The infinite life of pi - Reynaldo Lopes

1,604,875 views ・ 2013-07-10

TED-Ed


Dvaput kliknite na engleske titlove ispod za reprodukciju videozapisa.

Prevoditelj: Irma Komljenović Recezent: Ivan Stamenković
00:13
Try to measure a circle.
0
13823
1710
Pokušaj izmjeriti krug.
00:15
The diameter and radius are easy,
1
15557
1881
Promjer i polumjer je lagano izmjeriti,
00:17
they're just straight lines
2
17462
970
oni su samo ravne crte
00:18
you can measure with a ruler.
3
18456
1685
koje možeš mjeriti ravnalom.
00:20
But to get the circumference,
4
20165
1517
No da bi dobio opseg,
00:21
you'd need measuring tape or a piece of string,
5
21706
2562
trebale bi ti traka za mjerenje ili djelić strune,
00:24
unless there was a better way.
6
24292
2225
osim ako ne postoji bolji način.
00:26
Now, it's obvious
7
26541
1143
Pa, očito je
00:27
that a circle's circumference would get smaller or larger
8
27708
2912
da bi se opseg kruga smanjivao ili povećavao
00:30
along with its diameter,
9
30644
1593
s obzirom na promjer,
00:32
but the relationship goes further than that.
10
32261
2096
no njihov odnos je dublji od toga.
00:34
In fact, the ratio between the two,
11
34381
2537
Zapravo, omjer jednog i drugog,
00:36
the circumference divided by the diameter,
12
36942
2270
opseg podijeljen s promjerom,
00:39
will always be the same number,
13
39236
1805
uvijek će biti isti broj,
00:41
no matter how big or small the circle gets.
14
41065
3224
bez obzira na to koliko je krug velik ili malen.
00:44
Historians aren't sure when or how
15
44313
2275
Povjesničari nisu sigurni kada ili kako
00:46
this number was first discovered,
16
46612
1688
je taj broj prvi put otkriven,
00:48
but it's been known in some form
17
48324
1807
no poznat je u nekom obliku
00:50
for almost 4,000 years.
18
50155
2927
već skoro 4,000 godina.
00:53
Estimates of it appear in the works of ancient Greek,
19
53106
2524
Njegove procjene pojavljuju se u radovima drevnih grčkih,
00:55
Babylonian,
20
55654
1010
babilonskih,
00:56
Chinese,
21
56688
729
kineskih
00:57
and Indian mathematicians.
22
57441
1943
i indijskih matematičara.
00:59
And it's even believed to have been used
23
59408
1905
Čak se vjeruje da ga se koristilo
01:01
in building the Egyptian pyramids.
24
61337
2003
u izgradnji egipatskih piramida.
01:03
Mathematicians estimated it
25
63364
1557
Matematičari su ga procjenjivali
01:04
by inscribing polygons in circles.
26
64945
2554
upisivanjem mnogokuta u krugove.
01:07
And by the year 1400,
27
67523
1624
Do 1400. godine,
01:09
it had been calculated to as far as ten decimal places.
28
69171
4204
izračunato je 10 njegovih decimala.
01:13
So, when did they finally figure out the exact value
29
73399
2522
Pa, kada su napokon točno odredili njegovu vrijednost,
01:15
instead of just estimating?
30
75945
1806
umjesto procjenjivanja?
01:17
Actually, never!
31
77775
2235
Zapravo, nikad!
01:20
You see, the ratio
32
80034
1568
Vidiš, omjer
01:21
of a circle's circumference to its diameter
33
81626
2486
opsega kruga i njegova promjera
01:24
is what's known as an irrational number,
34
84136
2737
iracionalan je broj,
01:26
one that can never be expressed
35
86897
1477
što znači da ga se ne može izraziti
01:28
as a ratio of two whole numbers.
36
88398
3149
kao omjer dvaju cijelih brojeva.
01:31
You can come close,
37
91571
1517
Možeš biti blizu,
01:33
but no matter how precise the fraction is,
38
93112
2504
no bez obzira na preciznost razlomka,
01:35
it will always be just a tiny bit off.
39
95640
2725
uvijek će biti djelomično netočan.
01:38
So, to write it out in its decimal form,
40
98389
2289
Da bi ga napisao u decimalnom obliku,
01:40
you'd have an on-going series of digits
41
100702
1977
imao bi neprekidan niz znamenki
01:42
starting with
42
102703
957
počevši sa
01:43
3.14159
43
103684
2917
3.14159
01:46
and continuing
44
106625
1068
i nastavljajući se
01:47
forever!
45
107717
1704
zauvijek!
01:49
That's why, instead of trying to write out
46
109445
2000
Zato ga, umjesto da svaki put ispisujemo
01:51
an infinite number of digits every time,
47
111469
2259
beskonačan broj znamenki,
01:53
we just refer to it using the Greek letter pi.
48
113752
3917
