1
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Tasavvur qiling, milodan avvalgi yillar.

2
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Endi o'ylab ko'ring:

3
00:00:09,468 --> 00:00:12,721
Qanday qilib soatsiz vaqtni aniqlashgan?

4
00:00:12,721 --> 00:00:15,315
Barcha soatlar vaqt oqimini 
teng bo'laklarga bo'luvchi

5
00:00:15,315 --> 00:00:18,890
qandaydir takroriy shaklga asoslangan.

6
00:00:18,890 --> 00:00:20,688
Bunday takroriy shakillarni topish uchun

7
00:00:20,688 --> 00:00:22,918
samolarga yuzlanamiz.

8
00:00:22,918 --> 00:00:24,902
Har kuni, quyoshning chiqishi va botishi

9
00:00:24,902 --> 00:00:26,184
bunday shakllarning eng oddiysidir.

10
00:00:26,184 --> 00:00:28,760
Lekin uzoqroq vaqt bo'lagini kuzatish uchun

11
00:00:28,760 --> 00:00:30,811
uzoqroq takrorlanishlarga e'tibor beramiz.

12
00:00:30,811 --> 00:00:32,512
Buning uchun esa oyga yuzlanamiz.

13
00:00:32,512 --> 00:00:33,853
E'tibor bergan bo'lsangiz,

14
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oy kunlar osha to'lishadi va kichrayadi.

15
00:00:36,578 --> 00:00:37,894
To'lin oylar orasidagi kunlar sonini

16
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sanaydigan bo'lsak,

17
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u 29 kunga teng.

18
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Bir oydagi kunlar soni shundan 
kelib chiqqan bo'lsa kerak.

19
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Ammo 29 ni teng bo'laklarga 
bo'lishga harakat qilsak,

20
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bir muammoga duch kelamiz: 
buning iloji yo'q.

21
00:00:49,227 --> 00:00:51,676
29 ni teng bo'laklarga bo'lishning 
yagona yo'li

22
00:00:51,676 --> 00:00:54,819
uni 29 ta teng bo'lakka bo'lishdan iborat.

23
00:00:54,819 --> 00:00:57,102
29 soni tub son hisoblanadi.

24
00:00:57,102 --> 00:00:59,061
Uni bo'linmas deb tasavvur qiling.

25
00:00:59,061 --> 00:01:00,879
Agar son birdan boshqa

26
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teng bo'laklarga bo'linsa,

27
00:01:02,814 --> 00:01:04,621
biz uni 'murakkab son' deb ataymiz.

28
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Endi biz qiziqishimiz mumkin,

29
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"Dunyoda nechta tub son bo'lishi mumkin?

30
00:01:08,450 --> 00:01:10,398
Va ularning eng kattasi 
nechaga teng ekan?"

31
00:01:10,398 --> 00:01:13,744
Keling, barcha sonlarni 
ikkita guruhga bo'lamiz.

32
00:01:13,744 --> 00:01:15,611
Tub sonlar chap tomonda

33
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va murakkab sonlar o'ng tomonda.

34
00:01:17,648 --> 00:01:20,379
Boshida, u tomondan bu tomonga raqs 
tushayotganga o'xshaydilar.

35
00:01:20,379 --> 00:01:23,017
Ammo ularning joylashuvida 
aniq bir shakl mavjud emas.

36
00:01:23,017 --> 00:01:24,439
Keling, bunday shaklni ko'rish uchun

37
00:01:24,439 --> 00:01:26,077
zamonaviy usuldan foydalanamiz.

38
00:01:26,077 --> 00:01:29,047
Bu usul "Ulam spirali" deb nomlanadi.

39
00:01:29,047 --> 00:01:32,011
Boshida, barcha raqamlarni tartib bilan
o'sayotgan spiral

40
00:01:32,011 --> 00:01:34,043
ichiga joylab chiqamiz.

41
00:01:34,043 --> 00:01:37,164
Keyin, barcha tub sonlarni 
ko'k ranga bo'yab chiqamiz.

42
00:01:37,164 --> 00:01:41,290
Nihoyat, biz millionlab raqamlarni
ko'rish uchun uzoqlashamiz.

43
00:01:41,290 --> 00:01:42,860
Mana bu tugalmas tub sonlarning

44
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shakli hisoblanadi.

45
00:01:45,365 --> 00:01:47,967
Hayratlanarlisi shuki, bu shaklning 
tuliq strukturasi

46
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haligacha topilmagan.

47
00:01:50,314 --> 00:01:51,843
Nimanidir kashf etish arafasida
turganga o'xshaymiz...

48
00:01:51,843 --> 00:01:52,987
Keling, m.a. 300 yillarga,

49
00:01:52,987 --> 00:01:55,526
Qadimgi Gretsiyaga sayr qilamiz.

