ABOUT THE SPEAKER
Arthur Benjamin - Mathemagician
Using 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").

More profile about the speaker
Arthur Benjamin | Speaker | TED.com
TEDGlobal 2013

Arthur Benjamin: The magic of Fibonacci numbers

Čarolija Fibonaccijevih brojeva

Filmed:
7,057,274 views

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!)
- Mathemagician
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 learnnaučiti mathematicsmatematika?
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Zašto mi, zapravo, učimo matematiku?
00:15
EssentiallyU suštini, for threetri reasonsrazlozi:
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Tri su bitna razloga:
00:18
calculationračunanje,
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računanje,
00:19
applicationprimjena,
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primjena,
00:21
and last, and unfortunatelynažalost leastnajmanje
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i posljednje, a nažalost i najmanje važno
00:24
in termsUvjeti of the time we give it,
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u smislu vremena
koje joj posvećujemo,
00:26
inspirationinspiracija.
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nadahnuće.
00:28
MathematicsMatematika is the scienceznanost of patternsobrasci,
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Matematika je znanost o obrascima,
00:30
and we studystudija it to learnnaučiti how to think logicallylogički,
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i proučavamo je kako bismo
naučili misliti logički,
00:34
criticallykritički and creativelykreativno,
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kritički i stvaralački,
00:36
but too much of the mathematicsmatematika
that we learnnaučiti in schoolškola
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ali suviše matematike
koju u školi učimo
00:39
is not effectivelyučinkovito motivatedmotivirani,
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nije pravilno motivirana,
00:41
and when our studentsstudenti askpitati,
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i kad nas naši učenici pitaju,
00:43
"Why are we learningučenje this?"
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"Zašto ovo učimo?"
00:44
then they oftenčesto hearčuti that they'lloni će need it
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često čuju da će im to trebati
00:46
in an upcomingnadolazeće mathmatematika classklasa or on a futurebudućnost testtest.
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na sljedećem satu matematike,
ili u nekom testu sljedećeg mjeseca.
00:50
But wouldn'tne bi it be great
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Ali, ne bi li bilo sjajno
00:51
if everysvaki oncejednom in a while we did mathematicsmatematika
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kad bismo se s vremena
na vrijeme matematikom bavili
00:54
simplyjednostavno because it was funzabava or beautifullijep
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jednostavno zato što je ona
zabavna, prelijepa
00:57
or because it exciteduzbuđen the mindum?
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ili intelektualno uzbudljiva?
00:59
Now, I know manymnogi people have not
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Znam da mnogi ljudi nisu nikad imali
01:01
had the opportunityprilika to see how this can happendogoditi se,
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prigodu vidjeti kako bi to izgledalo,
01:03
so let me give you a quickbrz exampleprimjer
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pa mi dopustite da vam dam
jednostavan primjer,
01:05
with my favoriteljubimac collectionkolekcija of numbersbrojevi,
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primjer mojeg omiljenog skupa brojeva,
01:07
the FibonacciFibonacci numbersbrojevi. (ApplausePljesak)
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Fibonaccijevih brojeva.
(Pljesak)
01:10
Yeah! I alreadyveć have FibonacciFibonacci fansfanovi here.
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Odlično! I ovdje ima
ljubitelja Fibonaccijeviih brojeva.
01:12
That's great.
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To je odlično.
01:13
Now these numbersbrojevi can be appreciatedpoštovati
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Vrijednost tih brojeva
moguće je cijeniti
01:15
in manymnogi differentdrugačiji waysnačine.
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na mnogo različitih načina.
01:17
From the standpointstanovište of calculationračunanje,
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Promotrimo li ih iz kuta računanja,
01:20
they're as easylako to understandrazumjeti
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lako ih je razumjeti kao i
01:22
as one plusplus one, whichkoji is two.
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kao jedan plus jedan, što je dva..
01:24
Then one plusplus two is threetri,
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Potom, jedan plus dva je tri,
01:26
two plusplus threetri is fivepet, threetri plusplus fivepet is eightosam,
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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
IndeedDoista, the personosoba we call FibonacciFibonacci
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Doista, osoba koju nazivamo Fibonacci
01:33
was actuallyzapravo namedpod nazivom LeonardoLeonardo of PisaPisa,
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zvao se, zapravo, Leonardo od Pise,
01:36
