본문바로가기
자유게시판
수학을 주제로 떠들어 보세요!
[수학 정보] 0!=1인 이유
문제은행 2020.09.09 22:49 조회 1006
공유하기
페이스북 트위터 카카오톡 카카오스토리 네이버블로그 네이버밴드

팩토리얼은 보통 시작하는 수에 그 보다 낮은 수를 차례로 곱한다,예를 들어 4!의 값은 4x3x2x1=24가 된다.

 

자 이제 위의 논리를 1!에다 적용시키면 아래와 같이 1! 은 1 x (1-1)! 이 되고 그 결과는 다시 1! = 1 x 0! 로 나타낼 수 있다.

따라서 식의 0!는 반드시 1이 되어야 등식이 성립한다.

따라서 0!=1이 된다.

1!=1*(1-1)!

1!=1*0!

0!=1

추천 0 목록보기
댓글 15
댓글 작성하기
  •  
    Sin X Lv.5 2020.09.09 23:14

    실제로 gamma(1)=1 입니다

    댓글 작성하기 좋아요1 댓글수1
    •  
      문제은행 Lv.5 2020.09.09 23:33

      오호 그렇군요

      좋아요1
  •  
    khr519 Lv.8 2020.09.11 00:50

    1! 를 1*(1-1) 이라고 하신것은 알겠는데 거기에다 !를 왜 붙이죠?

    1!=1*(1-1) 이어야 하지 않나요?

    댓글 작성하기 좋아요1 댓글수5
    •  
      문제은행 Lv.5 2020.09.11 01:46

      아니죠

      예를들어 5!은 5*(5-1)이 아닌 5*(5-1)!이죠 계속 그렇게 가다보면

      5!=5*(5-1)!

      5!=5*4*(4-1)!

      ...

      5!=5*4*3*2*(2-1)!

      5!=5*4*3*2*1 이 되는겁니다

      똑같은 논리로 1!도 1*(1-1)! 이 되는거죠

      좋아요1
    •  
      아인수타인 Lv.12 2020.09.11 04:13

      그러니까 예를 들어 5!=5*4*3*2*1인데 4*3*2*1을 4!으로 묶으면 5!=5*4!이 되죠.

      좋아요0
    •  
      khr519 Lv.8 2020.09.11 04:34

      아 감사합니다

      근데 그럼 뭔가 꼬이는데...

      5!=5*4*3*2*(2-1)!=5*4*3*2*1*(1-1)=5*4*3*2*1*0=0 이렇게 되버리는 거 아닌가요...?

      좋아요0
    •  
      아인수타인 Lv.12 2020.09.11 04:39

      5×4×3×2×(2-1)!=5×4×3×2×1!=5×4×3×2×1×(1-1)!=5×4×3×2×1×0!이죠. (물론 (-1)!은 정의가 되지 않습니다. 감마함수로도요)

      좋아요0
    •  
      khr519 Lv.8 2020.09.11 04:41

      그렇군요

      좋아요0
  •  
    아인수타인 Lv.12 2020.09.11 04:12

    A PHP Error was encountered

    Severity: Warning

    Message: mkdir(): Permission denied

    Filename: libraries/Common.php

    Line Number: 202

    Backtrace:

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 202
    Function: mkdir

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 236
    Function: getLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
    Line: 90
    Function: parseLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
    Line: 343
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
    Line: 558
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
    Line: 315
    Function: require_once

    A PHP Error was encountered

    Severity: Warning

    Message: file_put_contents(/DATA/upload/polymath/latex/521597aa84104fa48ec57a9bf93044f3.gif): failed to open stream: No such file or directory

    Filename: libraries/Common.php

    Line Number: 213

    Backtrace:

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 213
    Function: file_put_contents

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 236
    Function: getLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
    Line: 90
    Function: parseLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
    Line: 343
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
    Line: 558
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
    Line: 315
    Function: require_once

    A PHP Error was encountered

    Severity: Warning

    Message: mkdir(): Permission denied

    Filename: libraries/Common.php

    Line Number: 202

    Backtrace:

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 202
    Function: mkdir

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 236
    Function: getLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
    Line: 90
    Function: parseLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
    Line: 343
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
    Line: 558
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
    Line: 315
    Function: require_once

    A PHP Error was encountered

    Severity: Warning

    Message: file_put_contents(/DATA/upload/polymath/latex/f685b13cdfa831009a823dfd7137d073.gif): failed to open stream: No such file or directory

    Filename: libraries/Common.php

    Line Number: 213

    Backtrace:

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 213
    Function: file_put_contents

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 236
    Function: getLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
    Line: 90
    Function: parseLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
    Line: 343
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
    Line: 558
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
    Line: 315
    Function: require_once

    저 방법도 되긴 하지만 약간만 더 엄밀성 있는 증명을 써 보자면,

     

    _nC_0은 n개의 물건 중 0개를 뽑는 경우의 수이므로 1가지이다. 그런데 조합 공식에 의해

    _nC_0=\frac{n!}{0!n!}=\frac{1}{0!}=1이므로 이를 만족하는 0!=1이다.

