Mathematics Appreciation
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Mathematics Appreciation. 数学欣赏. 主讲:张文俊. 第五章 数学之奇. 自言自语. 数学中不少结论由于其巧妙无比而令人赞叹,正是因为这一点,数学才有无穷的魅力。. 第一节 实数系统. [email protected] 实数集. 有理数集. 实数系统. In This Section. 一家人. 数系扩 充概述. 连续统 假设. Hilbert 的旅馆. 德国著名数学家大卫 • 希尔伯特曾经讲过一个精彩故事。在那里,希尔伯特成为一个旅馆的老板,这个旅馆不同于我们现实生活中的任何旅馆,它设有无穷多个房间。

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Mathematics Appreciation




[email protected]


In This Section

[email protected]


Hilbert


1


1.


1.


1.

(560480


1.

4


1.

  • 250

250


2.


2.

  • 1484Chuquet,1445--1500 4x23x

    x3/2(9/4-4)


2.

(Cardano,1501--1576)


2.

15451040


2.

1637 (imaginary)

(R.Descartes,1596--1661)


2.

1777i (-1)

(L.Euler,1707~1783)


2.

1748

eix cosx + i sinx


2.

17991831(a,b) a+bi

(Carl Friedrich Gauss17771855)


y

O

x

2.

(a,b) ~a+bi

b

a


2.

  • 18

  • 19


2.

1873(1833~1902)Bangbeili1530~1590


3


3.

1843

Hamilton, William Rowan, 18051865


3.

1847

(Cayley,Arthur. 1821-1895)


N Z

Q R

C

4


4.


4.

  • 0


1


5.

  • 11/231/2

  • e


6


6.


20

.

6.


6.


6.


6.


2


1.

2.


2.


x

1

1

0

2.

3.


3.

  • 1874


Georg Cantor; 18451918

  • 1845

  • 11 1867 1872

  • 1874


12345678

246810121416


12345678

1222324252627282



y

5

4

3

2

1

x

1

2

3

4

5

1(1 , 1)

2 (2 , 1)

3 (1 , 2)

4 (3 , 1)

5 (2 , 2)

6 (1 , 3)

=


123456

(1 , 1)(2 , 1)(1 , 2)(3 , 1)(2 , 2)(1 , 3)

= =


4. 0


4. 0

0


4. 0


4. 0

So small!

0!


0



3


1.


1.

19

G. Cantor, 1845---1918

(J. W. R. Dedekind, 18311916)

K. W. T. Weierstrass, 18151897


Georg Cantor; 18451918

  • 1845

  • 11 1867 1872

  • 1874


K.W.T., Weierstrass

  • K.T.W Weierstrass (18151897)

  • 18541856


R. (Dedekind, Richard __1916 )

  • R. (Dedekind, Richard) 183110 6

  • 1916 212

2.


2.

3.


3.

  • 1

  • Cauchy;

  • .


3.

2

4.


4.

  • 1-1


(0,1)


01

01



4


1.


1.

  • 00.

  • 0

2.


2.

0

  • 0 +n = 0

  • 0 +0 = 0 n0= 0

  • 00= 0 (0)n = 0


2.


2.

3.


3.

  • 0


n(0)n = 0

00 = 0

n0

0

4


4.

  • 1

  • 1 > 0


()

()


  • 0

  • 1

  • 0

  • 1

5.


5.

1

  • 1 +n +0 = 1

  • 1 + 1 = 1 n1= 1

  • 01= 1 (1)n = 1

6.


6.

  • 1

  • 1851

  • L=1/10n!=0.11000100000,

  • 1126241207205040


6.

  • e

  • eHermit1873, Lindemann1882.


5


01

1.


1.

MMM P(M) 2M.

M= , P(M)={ }

M={1}, P(M)={,M}

M={1,2}, P(M)={,{1},{2},M}

M={1,2,3},

P(M)={,{1},{2},{3},{1,2},{1,3},{2,3},M}


1.

M=, P(M)={} |M|= 0, |P(M)| = 1

M={1}, P(M)={,M} |M|= 1, |P(M)| = 2

M={1,2}, P(M)={,{1},{2},M}

|M|= 2, |P(M)| = 4

M={1,2,3},

P(M)={,{1},{2},{3},{1,2},{1,3},{2,3},M}

|M|= 3, |P(M)| = 8


1.

M

|P(M)| = 2|M|

2. Cantor


2. Cantor

CantorM

|P(M)|= 2|M| > |M|

P(M) M

3. 1


3. 1

  • 1= 2 0

    - + (0, 1) {01}N

    ak/2k, ak=0,1

4.


4.

0

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,

1/2 , 1/3 , 1/4 , 2/3 , 3/2 , 1/5 , 1/6 , 2/5 , 3/4 , 4/3 , 5/2 ,

22/7 , 113/355 , 52163/16604 , 17/12 ,


4.

1

=20


4.

2

=21


4.

3= 22

Je le vois, mais je ne le crois pas.

Cantor


6


0 < 101

1878Cantor

01


190023Cantor


Kurt Gdel; 1906 1978

1938


Paul Joseph Cohen; 1934__ ,

1963


100

Cantor

01?