and these numbersbrojevi appearpojaviti in his bookrezervirati "LiberLiber AbaciAbaci,"
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a ovi se brojevi pojavljuju u njegovoj knjizi "Liber Abaci",
01:39
whichkoji taughtučio the WesternZapadni worldsvijet
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iz koje je Zapadni svijet naučio
01:41
the methodsmetode of arithmeticaritmetika that we use todaydanas.
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aritmetičke metode koje danas koristimo.
01:44
In termsUvjeti of applicationsaplikacije,
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Što se primjene tiče,
01:45
FibonacciFibonacci numbersbrojevi appearpojaviti in naturepriroda
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Fibonaccijevi brojevi se u prirodi pojavljuju
01:48
surprisinglyiznenađujuče oftenčesto.
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iznenađujuće često.
01:49
The numberbroj of petalslatice on a flowercvijet
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Broj latica na cvijetu
01:51
is typicallytipično a FibonacciFibonacci numberbroj,
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obično je neki Fibonaccijev broj,
01:53
or the numberbroj of spiralsspirala on a sunflowersuncokret
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ili broj spirala na suncokretovom cvijetu,
01:56
or a pineappleananas
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ili na ananasovom plodu
01:57
tendsteži to be a FibonacciFibonacci numberbroj as well.
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također teži jednom od Fibonaccijevih brojeva.
02:00
In factčinjenica, there are manymnogi more
applicationsaplikacije of FibonacciFibonacci numbersbrojevi,
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Ustvari, u mnogo drugih slučajeva nalazimo Fibonaccijeve brojeve,
02:03
but what I find mostnajviše inspirationalinspirativna about them
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ali ono što ja u njima smatram najviše nadahnjujućim
02:06
are the beautifullijep numberbroj patternsobrasci they displayprikaz.
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jesu prelijepi brojevni obrasci koje prikazuju.
02:08
Let me showpokazati you one of my favoritesFavoriti.
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Pokazat ću vam jedan od svojih omiljenih.
02:11
SupposePretpostavimo da you like to squarekvadrat numbersbrojevi,
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Pretpostavimo da volite kvadrirati brojeve,
02:13
and franklyiskreno, who doesn't? (LaughterSmijeh)
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i, iskreno, tko ne voli? (Smijeh)
02:16
Let's look at the squarestrgovi
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Pogledajmo kvadrate
02:18
of the first fewnekoliko FibonacciFibonacci numbersbrojevi.
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prvih nekoliko Fibonaccijevih brojeva.
02:20
So one squaredna kvadrat is one,
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Dakle, jedan na kvadrat je jedan,
02:22
two squaredna kvadrat is fourčetiri, threetri squaredna kvadrat is ninedevet,
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dva na kvadrat je četiri, tri na kvadrat je devet,
02:24
fivepet squaredna kvadrat 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 surpriseiznenađenje
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Naravno, nije iznenađujuće
02:29
that when you adddodati consecutiveuzastopnih FibonacciFibonacci numbersbrojevi,
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kad pribrajanjem uzastopnih Fibonaccijevih brojeva
02:32
you get the nextSljedeći FibonacciFibonacci numberbroj. Right?
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dobijemo sljedeći Fibonaccijev broj. Zar ne?
02:34
That's how they're createdstvorio.
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Tako su i stvoreni.
02:35
But you wouldn'tne bi expectočekivati anything specialposeban
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Međutim, ne biste očekivali ništa osobito
02:37
to happendogoditi se when you adddodati the squarestrgovi togetherzajedno.
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krenete li zbrajati kvadrate.
02:40
But checkprovjeriti this out.
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Ali, pogledajte ovo.
02:42
One plusplus one givesdaje us two,
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Jedan plus jedan daje dva,
02:44
and one plusplus fourčetiri givesdaje us fivepet.
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a jedan plus četiri daje pet.
02:46
And fourčetiri plusplus ninedevet is 13,
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A četiri plus devet daju trinaest,
02:48
ninedevet plusplus 25 is 34,
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a devet plus 25 je 34,
02:52
and yes, the patternuzorak continuesnastavlja.
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i da, obrazac se nastavlja.
02:54
In factčinjenica, here'sevo anotherjoš one.
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Zapravo, evo vam još jednog.
02:56
SupposePretpostavimo da you wanted to look at
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Pretpostavimo da ste poželjeli sagledati
02:58
addingdodajući the squarestrgovi of
the first fewnekoliko FibonacciFibonacci numbersbrojevi.
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zbrajanje kvadrata prvih nekoliko
Fibonaccijevih brojeva.
03:00
Let's see what we get there.
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Pogledajmo što ćemo dobiti.
03:02
