9주차. visual mathematics 2 - kocwelearning.kocw.net/contents4/document/lec/2013/hufs/... ·...
TRANSCRIPT
![Page 1: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/1.jpg)
수의 세계
9주차. Visual Mathematics 2
![Page 2: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/2.jpg)
1. Tiling
• 쪽매붙이기
학습내용
학습목표
![Page 3: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/3.jpg)
수의 세계
9주차. Visual Mathematics 2
![Page 4: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/4.jpg)
1 쪽매붙이기--Tiling 1
쪽매 붙이기 : 특정한 조각들과 합동인 조각,
즉 쪽매들을 이용하여 평면을 빈틈없이 그리고
중복없이 채우는 행위 (tiling, mosaic,
tessellation, quilt)
![Page 5: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/5.jpg)
1 쪽매붙이기--Tiling 1
1) 정다각형 쪽매 -- 정사각형
![Page 6: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/6.jpg)
1 쪽매붙이기--Tiling 1
덕 산 수 덕 사
1) 정다각형 쪽매 -- 정사각형
![Page 7: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/7.jpg)
1 쪽매붙이기--Tiling 1
1) 정다각형 쪽매 -- 정삼각형
![Page 8: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/8.jpg)
1 쪽매붙이기--Tiling 1
1) 정다각형 쪽매 -- 정삼각형
![Page 9: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/9.jpg)
1 쪽매붙이기--Tiling 1
1) 정다각형 쪽매 -- 정육각형
![Page 10: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/10.jpg)
1 쪽매붙이기--Tiling 1
정칠각형 이상의 다각형으로는 쪽매붙이기가
불가능.
(정 n각형의 내각은 180° – 360°/n.
n≥7이면 세개 이상의 내각이 만날 때 도형들이
겹친다.)
![Page 11: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/11.jpg)
1 쪽매붙이기--Tiling 1
2) 다각형 쪽매 -- 사각형
![Page 12: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/12.jpg)
1 쪽매붙이기--Tiling 1
2) Alhambra Palace
Granada, Spain
![Page 13: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/13.jpg)
1 쪽매붙이기--Tiling 1
2) Alhambra Palace
Arabic patterns of tiling
![Page 14: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/14.jpg)
1 쪽매붙이기--Tiling 1
2) Alhambra Palace
Arabic patterns of tiling
![Page 15: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/15.jpg)
1 쪽매붙이기--Tiling 1
2) 쪽매 겹치기
![Page 16: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/16.jpg)
1 쪽매붙이기--Tiling 1
2) 쪽매 겹치기
투명 아크릴판 쪽매를 겹치면….
피타고라스 정리
![Page 17: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/17.jpg)
1 쪽매붙이기--Tiling 1
2) 쪽매 겹치기
투명 아크릴판 쪽매를 겹치면….
