TEDGlobal 2013
Arthur Benjamin: The magic of Fibonacci numbers
Čarolija Fibonaccijevih brojeva
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Matematika je logična, funkcionalna i jednostavno... impresivna. Matemagičar Arthur Benjamin istražuje skrivena svojstva tog tajanstvenog i čudesnog niza brojeva - Fibonaccijevog niza. (I podsjeća nas koliko i matematika može biti nadahnjujuća!)
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 mi, zapravo, učimo matematiku?
00:15
Essentially, for three reasons:
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Tri su bitna razloga:
00:18
calculation,
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računanje,
00:19
application,
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primjena,
00:21
and last, and unfortunately least
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i posljednje, a nažalost i najmanje važno
00:24
in terms of the time we give it,
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u smislu vremena
koje joj posvećujemo,
koje joj posvećujemo,
00:26
inspiration.
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nadahnuće.
00:28
Mathematics is the science of patterns,
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Matematika je znanost o obrascima,
00:30
and we study it to learn how to think logically,
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i proučavamo je kako bismo
naučili misliti logički,
naučili misliti logički,
00:34
critically and creatively,
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kritički i stvaralački,
00:36
but too much of the mathematics
that we learn in school
that we learn in school
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ali suviše matematike
koju u školi učimo
koju u školi učimo
00:39
is not effectively motivated,
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nije pravilno motivirana,
00:41
and when our students ask,
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i kad nas naši učenici pitaju,
00:43
"Why are we learning this?"
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"Zašto ovo učimo?"
00:44
then they often hear that they'll need it
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često čuju da će im to trebati
00:46
in an upcoming math class or on a future test.
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na sljedećem satu matematike,
ili u nekom testu sljedećeg mjeseca.
ili u nekom testu sljedećeg mjeseca.
00:50
But wouldn't it be great
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Ali, ne bi li bilo sjajno
00:51
if every once in a while we did mathematics
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kad bismo se s vremena
na vrijeme matematikom bavili
na vrijeme matematikom bavili
00:54
simply because it was fun or beautiful
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jednostavno zato što je ona
zabavna, prelijepa
zabavna, prelijepa
00:57
or because it excited the mind?
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ili intelektualno uzbudljiva?
00:59
Now, I know many people have not
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Znam da mnogi ljudi nisu nikad imali
01:01
had the opportunity to see how this can happen,
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prigodu vidjeti kako bi to izgledalo,
01:03
so let me give you a quick example
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pa mi dopustite da vam dam
jednostavan primjer,
jednostavan primjer,
01:05
with my favorite collection of numbers,
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primjer mojeg omiljenog skupa brojeva,
01:07
the Fibonacci numbers. (Applause)
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Fibonaccijevih brojeva.
(Pljesak)
(Pljesak)
01:10
Yeah! I already have Fibonacci fans here.
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Odlično! I ovdje ima
ljubitelja Fibonaccijeviih brojeva.
ljubitelja Fibonaccijeviih brojeva.
01:12
That's great.
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To je odlično.
01:13
Now these numbers can be appreciated
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Vrijednost tih brojeva
moguće je cijeniti
moguće je cijeniti
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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Promotrimo li ih iz kuta računanja,
01:20
they're as easy to understand
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lako ih je razumjeti kao i
01:22
as one plus one, which is two.
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kao jedan plus jedan, što je dva..
01:24
Then one plus two is three,
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Potom, jedan plus dva je tri,
01:26
two plus three is five, three plus five is eight,
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dva plus tri je pet, tri plus pet je osam,
01:29
and so on.
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i tako dalje.
01:31
Indeed, the person we call Fibonacci
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Doista, osoba koju nazivamo Fibonacci
01:33
was actually named Leonardo of Pisa,
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zvao se, zapravo, Leonardo od Pise,
01:36
and these numbers appear in his book "Liber Abaci,"
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a ovi se brojevi pojavljuju u njegovoj knjizi "Liber Abaci",
01:39
which taught the Western world
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iz koje je Zapadni svijet naučio
01:41
the methods of arithmetic that we use today.
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aritmetičke metode koje danas koristimo.
01:44
In terms of applications,
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Što se primjene tiče,
01:45
Fibonacci numbers appear in nature
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Fibonaccijevi brojevi se u prirodi pojavljuju
01:48
surprisingly often.
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iznenađujuće često.
01:49
The number of petals on a flower
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Broj latica na cvijetu
01:51
is typically a Fibonacci number,
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obično je neki Fibonaccijev broj,
01:53
or the number of spirals on a sunflower
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ili broj spirala na suncokretovom cvijetu,
01:56
or a pineapple
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ili na ananasovom plodu
01:57
tends to be a Fibonacci number as well.
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također teži jednom od Fibonaccijevih brojeva.
