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Elementary proof (view source)

Revision as of 00:08, April 8, 2012

610 bytes added ,  00:08, April 8, 2012
restore as before, because complex numbers are not simply ordered pairs; complex numbers do depend on a unique square root of -1
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The term "elementary techniques" in mathematics means use of traditional, very well-understood fields of mathematics like number theory.  Other fields of math that require additional assumptions, such as complex analysis, cannot be used in a proof based on elementary techniques.  Included in elementary techniques are objects, operations, and relations.  Sets, sequences and many types of geometry are not included.
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An '''elementary proof''' or '''elementary technique''' in mathematics is a [[proof]] that uses only [[real numbers]] or [[real analysis]] rather than the use of [[complex analysis]]<ref>http://mathworld.wolfram.com/ElementaryProof.html</ref> or reliance on less rigorous axioms, such as the [[Axiom of Choice]].  An elementary proof typically cannot be improved by expressing it in simpler form.
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For example, there was a proof of the prime number theorem relying on non-elementary techniques, but in 1949 and 1950 an "elementary proof" by Erdos and Selberg earned Selberg the highest prize in math, the Fields medal.
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The [[Prime Number Theorem]] has long been proven using complex analysis ([[Riemann Zeta function]]), but in 1949 and 1950 an elementary proof by [[Paul Erdos]] and [[Atle Selberg]] earned Selberg the highest prize in math, the [[Fields Medal]]. In contrast, [[Andrew Wiles]]' proof of [[Fermat's Last Theorem]] is not an elementary proof.<ref name="Occam">http://www.occampress.com/fermat.pdf Page 5</ref>
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== Significance of Elementary Proofs ==
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Elementary proofs are preferred over non-elementary proofs for at least two reasons:
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*elementary proofs minimize the underlying assumptions, as in avoiding the assumption that there is a unique, algebraically manipulable square root of negative one
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*elementary proofs are typically impossible to simplify further in a logically significant manner
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==References==
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{{reflist}}
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[[Category:Mathematics]]
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