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Tasavvur qiling, milodan avvalgi yillar.

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O'ylab ko'ring:

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Qanday qilib soatsiz, vaqtni aniqlashgan?

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Barcha soatlar, vaqt oqimini 
teng bo'laklarga bo'luvchi

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qandaydir takroriy shaklga asoslangan.

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Bunday takroriy shakillarni topish uchun

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samolarga yuzlanamiz.

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Har kuni, quyoshning chiqishi va botishi

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bunday shakllarning eng oddiysidir.

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Lekin, uzoqroq vaqt bo'lagini kuzatish uchun,

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uzoqroq takrorlanishlarga e'tibor beramiz.

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Buning uchun esa, oyga yuzlanamiz.

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E'tibor bergan bo'lsangiz,

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oy kunlar osha to'lishadi va kichrayadi.

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To'lin oylar orasidagi kunlar sonini

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sanaydigan bo'lsak,

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u 29 kunga teng.

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

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

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bir muammoga duch kelamiz: 
buning iloji yo'q.

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29 ni teng bo'laklarga bo'lishning 
yagona yo'li

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uni 29 ta teng bo'lakka bo'lishdan iborat.

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29 soni tub son hisoblanadi.

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Uni bo'linmas deb tasavvur qiling.

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Agar son birdan boshqa

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teng bo'laklarga bo'linsa,

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biz uni 'murakkab son' deb ataymiz.

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Endi biz qiziqishimiz mumkin,

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"Dunyoda nechta tub son bo'lishi mumkin -

30
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va ularning eng kattasi 
nechaga teng ekan?"

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Keling, barcha sonlarni 
ikkita guruhga bo'lamiz.

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Tub sonlar chap tomonda

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va murakkab sonlar o'ng tomonda.

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Boshida, u tomondan bu tomonga raqs 
tushayotganga o'xshaydilar.

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Ammo, ularning joylashuvida 
aniq bir shakl mavjud emas.

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Keling, bunday shaklni ko'rish uchun

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zamonaviy usuldan foydalanamiz.

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Bu usul "Ulam spirali" deb nomlanadi.

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Boshida, barcha raqamlarni tartib bilan,
o'sayotgan spiral

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ichiga joylab chiqamiz.

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Keyin, barcha tub sonlarni 
ko'k ranga bo'yab chiqamiz.

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Nihoyat, biz millionlab raqamlarni
ko'rish uchun uzoqlashamiz.

43
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Mana bu tugalmas tub sonlarning

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shakli hisoblanadi.

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Hayratlanarlisi shuki, bu shaklning 
tuliq strukturasi

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haligacha topilmagan.

47
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Nimanidir kashf etish arafasida
turganga o'xshaymiz.

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Keling, m.a. 300 yillarga

49
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Qadimgi Gretsiyaga sayr qilamiz.

50
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Buyuk faylasuf Aleksandryalik Evklid

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barcha sonlarni

52
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bu ikki guruhga ajralishini anglab yetadi.

53
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Dastlab, u istalgan sonni

54
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eng kichik teng sonlar guruhlarigacha

55
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bo'lish mumkinligini anglab yetadi.

56
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Va bu eng kichik sonlar esa, har doim,

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tub sonlardir.

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Shunday qilib u, barcha sonlar

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tub sonlardan qurilganini tushunib yetadi.

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Aniqrog'i, barcha sonlar olamini 
tasavvur qiling.

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tub sonlar haqida unuting.

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Endi, istalgan murakkab sonni olamiz,

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va bo'laklarga ajratamiz,

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va bu bo'laklar har doim 
tub sonlardir.

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Demak, Evklid istalgan raqam

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kichikroq tub sonlar guruhi orqali ifodalanishi 
mumkinligin tushunib yetgan.

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Ularni g'ishtlar deb tasavvur qiling.

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Qaysi son bo'lishidan qat'iy nazar

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uni kichiroq tub sonlarni qo'shish 
bilan yasash mumkin.

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Mana shu Evklid kashfiyotining 
asosi bo'lib,

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"Arifmetikaning asosiy nazariyasi" 
deb nomlanadi.

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