02:00
In fact, there are many more
applications of Fibonacci numbers,
applications of Fibonacci numbers,
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Ustvari, u mnogo drugih slučajeva nalazimo Fibonaccijeve brojeve,
02:03
but what I find most inspirational about them
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ali ono što ja u njima smatram najviše nadahnjujućim
02:06
are the beautiful number patterns they display.
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jesu prelijepi brojevni obrasci koje prikazuju.
02:08
Let me show you one of my favorites.
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Pokazat ću vam jedan od svojih omiljenih.
02:11
Suppose you like to square numbers,
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Pretpostavimo da volite kvadrirati brojeve,
02:13
and frankly, who doesn't? (Laughter)
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i, iskreno, tko ne voli? (Smijeh)
02:16
Let's look at the squares
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Pogledajmo kvadrate
02:18
of the first few Fibonacci numbers.
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prvih nekoliko Fibonaccijevih brojeva.
02:20
So one squared is one,
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Dakle, jedan na kvadrat je jedan,
02:22
two squared is four, three squared is nine,
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dva na kvadrat je četiri, tri na kvadrat je devet,
02:24
five squared is 25, and so on.
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pet na kvadrat je dvadeset i pet, i tako dalje.
02:27
Now, it's no surprise
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Naravno, nije iznenađujuće
02:29
that when you add consecutive Fibonacci numbers,
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kad pribrajanjem uzastopnih Fibonaccijevih brojeva
02:32
you get the next Fibonacci number. Right?
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dobijemo sljedeći Fibonaccijev broj. Zar ne?
02:34
That's how they're created.
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Tako su i stvoreni.
02:35
But you wouldn't expect anything special
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Međutim, ne biste očekivali ništa osobito
02:37
to happen when you add the squares together.
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krenete li zbrajati kvadrate.
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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Jedan plus jedan daje dva,
02:44
and one plus four gives us five.
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a jedan plus četiri daje pet.
02:46
And four plus nine is 13,
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A četiri plus devet daju trinaest,
02:48
nine plus 25 is 34,
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a devet plus 25 je 34,
02:52
and yes, the pattern continues.
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i da, obrazac se nastavlja.
02:54
In fact, here's another one.
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Zapravo, evo vam još jednog.
02:56
Suppose you wanted to look at
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Pretpostavimo da ste poželjeli sagledati
02:58
adding the squares of
the first few Fibonacci numbers.
the first few Fibonacci numbers.
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zbrajanje kvadrata prvih nekoliko
Fibonaccijevih brojeva.
Fibonaccijevih brojeva.
03:00
Let's see what we get there.
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Pogledajmo što ćemo dobiti.
03:02
So one plus one plus four is six.
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Dakle jedan plus jedan plus četiri je šest.
03:04
Add nine to that, we get 15.
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Dodamo li tome devet, dobit ćemo 15.
03:07
Add 25, we get 40.
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Dodajmo 25 i dobivamo 40.
03:09
Add 64, we get 104.
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Dodajmo 64 i dobivamo 104.
03:12
Now look at those numbers.
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Razmotrimo te brojeve.
03:14
Those are not Fibonacci numbers,
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To nisu Fiboonaccijevi brojevi,
03:16
but if you look at them closely,
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ali promotrite li ih pažljivije,
03:18
you'll see the Fibonacci numbers
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uočit ćete Fibonaccijeve brojeve
03:20
buried inside of them.
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skrivene u njima.
03:22
Do you see it? I'll show it to you.
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Vidite li ih?
Pokazat ću vam.
Pokazat ću vam.
03:24
Six is two times three, 15 is three times five,
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Šest je dva puta tri,
a 15 je tri puta pet,
a 15 je tri puta pet,
03:28
40 is five times eight,
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40 je pet puta osam,
03:30
two, three, five, eight, who do we appreciate?
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dva, tri, pet, osam,
volite me takvog tko sam?
volite me takvog tko sam?
03:33
(Laughter)
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(Smijeh)
03:34
Fibonacci! Of course.
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Fibonacci!
Naravno.
Naravno.
03:36
Now, as much fun as it is to discover these patterns,
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Koliko god bilo zabavno otkrivati ovakve obrasce,
03:40
it's even more satisfying to understand
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još je više ispunjavajuće uvidjeti
03:42
why they are true.
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zašto je tome tako.
03:44
Let's look at that last equation.
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Pogledajmo posljednju jednadžbu.
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 kvadrati brojeva jedan, jedan, dva, tri, pet i osam
03:50
add up to eight times 13?
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u zbroju bili jednaki umnošku osam i 13?
03:53
I'll show you by drawing a simple picture.
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Objasnit ću vam ovim
jednostavnim prikazom.
jednostavnim prikazom.
03:56
We'll start with a one-by-one square
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Započnimo s kvadratom
dimenzija jedan puta jedan
dimenzija jedan puta jedan
03:58
and next to that put another one-by-one square.
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i do njega stavimo još jedan
kvadrat dimenzija jedan puta jedan.
kvadrat dimenzija jedan puta jedan.
