TEDGlobal 2013
Arthur Benjamin: The magic of Fibonacci numbers
Artur Bendžamin (Arthur Benjamin): Magija Fibonačijevih brojeva
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Matematika je logična, funkcionalna i jednostavno... sjajna. Matemagičar Artur Bendžamin istražuje skrivena svojstva čudnog i predivnog skupa brojeva, Fibonačijevog niza. (I podseća vas da matematika može i da nadahnjuje!)
Arthur Benjamin - Mathemagician
Using daring displays of algorithmic trickery, lightning calculator and number wizard Arthur Benjamin mesmerizes audiences with mathematical mystery and beauty. Full bio
Using daring displays of algorithmic trickery, lightning calculator and number wizard Arthur Benjamin mesmerizes audiences with mathematical mystery and beauty. Full bio
Double-click the English transcript below to play the video.
00:12
So why do we learn mathematics?
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Zašto učimo matematiku?
00:15
Essentially, for three reasons:
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U suštini, iz tri razloga:
00:18
calculation,
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računanje,
00:19
application,
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primena
00:21
and last, and unfortunately least
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i poslednje i nažalost najmanje bitno,
00:24
in terms of the time we give it,
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što se tiče vremena
koje mu posvećujemo,
koje mu posvećujemo,
00:26
inspiration.
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nadahnuće.
00:28
Mathematics is the science of patterns,
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Matematika je nauka o šablonima
00:30
and we study it to learn how to think logically,
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i proučavamo je kako bismo saznali
kako da mislimo logički,
kako da mislimo logički,
00:34
critically and creatively,
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kritički i kreativno,
00:36
but too much of the mathematics
that we learn in school
that we learn in school
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ali previše matematike
koju učimo u školama
koju učimo u školama
00:39
is not effectively motivated,
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nije valjano motivisano
00:41
and when our students ask,
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i kada naši učenici pitaju:
00:43
"Why are we learning this?"
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"Zašto učimo ovo?",
00:44
then they often hear that they'll need it
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često čuju da će im to
biti potrebno
biti potrebno
00:46
in an upcoming math class or on a future test.
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na nekom budućem testu
ili času matematike.
ili času matematike.
00:50
But wouldn't it be great
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Ali zar ne bi bilo sjajno
00:51
if every once in a while we did mathematics
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kad bismo se, s vremena na vreme,
bavili matematikom
bavili matematikom
00:54
simply because it was fun or beautiful
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jednostavno jer je zabavna ili predivna
00:57
or because it excited the mind?
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ili zato što je uzbudljiva za um?
00:59
Now, I know many people have not
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Znam da dosta ljudi nije imalo
01:01
had the opportunity to see how this can happen,
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priliku da vidi kako ovo
može da se desi,
može da se desi,
01:03
so let me give you a quick example
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zato hajde da vam dam brz primer
01:05
with my favorite collection of numbers,
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sa mojim omiljenim skupom brojeva,
01:07
the Fibonacci numbers. (Applause)
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Fibonačijevim brojevima.
(Aplauz)
(Aplauz)
01:10
Yeah! I already have Fibonacci fans here.
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Da! Ovde već imam
Fibonačijeve fanove.
Fibonačijeve fanove.
01:12
That's great.
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To je sjajno.
01:13
Now these numbers can be appreciated
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Ove brojeve možete razumeti
01:15
in many different ways.
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na mnogo različitih načina.
01:17
From the standpoint of calculation,
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Sa stanovišta računanja,
01:20
they're as easy to understand
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jednako ih je lako razumeti
01:22
as one plus one, which is two.
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kao 1 + 1, što je 2.
01:24
Then one plus two is three,
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1 + 2 je onda 3,
01:26
two plus three is five, three plus five is eight,
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2 + 3 je 5,
3 + 5 je 8,
3 + 5 je 8,
01:29
and so on.
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i tako dalje.
01:31
Indeed, the person we call Fibonacci
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Zaista, osoba koju nazivamo Fibonači
01:33
was actually named Leonardo of Pisa,
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se zapravo zvala Leonardo od Pize
01:36
and these numbers appear in his book "Liber Abaci,"
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i ovi brojevi se pojavljuju
u njegovoj knjizi "Liber Abaci",
u njegovoj knjizi "Liber Abaci",
01:39
which taught the Western world
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koja je Zapadni svet naučila
01:41
the methods of arithmetic that we use today.
