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Elementary proof (view source)

Revision as of 22:48, December 23, 2006

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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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The term "elementary proof" or "elementary techniques" in mathematics means use of only [[real numbers]] rather than [[complex numbers]], which relies on manipulation of the imaginary square root of (-1).  Elementary proofs are preferred because they are do not require additional assumptions inherent in complex analysis, such as that there is a unique square root of (-1) that will yield consistent results.
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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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Mathematicians also consider elementary techniques to include objects, operations, and relations.  Sets, sequences and geometry are not included.
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The prime number theorem has long been proven using non-elementary techniques, but in 1949 and 1950 an elementary proof by [[Paul Erdos]] and [[Atle Selberg]] earned Selberg the highest prize in math, the Fields medal.
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