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

Arthur Benjamin: Magija Fibonačijevih brojeva

Filmed:
7,057,274 views

Matematika je logična, upotrebljiva i jednostavno... fenomenalna. Matemagičar Arthur Benjamin istražuje skrivene mogućnosti tog čudnog i divnog niza brojeva, Fibonačijevog niza (I podsjeća nas da matematika može biti inspirativna, također!).
- 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 learnuči mathematicsmatematika?
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Dakle, zašto učimo matematiku?
00:15
EssentiallyU suštini, for threetri reasonsrazloge:
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U suštini, iz tri razloga:
00:18
calculationproračun,
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računanje,
00:19
applicationaplikacija,
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primjena,
00:21
and last, and unfortunatelynažalost leastnajmanje
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i posljednji, nažalost najmanje važan
00:24
in termsuslovi of the time we give it,
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u smislu vremena koji mu posvetimo,
00:26
inspirationinspiracija.
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je inspiracija.
00:28
MathematicsMatematika is the sciencenauka of patternsobrasci,
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Matematika je nauka o uzorcima
00:30
and we studystudija it to learnuči how to think logicallylogički,
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i proučavamo je s ciljem da naučimo kako razmišljati logički,
00:34
criticallykritički and creativelykreativno,
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kritički i kreativno,
00:36
but too much of the mathematicsmatematika
that we learnuči in schoolškola
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ali matematika koju učimo u školi
00:39
is not effectivelyefikasno motivatedmotivirani,
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uglavnom neuspješno motiviše
00:41
and when our studentsstudenti askpitajte,
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i kada 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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obično čuju da će im to zatrebati
00:46
in an upcomingNadolazeći mathmatematika classklasa or on a futurebudućnost testtest.
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na narednom času matematike ili na budućem ispitu.
00:50
But wouldn'tne bi it be great
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Međutim, zar ne bi bilo divno
00:51
if everysvaki oncejednom in a while we did mathematicsmatematika
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kad bismo se s vremena na vrijeme bavili matematikom
00:54
simplyjednostavno because it was funzabava or beautifulprelepo
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jednostavno zato što je zabavna i lijepa
00:57
or because it exciteduzbuđeni the mindum?
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ili možda zato što je uspjela uzbuditi um?
00:59
Now, I know manymnogi people have not
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Znam da mnogi nisu
01:01
had the opportunityprilika to see how this can happenda se desi,
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uspjeli doživjeti to o čemu pričam,
01:03
so let me give you a quickbrzo exampleprimer
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pa zato dopustite da vam dam jednostavan primjer
01:05
with my favoriteomiljeni collectionkolekcija of numbersbrojevi,
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koristeći moju omiljenu kolekciju brojeva,
01:07
the FibonacciFibonacci numbersbrojevi. (ApplausePljesak)
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Fibonačijeve brojeve. (Aplauz)
01:10
Yeah! I alreadyveć have FibonacciFibonacci fansfanovi here.
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Tako je! Vidim da ovdje imamo Fibonačijeve obožavatelje.
01:12
That's great.
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To je divno.
01:13
Now these numbersbrojevi can be appreciatedpoštovati
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Značaj ovih brojeva se ogleda
01:15
in manymnogi differentdrugačiji waysnačina.
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na više načina.
01:17
From the standpointstanovište of calculationproračun,
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Sa stanovišta računanja,
01:20
they're as easylako to understandrazumijete
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jednostavno ih je razumjeti
01:22
as one plusplus one, whichšto is two.
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kao što je i to da je jedan i jedan jednako dva.
01:24
Then one plusplus two is threetri,
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Zatim, jedan i dva je tri,
01:26
two plusplus threetri is fivepet, threetri plusplus fivepet is eightosam,
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dva i tri je pet, tri i 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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Zaista, osoba koju zovemo Fibonači
01:33
was actuallyzapravo namedimenovan LeonardoLeonardo of PisaPisa,
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se ustvari zvala Leonardo od Pise,
01:36
and these numbersbrojevi appearpojaviti in his bookknjiga "LiberLiber AbaciAbaci,"
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a ovi brojevi se spominju u njegovoj knjizi "Liber Abaci" ("Knjiga računanja"),
01:39
whichšto taughtpredavao the WesternZapadni worldsvet
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koja je naučila zapadni svijet
01:41
the methodsmetode of arithmeticaritmetika that we use todaydanas.
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metodama aritmetike koje koristimo danas.
01:44
In termsuslovi of applicationsaplikacije,
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U smislu primjene,
01:45
FibonacciFibonacci numbersbrojevi appearpojaviti in naturepriroda
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Fibonačijevi brojevi se pojavljuju u prirodi
01:48
surprisinglyiznenađujuće oftenčesto.
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iznenađujuće često.
01:49
The numberbroj of petalslatice on a flowercvet
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Broj latica na cvijetu
01:51
is typicallyobično a FibonacciFibonacci numberbroj,
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je obično Fibonačijev broj,
01:53
or the numberbroj of spiralsspirala on a sunflowersuncokret
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ili broj spirala na suncokretu
01:56
or a pineappleananas
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ili ananasu
01:57
tendsteži to be a FibonacciFibonacci numberbroj as well.