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aritmetičkim metodama
koje danas koristimo.
koje danas koristimo.
01:44
In terms of applications,
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Što se tiče primene,
01:45
Fibonacci numbers appear in nature
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Fibonačijevi brojevi se u prirodi
01:48
surprisingly often.
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pojavljuju iznenađujuće često.
01:49
The number of petals on a flower
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Broj latica na cvetu
01:51
is typically a Fibonacci number,
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je tipično Fibonačijev broj,
01:53
or the number of spirals on a sunflower
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ili broj spirala na suncokretu
01:56
or a pineapple
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ili ananasu,
01:57
tends to be a Fibonacci number as well.
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to su takođe uglavnom
Fibonačijevi brojevi.
Fibonačijevi brojevi.
02:00
In fact, there are many more
applications of Fibonacci numbers,
applications of Fibonacci numbers,
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Zapravo postoji još dosta
primena Fibonačijevih brojeva,
primena Fibonačijevih brojeva,
02:03
but what I find most inspirational about them
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ali ono što je za mene najinspirativnije
u vezi sa njima
u vezi sa njima
02:06
are the beautiful number patterns they display.
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su predivni šabloni brojeva
koje oni prikazuju.
koje oni prikazuju.
02:08
Let me show you one of my favorites.
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Dozvolite da vam pokažem
jedan od meni omiljenih.
jedan od meni omiljenih.
02:11
Suppose you like to square numbers,
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Recimo da volite da kvadrirate brojeve,
02:13
and frankly, who doesn't? (Laughter)
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a iskreno, ko to ne voli?
(Smeh)
(Smeh)
02:16
Let's look at the squares
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Hajde da pogledamo kvadrate
02:18
of the first few Fibonacci numbers.
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prvih nekoliko Fibonačijevih brojeva.
02:20
So one squared is one,
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1 na kvadrat je 1,
02:22
two squared is four, three squared is nine,
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2 na kvadrat je 4.
3 na kvadrat je 9,
3 na kvadrat je 9,
02:24
five squared is 25, and so on.
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5 na kvadrat je 25,
i tako dalje.
i tako dalje.
02:27
Now, it's no surprise
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Ne iznenađuje činjenica
02:29
that when you add consecutive Fibonacci numbers,
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da kada dodate uzastopne
Fibonačijeve brojeve,
Fibonačijeve brojeve,
02:32
you get the next Fibonacci number. Right?
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dobijate sledeći Fibonačijev broj.
Zar ne?
Zar ne?
02:34
That's how they're created.
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Tako oni nastaju.
02:35
But you wouldn't expect anything special
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Ali ne biste očekivali da se desi
ništa posebno
ništa posebno
02:37
to happen when you add the squares together.
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kada saberete kvadratne vrednosti.
02:40
But check this out.
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Ali pogledajte ovo.
02:42
One plus one gives us two,
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1 + 1 nam daje 2,
02:44
and one plus four gives us five.
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i 1 + 4 daje 5.
02:46
And four plus nine is 13,
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4 + 9 je 13,
02:48
nine plus 25 is 34,
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9 + 25 je 34,
02:52
and yes, the pattern continues.
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i da, šablon se nastavlja.
02:54
In fact, here's another one.
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Zapravo, evo još jednog.
02:56
Suppose you wanted to look at
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Recimo da želite da pogledate
02:58
adding the squares of
the first few Fibonacci numbers.
the first few Fibonacci numbers.
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sabiranje kvadratnih vrednosti
prvih nekoliko Fibonačijevih brojeva.
prvih nekoliko Fibonačijevih brojeva.
03:00
Let's see what we get there.
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Da vidimo šta tu dobijamo.
03:02
So one plus one plus four is six.
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1 + 1 + 4 je 6.
03:04
Add nine to that, we get 15.
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Tome dodajte 9, to je 15.
03:07
Add 25, we get 40.
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Dodajte 25, to je 40.
03:09
Add 64, we get 104.
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Dodajte 64 i to je 104.
03:12
Now look at those numbers.
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Sada pogledajte te brojke.