So one plusplus one plusplus fourčetiri is sixšest.
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Dakle jedan plus jedan plus četiri je šest.
03:04
AddDodati ninedevet to that, we get 15.
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Dodamo li tome devet, dobit ćemo 15.
03:07
AddDodati 25, we get 40.
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Dodajmo 25 i dobivamo 40.
03:09
AddDodati 64, we get 104.
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Dodajmo 64 i dobivamo 104.
03:12
Now look at those numbersbrojevi.
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Razmotrimo te brojeve.
03:14
Those are not FibonacciFibonacci numbersbrojevi,
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To nisu Fiboonaccijevi brojevi,
03:16
but if you look at them closelytijesno,
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ali promotrite li ih pažljivije,
03:18
you'llvi ćete see the FibonacciFibonacci numbersbrojevi
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uočit ćete Fibonaccijeve brojeve
03:20
buriedpokopan insideiznutra of them.
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skrivene u njima.
03:22
Do you see it? I'll showpokazati it to you.
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Vidite li ih?
Pokazat ću vam.
03:24
SixŠest is two timesputa threetri, 15 is threetri timesputa fivepet,
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Šest je dva puta tri,
a 15 je tri puta pet,
03:28
40 is fivepet timesputa eightosam,
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40 je pet puta osam,
03:30
two, threetri, fivepet, eightosam, who do we appreciatecijeniti?
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dva, tri, pet, osam,
volite me takvog tko sam?
03:33
(LaughterSmijeh)
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(Smijeh)
03:34
FibonacciFibonacci! Of coursenaravno.
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Fibonacci!
Naravno.
03:36
Now, as much funzabava as it is to discoverotkriti these patternsobrasci,
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Koliko god bilo zabavno otkrivati ovakve obrasce,
03:40
it's even more satisfyingzadovoljavajući to understandrazumjeti
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još je više ispunjavajuće uvidjeti
03:42
why they are truepravi.
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zašto je tome tako.
03:44
Let's look at that last equationjednadžba.
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Pogledajmo posljednju jednadžbu.
03:46
Why should the squarestrgovi of one, one,
two, threetri, fivepet and eightosam
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Zašto bi kvadrati brojeva jedan, jedan, dva, tri, pet i osam
03:50
adddodati up to eightosam timesputa 13?
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u zbroju bili jednaki umnošku osam i 13?
03:53
I'll showpokazati you by drawingcrtanje a simplejednostavan pictureslika.
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Objasnit ću vam ovim
jednostavnim prikazom.
03:56
We'llMi ćemo startpočetak with a one-by-onejedan po jedan squarekvadrat
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Započnimo s kvadratom
dimenzija jedan puta jedan
03:58
and nextSljedeći to that put anotherjoš one-by-onejedan po jedan squarekvadrat.
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i do njega stavimo još jedan
kvadrat dimenzija jedan puta jedan.
04:02
TogetherZajedno, they formoblik a one-by-twojedan od dva rectanglepravokutnik.
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Zajedno, oni čine
pravokutnik dimenzija jedan puta dva.
04:06
BeneathIspod that, I'll put a two-by-twopo dvije squarekvadrat,
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Ispod njih, nacrtat ću
kvadrat dimenzija dva puta dva,
04:08
and nextSljedeći to that, a three-by-threetri po tri squarekvadrat,
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a do njih, kvadrat tri puta tri,.
04:11
beneathispod that, a five-by-fivepet za pet squarekvadrat,
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Ispod njih, kvadrat pet puta pet,
04:13
and then an eight-by-eightosam od osam squarekvadrat,
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a potom kvadrat osam puta osam,
04:15
creatingstvaranje one giantgigantski rectanglepravokutnik, right?
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kreirajući tako jedan ogroman pravokutnik, zar ne?
04:18
Now let me askpitati you a simplejednostavan questionpitanje:
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Postavit ću vam jednostavno pitanje:
04:20
what is the areapodručje of the rectanglepravokutnik?
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Kolika je površina pravokutnika?
04:23
Well, on the one handruka,
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S jedne strane,
04:25
it's the sumiznos of the areaspodručja
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ona je suma površina
04:28
of the squarestrgovi insideiznutra it, right?
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ucrtanih kvadrata, zar ne?
04:30
Just as we createdstvorio it.
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Tako je pravokutnik i nastao.
04:31
It's one squaredna kvadrat plusplus one squaredna kvadrat
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Dakle, jedan na kvadrat plus jedan na kvadrat,
04:33
plusplus two squaredna kvadrat plusplus threetri squaredna kvadrat
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plus dva na kvadrat, plus tri na kvadrat,
04:35
plusplus fivepet squaredna kvadrat plusplus eightosam squaredna kvadrat. Right?
