"무리수와 초월수"의 두 판 사이의 차이

수학노트
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<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">린데만-바이어슈트라스 정리</h5>
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">린데만-바이어슈트라스 정리</h5>
  
 
* [[린데만-바이어슈트라스 정리]]<br>
 
* [[린데만-바이어슈트라스 정리]]<br>
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<h5>겔퐁드-슈나이더 정리</h5>
 
<h5>겔퐁드-슈나이더 정리</h5>
  
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:If α and β are [[algebraic number]]s (with α≠0 and <math>\log \alpha</math> any non-zero logarithm of α), and if β is not a [[rational number]], then any value of <math>\alpha^{\beta} = \exp\{\beta \log \alpha\}</math> is a [[transcendental number]]. ===Comments=== * The values of <math>\alpha</math> and <math>\beta</math> are not restricted to [[real number]]s; all [[complex number]]s are allowed. * In general, <math>\alpha^{\beta} = \exp\{\beta \log \alpha\}</math> is [[multivalued function|multivalued]], where "log" stands for the [[complex logarithm]]. This accounts for the phrase "any value of" in the theorem's statement. * An equivalent formulation of the theorem is the following: if <math>\alpha</math> and <math>\gamma</math> are nonzero algebraic numbers, and we take any non-zero logarithm of α, then <math>(\log \gamma)/(\log \alpha)</math> is either rational or transcendental. * If the restriction that <math>\beta</math> be algebraic is removed, the statement does not remain true in general (choose <math>\alpha=3</math> and <math>\beta=\log 2/\log 3</math>, which is transcendental, then <math>\alpha^{\beta}=2</math> is algebraic). A characterization of the values for α and β which yield a transcendental α<sup>β</sup> is not known.
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:If <math>\alpha</math> and <math>\beta</math> are algebraic numbers (with α≠0 and <math>\log \alpha</math> any non-zero logarithm of α), and if β is not a [[rational number]], then any value of <math>\alpha^{\beta} = \exp\{\beta \log \alpha\}</math> is a [[transcendental number]]. ===Comments=== * The values of <math>\alpha</math> and <math>\beta</math> are not restricted to [[real number]]s; all [[complex number]]s are allowed. * In general, <math>\alpha^{\beta} = \exp\{\beta \log \alpha\}</math> is [[multivalued function|multivalued]], where "log" stands for the [[complex logarithm]]. This accounts for the phrase "any value of" in the theorem's statement. * An equivalent formulation of the theorem is the following: if <math>\alpha</math> and <math>\gamma</math> are nonzero algebraic numbers, and we take any non-zero logarithm of α, then <math>(\log \gamma)/(\log \alpha)</math> is either rational or transcendental. * If the restriction that <math>\beta</math> be algebraic is removed, the statement does not remain true in general (choose <math>\alpha=3</math> and <math>\beta=\log 2/\log 3</math>, which is transcendental, then <math>\alpha^{\beta}=2</math> is algebraic). A characterization of the values for α and β which yield a transcendental α<sup>β</sup> is not known.
  
 
 
 
 
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<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">상위 주제</h5>
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">상위 주제</h5>
  
 
 
 
 
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<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">재미있는 사실</h5>
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">재미있는 사실</h5>
  
 
 
 
 
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<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">많이 나오는 질문과 답변</h5>
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">많이 나오는 질문과 답변</h5>
  
 
*  네이버 지식인<br>
 
*  네이버 지식인<br>
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<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">관련된 고교수학 또는 대학수학</h5>
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련된 고교수학 또는 대학수학</h5>
  
 
* [[대수적수론]]<br>
 
* [[대수적수론]]<br>
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<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">관련된 다른 주제들</h5>
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련된 다른 주제들</h5>
  
 
* [[파이 π는 초월수이다|파이는 초월수이다]]<br>
 
* [[파이 π는 초월수이다|파이는 초월수이다]]<br>
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* Baker's theorem
 
* Baker's theorem
  
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* [[수학사연표 (역사)|수학사연표]]
 
* [[수학사연표 (역사)|수학사연표]]
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<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">관련도서 및 추천도서</h5>
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련도서 및 추천도서</h5>
  