03:14
Those are not Fibonacci numbers,
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To nisu Fibonačijevi brojevi,
03:16
but if you look at them closely,
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ali ako ih pogledate pažljivo,
03:18
you'll see the Fibonacci numbers
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videćete Fibonačijeve brojeve
03:20
buried inside of them.
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sakrivene unutar njih.
03:22
Do you see it? I'll show it to you.
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Vidite li ih? Pokazaću vam.
03:24
Six is two times three, 15 is three times five,
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6 je 2 puta 3, 15 je 3 puta 5,
03:28
40 is five times eight,
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40 je 5 puta 8,
03:30
two, three, five, eight, who do we appreciate?
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2, 3, 5, 8.
Kome odajemo priznanje?
Kome odajemo priznanje?
03:33
(Laughter)
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(Smeh)
03:34
Fibonacci! Of course.
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Fibonačiju! Naravno.
03:36
Now, as much fun as it is to discover these patterns,
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Koliko god da je zabavno
otkrivati ove šablone,
otkrivati ove šablone,
03:40
it's even more satisfying to understand
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još je veće zadovoljstvo razumeti
03:42
why they are true.
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zašto su tačni.
03:44
Let's look at that last equation.
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Hajde da pogledamo
poslednju jednačinu.
poslednju jednačinu.
03:46
Why should the squares of one, one,
two, three, five and eight
two, three, five and eight
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Zašto bi kvadratne vrednosti
brojeva 1, 1, 2, 3, 5 i 8
brojeva 1, 1, 2, 3, 5 i 8
03:50
add up to eight times 13?
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sabrane, dale 8 puta 13?
03:53
I'll show you by drawing a simple picture.
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Pokazaću vam uz pomoć
jednostavnog crteža.
jednostavnog crteža.
03:56
We'll start with a one-by-one square
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Počećemo sa kvadratom
dimenzija 1x1
dimenzija 1x1
03:58
and next to that put another one-by-one square.
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i pored ćemo dodati
još jedan kvadrat dimenzija 1x1.
još jedan kvadrat dimenzija 1x1.
04:02
Together, they form a one-by-two rectangle.
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Zajedno daju pravougaonik
dimenzija 1x2.
dimenzija 1x2.
04:06
Beneath that, I'll put a two-by-two square,
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Ispod toga, dodaću kvadrat
dimenzija 2x2,
dimenzija 2x2,
04:08
and next to that, a three-by-three square,
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a pored toga,
kvadrat dimenzija 3x3,
kvadrat dimenzija 3x3,
04:11
beneath that, a five-by-five square,
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ispod toga, kvadrat dimenzija 5x5
04:13
and then an eight-by-eight square,
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i onda kvadrat dimenzija 8x8,
04:15
creating one giant rectangle, right?
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stvarajući jedan
ogromni pravougaonik, zar ne?
ogromni pravougaonik, zar ne?
04:18
Now let me ask you a simple question:
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Dozvolite da vam postavim
jednostavno pitanje:
jednostavno pitanje:
04:20
what is the area of the rectangle?
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koja je površina pravouganika?
04:23
Well, on the one hand,
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Sa jedne strane,
04:25
it's the sum of the areas
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to je zbir površina
04:28
of the squares inside it, right?
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kvadrata unutar njega, zar ne?
04:30
Just as we created it.
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Baš kao što smo ga napravili.
04:31
It's one squared plus one squared
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To je 1 na kvadrat + 1 na kvadrat
04:33
plus two squared plus three squared
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+ 2 na kvadrat + 3 na kvadrat
04:35
plus five squared plus eight squared. Right?
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+ 5 na kvadrat + 8 na kvadrat.
Zar ne?
Zar ne?
04:38
That's the area.
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To je površina.
04:40
On the other hand, because it's a rectangle,
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Sa druge strane, zbog toga
što je to pravougonik,
što je to pravougonik,
04:42
the area is equal to its height times its base,
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površinu dobijemo kada pomnožimo
visinu i osnovu,
visinu i osnovu,
04:46
and the height is clearly eight,
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a visina je očigledno 8
04:48
and the base is five plus eight,
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dok je osnova 5 + 8,
04:51
which is the next Fibonacci number, 13. Right?
