functions such as continuity.

The first serious consideration of the real numbers was by Archimedes and followed by other Greeks such as Euclid, Pappus, and Zeno. To honor Archimedes' contribution, real analysts have named a property of the real numbers the Archimedean property. Real analysis remained in geometry's shadow until the development of the subfield of calculus. This subject coordinatized all geometry known at the time, subsuming it into its scope.

The foundations of real analysis were shaken at the end of the 19th century with the work of Richard Dedekind, who relaid the Archimedean-style groundwork with his own radical concepts. His Dedekind cuts undercut the assumption of the continuity of the real line, by *cutting* at *gaps* between points. Mathematicians were worried that his techniques used the dubious Axiom of Choice and seemingly non-elementary methods. However, the unifying ideas of Cauchy, specifically that of the Cauchy sequence and completeness, helped eliminate doubts and gain acceptance for Dedekind's ideas among real analysts. Dedekind cuts are now viewed as a solid foundation for real analysis, more than Archimedes' ideas ever were.^{[1]}