Difference between revisions of "Elementary proof"
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(→Significance of Elementary Proofs: remove first reason: complex numbers do not require a unique sqrt of -1; rather, they are technically ordered pairs (a, b) with + and * s.t. (0,1)(0,1)=(-1,0)) |
(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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== Significance of Elementary Proofs == | == Significance of Elementary Proofs == | ||
| − | Elementary proofs are preferred over non-elementary proofs for at least | + | Elementary proofs are preferred over non-elementary proofs for at least two reasons: |
| + | *elementary proofs minimize the underlying assumptions, as in avoiding the assumption that there is a unique, algebraically manipulable square root of negative one | ||
*elementary proofs are typically impossible to simplify further in a logically significant manner | *elementary proofs are typically impossible to simplify further in a logically significant manner | ||
Latest revision as of 00:08, April 8, 2012
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[1] 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.
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.[2]
Significance of Elementary Proofs
Elementary proofs are preferred over non-elementary proofs for at least two reasons:
- elementary proofs minimize the underlying assumptions, as in avoiding the assumption that there is a unique, algebraically manipulable square root of negative one
- elementary proofs are typically impossible to simplify further in a logically significant manner