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[수학 정보] 0!=1인 이유

문제은행 2020.09.09 22:49 조회 1030

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

이전글 대45를 풀기 위한 배경지식 (결혼 정리)

다음글 1 / 0 * 0?

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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

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저 방법도 되긴 하지만 약간만 더 엄밀성 있는 증명을 써 보자면,

$_nC_0$ 은 n개의 물건 중 0개를 뽑는 경우의 수이므로 1가지이다. 그런데 조합 공식에 의해

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

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- 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

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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.

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- 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~~

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- 문제은행 Lv.5 2021.01.31 07:07
  
  ㅇㅁㅇ
  
  좋아요0

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