04:02
Together, they form a one-by-two rectangle.
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Zajedno, oni čine
pravokutnik dimenzija jedan puta dva.
pravokutnik dimenzija jedan puta dva.
04:06
Beneath that, I'll put a two-by-two square,
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Ispod njih, nacrtat ću
kvadrat dimenzija dva puta dva,
kvadrat dimenzija dva puta dva,
04:08
and next to that, a three-by-three square,
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a do njih, kvadrat tri puta tri,.
04:11
beneath that, a five-by-five square,
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Ispod njih, kvadrat pet puta pet,
04:13
and then an eight-by-eight square,
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a potom kvadrat osam puta osam,
04:15
creating one giant rectangle, right?
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kreirajući tako jedan ogroman pravokutnik, zar ne?
04:18
Now let me ask you a simple question:
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Postavit ću vam jednostavno pitanje:
04:20
what is the area of the rectangle?
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Kolika je površina pravokutnika?
04:23
Well, on the one hand,
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S jedne strane,
04:25
it's the sum of the areas
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ona je suma površina
04:28
of the squares inside it, right?
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ucrtanih kvadrata, zar ne?
04:30
Just as we created it.
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Tako je pravokutnik i nastao.
04:31
It's one squared plus one squared
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Dakle, jedan na kvadrat plus jedan na kvadrat,
04:33
plus two squared plus three squared
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plus dva na kvadrat, plus tri na kvadrat,
04:35
plus five squared plus eight squared. Right?
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plus pet na kvadrat, plus osam na kvadrat.
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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S druge strane, budući da se radi o pravokutniku,
04:42
the area is equal to its height times its base,
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površina je jednaka umnošku
njegove visine i njegove baze,
njegove visine i njegove baze,
04:46
and the height is clearly eight,
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pri čemu je visina očito osam
04:48
and the base is five plus eight,
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a baza je pet plus osam,
04:51
which is the next Fibonacci number, 13. Right?
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što je sljedeći
Fibonaccijev broj, 13.Zar ne?
Fibonaccijev broj, 13.Zar ne?
04:55
So the area is also eight times 13.
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Prema tome, površina je osam puta 13.
04:58
Since we've correctly calculated the area
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Budući da smo ispravno 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 trebaju biti isti brojevi,
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 etto zašto kvadrati brojeva
jedan, jedan, dva, tri, pet i osam
jedan, jedan, dva, tri, pet i osam
05:08
add up to eight times 13.
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zbrojeni daju osam puta 13.
05:10
Now, if we continue this process,
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Nastavimo li ovaj postupak,
05:12
we'll generate rectangles of the form 13 by 21,
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stvorit ćemo pravokutnike
oblika 13 puta 21,
oblika 13 puta 21,
05:16
21 by 34, and so on.
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21 puta 34, i tako dalje.
05:19
Now check this out.
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A razmotrimo ovo.
05:20
If you divide 13 by eight,
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Podijelimo li 13 sa osam,
05:22
you get 1.625.
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dobit ćemo 1,625.
05:24
And if you divide the larger number
by the smaller number,
by the smaller number,
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I dijelimo li veći broj
s manjim brojem,
s manjim brojem,
05:28
then these ratios get closer and closer
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primijetit ćemo da se
količnici sve više približavaju
količnici sve više približavaju
05:31
to about 1.618,
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broju 1,618,
05:33
known to many people as the Golden Ratio,
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mnogim ljudima znanom
kao Zlatni omjer,
kao Zlatni omjer,
05:37
a number which has fascinated mathematicians,
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broj koji je stoljećima očaravao matematičare,
05:39
scientists and artists for centuries.
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znanstvenike i umjetnike stoljećima.
05:42
Now, I show all this to you because,
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Sve vam ovo pokazujem zato što,
05:45
like so much of mathematics,
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kao toliko toga u matematici,
05:47
there's a beautiful side to it
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ovo posjeduje osobitu ljepotu kojoj,
05:49
that I fear does not get enough attention
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bojim se, ne poklanjamo dovoljno pozornosti
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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Mnogo vremena provodimo učeći o računanju,
05:55
but let's not forget about application,
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ali ne zaboravimo na primjenu,
05:58
including, perhaps, the most
important application of all,
important application of all,
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uključujući, možda, i najvažniju
od svih mogućih primjena,
od svih mogućih primjena,
06:01
learning how to think.
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učiti kako misliti.
06:03
If I could summarize this in one sentence,
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Kad bih ovo mogao sažeti u jednoj rečenici,
06:05
it would be this:
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bila bi to ova:
06:07
Mathematics is not just solving for x,
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Matematika ne služi samo za rješavanje x-a,
06:10
it's also figuring out why.
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već i razotkrivanje onoga zašto.
06:13
Thank you very much.
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Hvala vam puno.
06:15
(Applause)
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(Pljesak)
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