피타고라스 정리
![Page 18: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/18.jpg)
수의 세계
9주차. Visual Mathematics 2
![Page 19: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/19.jpg)
1 말이 필요없는 증명 -- 급수 2
1) Sum of cubes
2
333
2
)1(21
nnn
![Page 20: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/20.jpg)
1 말이 필요없는 증명 -- 급수 2
1) Sum of cubes
![Page 21: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/21.jpg)
1 말이 필요없는 증명 -- 급수 2
2) Series
13 + 23 + ×××+ n3 =n(n +1)
2
æ
èçö
ø÷
2
®¥
1
12+
1
22+
1
32+ ×××+
1
n2®
p 2
6
n®¥As
1-1+1-1+
![Page 22: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/22.jpg)
1 말이 필요없는 증명 -- 급수 2
2) 등비급수
1
2+
1
22+
1
23+ ×××+
1
2n®1
![Page 23: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/23.jpg)
1 말이 필요없는 증명 -- 급수 2
2) 등비급수
1
4+
1
42+
1
43+ ×××+
1
4n®
1
3
![Page 24: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/24.jpg)
1 말이 필요없는 증명 -- 급수 2
2) 등비급수 – 계산
1
4nn=1
¥
å = limn®¥
1
4+
1
42+ +
1
4næ
èçö
ø÷=
1
3
![Page 25: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/25.jpg)
1 말이 필요없는 증명 -- 급수 2
2) 등비급수 – 계산
1
4nn=1
¥
å =1
3
1
4+
1
42+
1
43+ ×××+
1
4n= sn -1®
1
3
![Page 26: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/26.jpg)
1 말이 필요없는 증명 -- 급수 2
2) 등비급수 – 아르키메데스의 증명
1
4nn=1
¥
å =1
3
![Page 27: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/27.jpg)
1 말이 필요없는 증명 -- 급수 2
Sum of reciprocal squares
limn®¥
1
12+
1
22+
1
32+ ×××+
1
n2
æ
èçö
ø÷< ¥
![Page 28: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/28.jpg)
1 말이 필요없는 증명 -- 급수 2
Sum of reciprocal of squares of all primes
limp®¥
1
22+
1
32+ ×××+
1
p2
æ
èçö
ø÷< ¥
(P는 소수)
limp®¥
1
2+
1
3+ ×××+
1
p
æ
èçö
ø÷®¥
Sum of reciprocal of all primes
Euler
→ →
![Page 29: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/29.jpg)
1 말이 필요없는 증명 -- 급수 2
Sum of reciprocals – Harmonic series
limn®¥
1
2+
1
3+ ×××+
1
n
æ
èçö
ø÷®¥
→ →
![Page 30: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/30.jpg)
1 말이 필요없는 증명 -- 급수 2
Sum of reciprocals – Harmonic series
limn®¥
1
2+
1
3+ ×××+
1
n
æ
èçö
ø÷®¥
n
nn
ln
1
3
1
2
1ln
![Page 31: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/31.jpg)
1 말이 필요없는 증명 -- 급수 2
Sums of geometric series
1
rn= limn®¥
1+1
r+
1
r2+
1
r3+ ×××+
1
rnæ
èçö
ø÷n=0
¥
å
If r≥1,
1+1
r+
1
r2+
1
r3+ ×××+
1
rn>1+1+1+ +1
If r<1,
1+1
r+
1
r2+
1
r3+ ×××+
1
rn=
1- rn+1
1- r®
1
1- r
![Page 32: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/32.jpg)
1 말이 필요없는 증명 -- 급수 2
Sums of geometric series
![Page 33: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/33.jpg)
1 말이 필요없는 증명 -- 급수 2
Differential geometric series
![Page 34: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/34.jpg)
수의 세계
9주차. Visual Mathematics 2
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[문제1] 아래 그림들이 나타내는 무한급수와 그
합을 구하라.
평가하기
![Page 36: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/36.jpg)
[문제2] “L”자 모양으로 생긴 아래 그림을 각 각
2, 3, 4, 8, 12개의 합동인 모양으로 나누어 보시요.
평가하기
![Page 37: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/37.jpg)
[문제3] 아래 그림을 이용하여 제곱수의 합에 대한
다음 공식을 증명하라.
평가하기
3 12 + 22 + + n2( ) = n n +1
2
æ
èçö
ø÷n +1( )
![Page 38: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/38.jpg)
수의 세계
9주차. Visual Mathematics 2
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정리하기
1강 . 쪽매붙이기
- 쪽매붙이기
![Page 40: 9주차. Visual Mathematics 2 - KOCWelearning.kocw.net/contents4/document/lec/2013/Hufs/... · 2013-10-11 · Visual Mathematics 2 . 1 쪽매붙이기--Tiling 쪽매 붙이기 : 특정한](https://reader031.vdocuments.us/reader031/viewer/2022040917/5e91fd10c84a2048fb720fd8/html5/thumbnails/40.jpg)
정리하기
2강. 무한급수
- 무한급수의 합
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