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plus pet na kvadrat, plus osam na kvadrat.
04:38
That's the areapodručje.
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To je površina.
04:40
On the other handruka, because it's a rectanglepravokutnik,
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S druge strane, budući da se radi o pravokutniku,
04:42
the areapodručje is equaljednak to its heightvisina timesputa its basebaza,
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površina je jednaka umnošku
njegove visine i njegove baze,
04:46
and the heightvisina is clearlyjasno eightosam,
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pri čemu je visina očito osam
04:48
and the basebaza is fivepet plusplus eightosam,
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a baza je pet plus osam,
04:51
whichkoji is the nextSljedeći FibonacciFibonacci numberbroj, 13. Right?
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što je sljedeći
Fibonaccijev broj, 13.Zar ne?
04:55
So the areapodručje is alsotakođer eightosam timesputa 13.
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Prema tome, površina je osam puta 13.
04:58
SinceOd we'veimamo correctlyispravno calculatedizračunava se the areapodručje
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Budući da smo ispravno izračunali površinu
05:00
two differentdrugačiji waysnačine,
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na dva različita načina,
05:02
they have to be the sameisti numberbroj,
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to trebaju biti isti brojevi,
05:04
and that's why the squarestrgovi of one,
one, two, threetri, fivepet and eightosam
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i etto zašto kvadrati brojeva
jedan, jedan, dva, tri, pet i osam
05:08
adddodati up to eightosam timesputa 13.
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zbrojeni daju osam puta 13.
05:10
Now, if we continuenastaviti this processpostupak,
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Nastavimo li ovaj postupak,
05:12
we'lldobro generategenerirati rectanglespravokutnika of the formoblik 13 by 21,
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stvorit ćemo pravokutnike
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 checkprovjeriti this out.
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A razmotrimo ovo.
05:20
If you dividepodijeliti 13 by eightosam,
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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 dividepodijeliti the largerveći numberbroj
by the smallermanji numberbroj,
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I dijelimo li veći broj
s manjim brojem,
05:28
then these ratiosomjeri get closerbliže and closerbliže
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primijetit ćemo da se
količnici sve više približavaju
05:31
to about 1.618,
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broju 1,618,
05:33
knownznan to manymnogi people as the GoldenZlatni RatioOmjer,
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mnogim ljudima znanom
kao Zlatni omjer,
05:37
a numberbroj whichkoji has fascinatedopčinjen mathematiciansmatematičari,
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broj koji je stoljećima očaravao matematičare,
05:39
scientistsznanstvenici and artistsizvođači for centuriesstoljeća.
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znanstvenike i umjetnike stoljećima.
05:42
Now, I showpokazati all this to you because,
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Sve vam ovo pokazujem zato što,
05:45
like so much of mathematicsmatematika,
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kao toliko toga u matematici,
05:47
there's a beautifullijep sidestrana to it
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ovo posjeduje osobitu ljepotu kojoj,
05:49
that I fearstrah does not get enoughdovoljno attentionpažnja
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bojim se, ne poklanjamo dovoljno pozornosti
05:51
in our schoolsškola.
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u našim školama.
05:52
We spendprovesti lots of time learningučenje about calculationračunanje,
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Mnogo vremena provodimo učeći o računanju,
05:55
but let's not forgetzaboraviti about applicationprimjena,
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ali ne zaboravimo na primjenu,
05:58
includinguključujući, perhapsmožda, the mostnajviše
importantvažno applicationprimjena of all,
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uključujući, možda, i najvažniju
od svih mogućih primjena,
06:01
learningučenje how to think.
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učiti kako misliti.
06:03
If I could summarizerezimirati this in one sentencekazna,
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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
MathematicsMatematika is not just solvingrješavanje for x,
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Matematika ne služi samo za rješavanje x-a,
06:10
it's alsotakođer figuringfiguring 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
(ApplausePljesak)
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(Pljesak)
Translated by Mladen Barešić
Reviewed by Senzos Osijek

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ABOUT THE SPEAKER
Arthur Benjamin - Mathemagician
Using 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").

More profile about the speaker
Arthur Benjamin | Speaker | TED.com