About: Proof that e is transcendental   Sponge Permalink

An Entity of Type : owl:Thing, within Data Space : 134.155.108.49:8890 associated with source dataset(s)

The first proof that the base of the natural logarithms, e, is transcendental dates from 1873. We will now follow the strategy of David Hilbert (1862–1943) who gave a simplification of the original proof of Charles Hermite. The idea is the following: Assume, for purpose of finding a contradiction, that e is algebraic. Then there exists a finite set of integer coefficients satisfying the equation: and such that and are both non-zero. We have arrived at the equation: which can now be written in the form where is a non-zero integer and is not. To show that for sufficiently large k for every real number G

AttributesValues
rdfs:label
  • Proof that e is transcendental
rdfs:comment
  • The first proof that the base of the natural logarithms, e, is transcendental dates from 1873. We will now follow the strategy of David Hilbert (1862–1943) who gave a simplification of the original proof of Charles Hermite. The idea is the following: Assume, for purpose of finding a contradiction, that e is algebraic. Then there exists a finite set of integer coefficients satisfying the equation: and such that and are both non-zero. We have arrived at the equation: which can now be written in the form where is a non-zero integer and is not. To show that for sufficiently large k for every real number G
dcterms:subject
abstract
  • The first proof that the base of the natural logarithms, e, is transcendental dates from 1873. We will now follow the strategy of David Hilbert (1862–1943) who gave a simplification of the original proof of Charles Hermite. The idea is the following: Assume, for purpose of finding a contradiction, that e is algebraic. Then there exists a finite set of integer coefficients satisfying the equation: and such that and are both non-zero. Depending on the value of n, we specify a sufficiently large positive integer k (to meet our needs later), and multiply both sides of the above equation by , where the notation will be used in this proof as shorthand for the integral: We have arrived at the equation: which can now be written in the form where The plan of attack now is to show that for k sufficiently large, the above relations are impossible to satisfy because is a non-zero integer and is not. The fact that is a nonzero integer results from the relation which is valid for any positive integer j and can be proved using integration by parts and mathematical induction. To show that for sufficiently large k we first note that is the product of the functions and . Using upper bounds for and on the interval [0,n] and employing the fact for every real number G is then sufficient to finish the proof. A similar strategy, different from Lindemann's original approach, can be used to show that the number π is transcendental. Besides the gamma-function and some estimates as in the proof for e, facts about symmetric polynomials play a vital role in the proof.
Alternative Linked Data Views: ODE     Raw Data in: CXML | CSV | RDF ( N-Triples N3/Turtle JSON XML ) | OData ( Atom JSON ) | Microdata ( JSON HTML) | JSON-LD    About   
This material is Open Knowledge   W3C Semantic Web Technology [RDF Data] Valid XHTML + RDFa
OpenLink Virtuoso version 07.20.3217, on Linux (x86_64-pc-linux-gnu), Standard Edition
Data on this page belongs to its respective rights holders.
Virtuoso Faceted Browser Copyright © 2009-2012 OpenLink Software