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što je sledeći Fibonačijev broj, 13.
Zar ne?
Zar ne?
04:55
So the area is also eight times 13.
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Površina je takođe 8 puta 13.
04:58
Since we've correctly calculated the area
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Pošto smo tačno izračunali površinu
05:00
two different ways,
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na dva različita načina,
05:02
they have to be the same number,
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to mora da bude isti broj
05:04
and that's why the squares of one,
one, two, three, five and eight
one, two, three, five and eight
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i zbog toga kvadradne vrednosti
brojeva 1, 1, 2, 3, 5 i 8
brojeva 1, 1, 2, 3, 5 i 8
05:08
add up to eight times 13.
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sabrane daju 8 puta 13.
05:10
Now, if we continue this process,
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Ako nastavimo ovaj proces
05:12
we'll generate rectangles of the form 13 by 21,
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dobićemo pravouganike formata 13x21,
05:16
21 by 34, and so on.
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21x34 i tako dalje.
05:19
Now check this out.
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Pogledajte sada ovo.
05:20
If you divide 13 by eight,
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Ako podelite 13 sa 8
05:22
you get 1.625.
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dobijate 1,625.
05:24
And if you divide the larger number
by the smaller number,
by the smaller number,
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A ako veći broj podelite manjim brojem,
05:28
then these ratios get closer and closer
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ove srazmere se sve više približavaju
05:31
to about 1.618,
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vrednosti oko 1,618,
05:33
known to many people as the Golden Ratio,
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što je mnogima poznato
kao Zlatni presek,
kao Zlatni presek,
05:37
a number which has fascinated mathematicians,
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broj koji vekovima fascinira
05:39
scientists and artists for centuries.
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matematičare, naučnike i umetnike.
05:42
Now, I show all this to you because,
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Ovo sve vam pokazujem zato što,
05:45
like so much of mathematics,
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baš kao u dobrom delu matematike,
05:47
there's a beautiful side to it
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postoji prelepa strana toga
05:49
that I fear does not get enough attention
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za koju se bojim
da ne dobija dovoljno pažnje
da ne dobija dovoljno pažnje
05:51
in our schools.
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u našim školama.
05:52
We spend lots of time learning about calculation,
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Puno vremena provodimo
učeći o računanju,
učeći o računanju,
05:55
but let's not forget about application,
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ali ne zaboravimo na primenu,
05:58
including, perhaps, the most
important application of all,
important application of all,
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uključujući možda i
najbitniju primenu od svih,
najbitniju primenu od svih,
06:01
learning how to think.
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učenje kako se misli.
06:03
If I could summarize this in one sentence,
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Kada bih ovo mogao da sažmem
u jednu rečenicu,
u jednu rečenicu,
06:05
it would be this:
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to bi bilo sledeće:
06:07
Mathematics is not just solving for x,
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matematika ne znači samo
pronaći vrednost x,
pronaći vrednost x,
06:10
it's also figuring out why.
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već takođe i otkriti zašto.
06:13
Thank you very much.
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Hvala vam mnogo.
06:15
(Applause)
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(Aplauz)
ABOUT THE SPEAKER
Arthur Benjamin - MathemagicianUsing daring displays of algorithmic trickery, lightning calculator and number wizard Arthur Benjamin mesmerizes audiences with mathematical mystery and beauty.
Why you should listen
Arthur Benjamin makes numbers dance. In his day job, he's a professor of math at Harvey Mudd College; in his other day job, he's a "Mathemagician," taking the stage in his tuxedo to perform high-speed mental calculations, memorizations and other astounding math stunts. It's part of his drive to teach math and mental agility in interesting ways, following in the footsteps of such heroes as Martin Gardner.
Benjamin is the co-author, with Michael Shermer, of Secrets of Mental Math (which shares his secrets for rapid mental calculation), as well as the co-author of the MAA award-winning Proofs That Really Count: The Art of Combinatorial Proof. For a glimpse of his broad approach to math, see the list of research talks on his website, which seesaws between high-level math (such as his "Vandermonde's Determinant and Fibonacci SAWs," presented at MIT in 2004) and engaging math talks for the rest of us ("An Amazing Mathematical Card Trick").
Arthur Benjamin | Speaker | TED.com