 
*  도서내검색<br>
 
*  도서내검색<br>
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<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">참고할만한 자료</h5>
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">참고할만한 자료</h5>
  
 
*   <br>
 
*   <br>
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* 무리수이야기<br>
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* [http://navercast.naver.com/science/math/561 무리수이야기]<br>
 
** 정경훈
 
** 정경훈
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** 네이버 오늘의 과학, 
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** 네이버 오늘의 과학, 2009-6-9
 
* [http://ko.wikipedia.org/wiki/%EC%B4%88%EC%9B%94%EC%88%98 http://ko.wikipedia.org/wiki/초월수]
 
* [http://ko.wikipedia.org/wiki/%EC%B4%88%EC%9B%94%EC%88%98 http://ko.wikipedia.org/wiki/초월수]
 
* http://en.wikipedia.org/wiki/Gelfond-Schneider_theorem
 
* http://en.wikipedia.org/wiki/Gelfond-Schneider_theorem
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<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">관련기사</h5>
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련기사</h5>
  
 
*  네이버 뉴스 검색 (키워드 수정)<br>
 
*  네이버 뉴스 검색 (키워드 수정)<br>
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<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">블로그</h5>
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">블로그</h5>
  
 
* 구글 블로그 검색 http://blogsearch.google.com/blogsearch?q=
 
* 구글 블로그 검색 http://blogsearch.google.com/blogsearch?q=
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<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">이미지 검색</h5>
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">이미지 검색</h5>
  
 
* http://commons.wikimedia.org/w/index.php?title=Special%3ASearch&search=
 
* http://commons.wikimedia.org/w/index.php?title=Special%3ASearch&search=
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<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">동영상</h5>
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">동영상</h5>
  
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* http://www.youtube.com/results?search_type=&search_query=<br>
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* http://www.youtube.com/results?search_type=&search_query=

2009년 6월 25일 (목) 18:18 판

간단한 소개
  • 복소수 중에서 어떠한 유리수 계수방정식도 만족시킬 수 없는 수를 초월수라 함
    • 유리수 계수방정식은 적당한 정수를 곱하여 다음과 같은 형태의 정수계수방정식으로 표현할 수도 있음.
      \(a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \cdots + a_1 x + a_0 = 0, a_i \in \mathbb{Z}\)
    • 복소수 중에서 어떠한 정수계수방정식도 만족시킬 수 없는 수를 초월수라 해도 무방
  • 대수적수론 에 비해 훨씬 어렵고, 체계적인 이론이 확립되어 있지 않음.

 

 

린데만-바이어슈트라스 정리

 

겔퐁드-슈나이더 정리
If \(\alpha\) and \(\beta\) are algebraic numbers (with α≠0 and \(\log \alpha\) any non-zero logarithm of α), and if β is not a rational number, then any value of \(\alpha^{\beta} = \exp\{\beta \log \alpha\}\) is a transcendental number. ===Comments=== * The values of \(\alpha\) and \(\beta\) are not restricted to real numbers; all complex numbers are allowed. * In general, \(\alpha^{\beta} = \exp\{\beta \log \alpha\}\) is multivalued, where "log" stands for the complex logarithm. This accounts for the phrase "any value of" in the theorem's statement. * An equivalent formulation of the theorem is the following: if \(\alpha\) and \(\gamma\) are nonzero algebraic numbers, and we take any non-zero logarithm of α, then \((\log \gamma)/(\log \alpha)\) is either rational or transcendental. * If the restriction that \(\beta\) be algebraic is removed, the statement does not remain true in general (choose \(\alpha=3\) and \(\beta=\log 2/\log 3\), which is transcendental, then \(\alpha^{\beta}=2\) is algebraic). A characterization of the values for α and β which yield a transcendental αβ is not known.

 

 

베이커의 정리

 

 

상위 주제

 

 

 

하위페이지

 

 

재미있는 사실

 

 

많이 나오는 질문과 답변

 

관련된 고교수학 또는 대학수학

 

관련된 다른 주제들

 

 

관련도서 및 추천도서

 

참고할만한 자료

 

관련기사

 

 

블로그

 

이미지 검색

 

동영상
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