    댓글 작성하기 좋아요0 댓글수2
    •  
      khr519 Lv.8 2020.09.11 04:36 비밀댓글
      비밀 댓글이 등록 되었습니다!
    •  
      문제은행 Lv.5 2020.09.11 06:19

      ㅎㄷㄷ 와우 감사합니다

      좋아요0
  •  
    매스파이 Lv.8 2020.09.11 23:19

    A PHP Error was encountered

    Severity: Warning

    Message: mkdir(): Permission denied

    Filename: libraries/Common.php

    Line Number: 202

    Backtrace:

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 202
    Function: mkdir

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 236
    Function: getLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
    Line: 90
    Function: parseLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
    Line: 343
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
    Line: 558
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
    Line: 315
    Function: require_once

    A PHP Error was encountered

    Severity: Warning

    Message: file_put_contents(/DATA/upload/polymath/latex/0e68bebfecaf335de8a9d874e420aa65.gif): failed to open stream: No such file or directory

    Filename: libraries/Common.php

    Line Number: 213

    Backtrace:

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 213
    Function: file_put_contents

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 236
    Function: getLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
    Line: 90
    Function: parseLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
    Line: 343
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
    Line: 558
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
    Line: 315
    Function: require_once

    A PHP Error was encountered

    Severity: Warning

    Message: mkdir(): Permission denied

    Filename: libraries/Common.php

    Line Number: 202

    Backtrace:

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 202
    Function: mkdir

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 236
    Function: getLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
    Line: 90
    Function: parseLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
    Line: 343
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
    Line: 558
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
    Line: 315
    Function: require_once

    A PHP Error was encountered

    Severity: Warning

    Message: file_put_contents(/DATA/upload/polymath/latex/ce7a8984569aa4b9c9b584d7c28fd73f.gif): failed to open stream: No such file or directory

    Filename: libraries/Common.php

    Line Number: 213

    Backtrace:

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 213
    Function: file_put_contents

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 236
    Function: getLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
    Line: 90
    Function: parseLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
    Line: 343
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
    Line: 558
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
    Line: 315
    Function: require_once

    A PHP Error was encountered

    Severity: Warning

    Message: mkdir(): Permission denied

    Filename: libraries/Common.php

    Line Number: 202

    Backtrace:

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 202
    Function: mkdir

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 236
    Function: getLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
    Line: 90
    Function: parseLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
    Line: 343
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
    Line: 558
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
    Line: 315
    Function: require_once

    A PHP Error was encountered

    Severity: Warning

    Message: file_put_contents(/DATA/upload/polymath/latex/efe000a9724b6944ff4b4b8f41d2fcd9.gif): failed to open stream: No such file or directory

    Filename: libraries/Common.php

    Line Number: 213

    Backtrace:

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 213
    Function: file_put_contents

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 236
    Function: getLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
    Line: 90
    Function: parseLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
    Line: 343
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
    Line: 558
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
    Line: 315
    Function: require_once

    이렇게도 증명할수 있습니다.n!=\Gamma(n+1)

    \therefore 0!=1

    이고 음수에 대해서도 팩토리얼을 정의할 수 있습니다.

    예를 들어서 -1!=\Gamma(0)=\infty 뭐 이런 식으로요.

    댓글 작성하기 좋아요0 댓글수1
    •  
      문제은행 Lv.5 2020.09.11 23:35

      오호 감사합니다

      좋아요0
  •  
    유한의끝도못본남자 Lv.7 2021.01.20 22:59

    아닌뒈요

     !=는 같지않다(회로에서 not gate)라는 뜻이므로 0 != 1이 참인뒈요

    댓글 작성하기 좋아요0 댓글수1
  • ㈜동아사이언스 | 등록번호 : 서울 아04396 | 등록일자 : 2017년 3월 2일 | 제호 : 동아사이언스 | 발행인 : 장경애 | 편집인 : 김민수
  • ㈜동아사이언스 폴리매스 | 대표 : 장경애 | 개인정보관리책임자 : 최수정(privacy@dongascience.com) | 청소년보호책임자 : 박무성
  • (03737) 서울시 서대문구 충정로 29, 10층 ｜ 문의 : 02-6749-2000 ｜팩스 : 02-3148-0809 ｜사업자등록번호 : 101-81-62201 (사업자정보확인)
  • 통신판매업신고번호 : 제2013-서울용산-01101호 ｜온라인 결제를 위한 PG사의 구매안전서비스(가입사실 확인)
  • Copyright © Dongascience. All right reserved.

  • 폴리매스 문제는 과학기술진흥기금 및 복권기금의 재원으로 운영되고, 과학기술정보통신부와 한국과학창의재단의 지원을 받아 수행된 성과물로 우리나라의 과학기술 발전과 사회적 가치 증진에 기여하고 있습니다.

  • ☎문의 02-6749-3911