GodelCohen:


[email protected]


God must be a geometer

Galileo



1





(1)

(2)


( ) (Thales of Miletus)} 625 547


(Pythagoras) 560 480


Euclid, 330---275


(3) (Descartes) (Fermat)


P. de (Fermat, Pierre de) 1601 8 20 --- - (Beaumont-de-Lomagne) 1665 1 12 (Castres)


(Descartes, Ren'e) 1596 3 31 (Touraine) (La Haye) (-- ) 1650 2 11


4 (orthogonal invariance) (coordinate-free analytical geometry) Hamilton Grassmann 3-


(HamiltonWilliam Rowan)1805 8 4 1865 9 2 (Dublin)


H.G. (GrassmannHermann Gunter)

1809 4 15 ( )1877 926


2





600

,

500

400

300

200


400

800

1200

1600


(Element)1482


151234657891011---15

  • 15621633Matteo Ricci 1552---161016076

  • (18111882)(A. Wylie, 18151887)18579


  • (Matteo Ricci; 1552 1610)

  • 1606 6


  • 1562 1633

  • 41 6

  • 1606 6


20



2355465


1.

2. a=c, b=c, a=b

3.


4.



1.

2.

3.

4.

5.


1.


2.


3.


P

R

ADC = PSQ

S

Q

4.

A

B

C

D


5.

1

a

b

a + b < 180

2



3


2200


4


  • 9


  • 180



19Gauss, C. F., 1777--1855ye . .1793---1856G. B. Riemann,1826--1866


  • ;

  • ;

  • 19


Gauss, C. F., 1777--1855


  • 1792

  • 1794

  • 1799

  • 1824


180



John Bolyai, 1802--18601832


(Bolyai Janos) 1802 1215 (Kolozsvar)()1860 1 17 ()


  • (Bolyai F) F. Bolyai

    1775-1856


1793---185618151823


(Lobachevskii , Nikolai Ivanovich) 1792 12 1 ( 11 20 ) ()1856 2 14


1826211

  • 1826211





1854Geord Bernhard Riemann,1826--1866




1868



(RiemannGeorg Friedrich Bernhard)1826 9 17 (Breselenz)1866 7 20 (Selasca)




4






Poncelet, 17881867S


J.-V. (PonceletJean--Victor)1788 7 1 (Metz)1867 1222


1868Beltrami, 18351899

(y)[tractrix]



(BeltramiEugenio) 18351116 (Cremona)1899 6 4


Jules Herni Poincar; 1854 1912l1 l2 180


P

l1

l2


(PoincareJules Henri)18544 29 (Nancy)1912 7 17


1870Klein, 18491925




5





6


:

1.


  • 1899

  • :;

  • :;

  • :



2.

__2030,


(Shapley)

1

2

3


3.


4.

Poincare, Minkouski, Hilbert



[email protected]



1



5510


23 9919915



1.


15


4

9

2

3

5

7

8

1

6


n2nnn


1n2n


2.



44812;

61014


:


2

1275310

105125



  • 31

  • 4880

  • 5275305224

  • 7363916800

  • 810


2


3


1.

12753103


1

9

4

9

2

4

4

2

2

3

5

7

3

7

5

5

7

3

8

1

6

8

8

6

6

1

9

5795


1

6

2

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7

3

16

12

8

4

21

17

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23

19

15

24

20

25

25


25

24

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3

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12

8

16

5

17

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15


2. De La Loubre

2n+12n+11


kk+1

24

61116


17

24

1

8

15

23

5

7

14

16

13

4

6

20

22

10

12

19

21

3

11

18

2

9

25



3. (Hire)


n p = 12nn(p-1)

103810320540

n4


1

1

3

2

4

4

4

2

2

3

3

1

4

2

2

3

3

1

1

1

3

2

4

4

11nn,A

2A1n,10,B

A

B


0

1

4

12

4

12

1

0

8

3

2

4

2

4

3

8

4

2

3

8

3

8

2

4

4

12

1

0

1

0

4

12

3BCC

4CD

C

D


1

1

0

3

15

12

2

14

12

4

4

0

12

4

8

2

6

4

3

4

7

1

9

8

4

4

8

2

8

10

3

11

8

1

5

4

12

13

1

3

0

3

2

2

0

4

12

16

5BDnE

E


4.


np=12n2n2 + 1 p.

1163148164560, 1055

8


1n2n2


1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64


n


64

2

3

61

60

6

7

57

9

55

54

12

13

51

50

16

17

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46

20

21

43

42

24

40

26

27

37

36

30

31

33

32

34

35

29

28

38

39

25

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23

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15

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1. Albrecht Drer

1514Albrecht Durer4


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134


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268748


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2

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22

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3

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8

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4.


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13

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4

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1

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69

6

51


14182=92+1

299


120

4

9

135

2

114

117

3

5

123

7

129

8

132

1

111

6

126

393331111353


15

10

3

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4

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9

14

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5.

Francis L. Miksa53600


6.

2601118020200


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8402058068231856000


46

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203

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184

189

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100

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8.

nn19103

n=3338


3

18

11

9

15

14

17

1

13

8

6

4

10

7

3

5

12

16

2

19


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