50
00:01:55,526 --> 00:01:58,183
Buyuk faylasuf, Aleksandryalik Evklid,

51
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barcha sonlarni

52
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bu ikki guruhga ajralishini anglab yetadi.

53
00:02:02,607 --> 00:02:04,896
Dastlab, u istalgan sonni

54
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kichik bo'linmas teng 
sonlar guruhlarigacha

55
00:02:07,078 --> 00:02:10,599
bo'lish mumkinligini anglab yetadi.

56
00:02:10,599 --> 00:02:12,921
Va bu eng kichik sonlar esa, har doim

57
00:02:12,921 --> 00:02:15,760
tub sonlardir.

58
00:02:15,760 --> 00:02:17,148
Shunday qilib, u barcha sonlar

59
00:02:17,148 --> 00:02:20,542
tub sonlardan qurilganini tushunib yetadi.

60
00:02:20,542 --> 00:02:23,317
Aniqrog'i, barcha sonlar olamini 
tasavvur qiling,

61
00:02:23,317 --> 00:02:25,674
tub sonlar haqida unuting.

62
00:02:25,674 --> 00:02:28,037
Endi istalgan murakkab sonni olamiz

63
00:02:28,037 --> 00:02:30,518
va bo'laklarga ajratamiz

64
00:02:30,518 --> 00:02:33,354
va bu bo'laklar, har doim 
tub sonlardir.

65
00:02:33,354 --> 00:02:34,774
Demak, Evklid istalgan raqam

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kichikroq tub sonlar guruhi orqali ifodalanishi 
mumkinligin tushunib yetgan.

67
00:02:37,675 --> 00:02:39,145
Ularni g'ishtlar deb tasavvur qiling.

68
00:02:40,221 --> 00:02:41,969
Qaysi son bo'lishidan qat'iy nazar,

69
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uni kichiroq tub sonlarni qo'shish 
bilan yasash mumkin.

70
00:02:46,157 --> 00:02:47,972
Mana shu Evklid kashfiyotining 
asosi bo'lib,

71
00:02:48,042 --> 00:02:50,759
"Arifmetikaning asosiy nazariyasi" 
deb nomlanadi.

72
00:02:50,759 --> 00:02:52,013
Unga ko'ra,

73
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istalgan raqamni, aytaylik, 30 ni olamiz

74
00:02:53,934 --> 00:02:55,501
va uning tub ko'paytuvchilarini topamiz.

75
00:02:55,501 --> 00:02:57,233
30 teng bo'linadi.

76
00:02:57,233 --> 00:02:59,763
Buni biz "ko'paytuvchilarga ajratish" 
deb ataymiz.

77
00:02:59,763 --> 00:03:01,624
Bu bizga tub ko'paytuvchilarni 
topish imkonini beradi.

78
00:03:01,624 --> 00:03:05,811
Bizning holatda 2,3 va 5
30 ning tub kupaytuvchilaridir.

79
00:03:05,811 --> 00:03:07,906
Evklid yana shuni tushunib yetdiki, 
sonning tub ko'paytuvchilarini

80
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bir necha bor ko'paytirish orqali

81
00:03:10,714 --> 00:03:12,739
dastlabki sonni keltirib chiqarish 
mumkin ekan.

82
00:03:12,739 --> 00:03:13,780
30 sonini yasash uchun esa 
uning tub ko'paytuvchilarini

83
00:03:13,780 --> 00:03:16,178
bir martadan ko'paytirish kifoya.

84
00:03:16,178 --> 00:03:20,158
2 x 3 x 5
30 soning tub kupaytuvchilaridir.

85
00:03:20,158 --> 00:03:23,153
Bularni o'ziga hos kalit yoki 
kombinatsiya deyish mumkin.

86
00:03:23,153 --> 00:03:24,887
30 sonini boshqa tub son guruhlari

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ko'paytmasi orqali yasashning

88
00:03:27,110 --> 00:03:28,792
imkoni yo'q.

89
00:03:28,792 --> 00:03:31,276
Shunday qilib, istalgan son 
faqat va faqat

90
00:03:31,276 --> 00:03:34,046
bitta yo'l bilan 
tub ko'paytuvchilarga ajraladi.

91
00:03:34,046 --> 00:03:36,299
Misol uchun, har bir sonni

92
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alohida qulf deb tasavvur qiling.

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Har bir qulfning (sonning) kaliti

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uning tub ko'paytuvchilari bo'ladi.

95
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Hech bir qulf bir xil kalitga ega emas.

96
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Hech bir son bir xil tub ko'paytuvchilardan 
tashkil topmaydi.