jednostavno zovemo grčkim slovom pi.
01:57
Nowadays, we test the speed of computers
49
117693
1905
Danas testiramo brzinu računala
01:59
by having them calculate pi,
50
119622
1993
dajući im da računaju pi,
02:01
and quantum computers have been able
51
121639
1715
a kvantna računala uspijevaju
02:03
to calculate it up to two quadrillion digits.
52
123378
3449
izračunati i dvije bilijarde znamenki.
02:06
People even compete to see
53
126851
1239
Ljudi se čak natječu
02:08
how many digits they can memorize
54
128114
1681
u učenju znamenki napamet,
02:09
and have set records for remembering
55
129819
1807
postavivši rekorde za
02:11
over 67,000 of them.
56
131650
3480
više od 67,000 zapamćenih znamenki.
02:15
But for most scientific uses,
57
135154
1745
No za većinu znanstvenih upotreba
02:16
you only need the first forty or so.
58
136923
2796
trebaš prvih četrdesetak.
02:19
And what are these scientific uses?
59
139743
2184
Koje su to znanstvene upotrebe?
02:21
Well, just about any calculations involving circles,
60
141951
3322
Pa svaka u kojoj ima izračuna koji uključuju krugove,
02:25
from the volume of a can of soda
61
145297
1780
od volumena limenke soka
02:27
to the orbits of satellites.
62
147101
1927
do orbite satelita.
02:29
And it's not just circles, either.
63
149052
1800
Ne radi se samo o krugovima.
02:30
Because it's also useful in studying curves,
64
150876
2368
Zbog svoje primjene u proučavanju krivulja,
02:33
pi helps us understand periodic or oscillating systems
65
153268
3555
pi nam pomaže shvatiti periodične ili oscilirajuće sustave
02:36
like clocks,
66
156847
849
kao što su satovi,
02:37
electromagnetic waves,
67
157720
1522
elektromagnetski valova,
02:39
and even music.
68
159266
1684
čak i glazba.
02:40
In statistics, pi is used in the equation
69
160974
2225
U statistici pi se koristi u jednadžbi
02:43
to calculate the area under a normal distribution curve,
70
163223
3354
za računanje površine ispod krivulje normalne raspodjele,
02:46
which comes in handy for figuring out distributions
71
166601
2103
što pomaže u određivanju raspodjela
02:48
of standardized test scores,
72
168728
1494
u rezultatima standardiziranih ispita,
02:50
financial models,
73
170246
1136
financijskim modelima
02:51
or margins of error in scientific results.
74
171406
3016
ili dopuštenoj pogreški u znanstvenim rezultatima.
02:54
As if that weren't enough,
75
174446
1305
Kao da i to nije dovoljno,
02:55
pi is used in particle physics experiments,
76
175775
2978
pi se koristi u pokusima fizike elementarnih čestica,
02:58
such as those using the Large Hadron Collider,
77
178777
2658
poput onih u kojima se koristi veliki hadronski sudarač,
03:01
not only due to its round shape,
78
181459
1748
ne samo zbog njegova okrugla oblika,
03:03
but more subtly,
79
183231
1064
nego i suptilnije,
03:04
because of the orbits in which tiny particles move.
80
184319
3246
zbog orbita u kojima se sitne čestice kreću.
03:07
Scientists have even used pi
81
187589
2136
Znanstvenici su čak iskoristili pi
03:09
to prove the illusive notion
82
189749
1483
da dokažu varljivu ideju
03:11
that light functions as both a particle
83
191256
2611
da svjetlost funkcionira kao čestica
03:13
and an electromagnetic wave,
84
193891
2310
i elektormagnetski val
03:16
and, perhaps most impressively,
85
196225
1891
te, možda najimpresivnije,
03:18
to calculate the density of our entire universe,
86
198140
3506
da izračunaju gustoću cijelog našeg svemira,
03:21
which, by the way,
87
201670
1560
koji, usput,
03:23
still has infinitely less stuff in it
88
203254
3159
ima beskonačno mnogo manje stvari u sebi
03:26
than the total number of digits in pi.
89
206437
3500
nego ukupan broj znamenki pi.
O ovoj web stranici

Ova stranica će vas upoznati s YouTube videozapisima koji su korisni za učenje engleskog jezika. Vidjet ćete lekcije engleskog koje vode vrhunski profesori iz cijelog svijeta. Dvaput kliknite na engleske titlove prikazane na svakoj video stranici da biste reproducirali video s tog mjesta. Titlovi se pomiču sinkronizirano s reprodukcijom videozapisa. Ako imate bilo kakvih komentara ili zahtjeva, obratite nam se putem ovog obrasca za kontakt.

https://forms.gle/WvT1wiN1qDtmnspy7