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također teži da bude Fibonačijev broj.
02:00
In factčinjenica, there are manymnogi more
applicationsaplikacije of FibonacciFibonacci numbersbrojevi,
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Ustvari, postoje mnoge druge primjene Fibonačijevih brojeva,
02:03
but what I find mostnajviše inspirationalinspirativni about them
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ali ono sto smatram najinspirativnijim
02:06
are the beautifulprelepo numberbroj patternsobrasci they displaydisplay.
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su divni šabloni brojeva koje predstavljaju.
02:08
Let me showshow you one of my favoritesFavoriti.
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Sad ću vam pokazati jedan od mojih 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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a realno, ko ne voli? (Smijeh)
02:16
Let's look at the squareskvadrati
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Pogledajmo kvadrate
02:18
of the first fewnekoliko FibonacciFibonacci numbersbrojevi.
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prvih nekoliko Fibonačijevih brojeva.
02:20
So one squaredkvadratna is one,
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Dakle, kvadrat broja jedan je jedan,
02:22
two squaredkvadratna is fourčetiri, threetri squaredkvadratna is ninedevet,
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kvadrat broja dva je četiri, tri na kvadrat je devet,
02:24
fivepet squaredkvadratna is 25, and so on.
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pet na kvadrat je 25, itd.
02:27
Now, it's no surpriseiznenađenje
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Nije nikakvo iznenađenje
02:29
that when you adddodati consecutiveuzastopnih FibonacciFibonacci numbersbrojevi,
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da sabiranjem dva uzastopna Fibonačijeva broja,
02:32
you get the nextsledeći FibonacciFibonacci numberbroj. Right?
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dobijemo sljedeći Fibonačijev broj, je li tako?
02:34
That's how they're createdstvoreno.
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Tako se oni i kreiraju.
02:35
But you wouldn'tne bi expectocekujem anything specialposeban
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Međutim, ne biste očekivali nista posebno
02:37
to happenda se desi when you adddodati the squareskvadrati togetherzajedno.
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da se dogodi u slučaju sabiranja njihovih kvadrata.
02:40
But checkproveri this out.
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Ali, pogledajte ovo.
02:42
One plusplus one givesdaje us two,
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Jedan i jedan je dva,
02:44
and one plusplus fourčetiri givesdaje us fivepet.
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a jedan i četiri je pet.
02:46
And fourčetiri plusplus ninedevet is 13,
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Četiri i devet je 13,
02:48
ninedevet plusplus 25 is 34,
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devet i 25 je 34,
02:52
and yes, the patternobrazac continuesnastavlja.
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i da, šablon se nastavlja.
02:54
In factčinjenica, here'sevo anotherdrugi one.
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Ustvari, evo jos jednog.
02:56
SupposePretpostavimo da you wanted to look at
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Pretpostavimo da ste htjeli pokušati
02:58
addingdodavanje the squareskvadrati of
the first fewnekoliko FibonacciFibonacci numbersbrojevi.
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sabrati kvadrate prvih nekoliko Fibonačijevih brojeva.
03:00
Let's see what we get there.
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Pogledajmo šta smo dobili ovdje.
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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Ako dodamo devet na to, dobit ćemo 15.
03:07
AddDodati 25, we get 40.
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Dodavanjem 25, dobijamo 40.
03:09
AddDodati 64, we get 104.
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Dodavanjem 64, dobijamo 104.
03:12
Now look at those numbersbrojevi.
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Sada pogledajte ove brojeve.
03:14
Those are not FibonacciFibonacci numbersbrojevi,
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Ovo nisu Fibonačijevi brojevi,
03:16
but if you look at them closelyblisko,
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ali ako ih bolje pogledate,
03:18
you'llti ćeš see the FibonacciFibonacci numbersbrojevi
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vidjet ćete Fibonačijeve brojeve
03:20
buriedpokopan insideunutra of them.
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unutar ovih brojeva.
03:22
Do you see it? I'll showshow it to you.
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Vidite li? Pokazat ću vam.
03:24
SixŠest is two timesputa threetri, 15 is threetri timesputa fivepet,
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Šest je dva pomnoženo sa tri, 15 je tri pomnoženo sa pet,
03:28
40 is fivepet timesputa eightosam,
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40 je pet pomnoženo sa osam,
03:30
two, threetri, fivepet, eightosam, who do we appreciatecenite?
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dva, tri, pet, osam, pogodi ko sam?
03:33
(LaughterSmijeh)
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(Smijeh)
03:34
FibonacciFibonacci! Of coursekurs.
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Fibonači, naravno!
03:36
Now, as much funzabava as it is to discoverotkriti these patternsobrasci,
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Koliko god da je zabavno otkriti ove šablone,
03:40
it's even more satisfyingzadovoljavajući to understandrazumijete
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još je bolje shvatiti
03:42
why they are trueistinito.
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zašto oni postoje.
03:44
Let's look at that last equationjednačina.
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Pogledajmo posljednju jednačinu.
03:46
Why should the squareskvadrati of one, one,
two, threetri, fivepet and eightosam
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Zašto bi zbir kvadrata od jedan, jedan, dva, tri, pet i osam
03:50
adddodati up to eightosam timesputa 13?
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bio jednak rezultatu proizvoda brojeva osam i 13?
03:53
I'll showshow you by drawingcrtanje a simplejednostavno pictureslika.
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Pokazat ću vam pomoću jednostavne slike.
03:56
We'llMi ćemo startpočnite with a one-by-onejedan po jedan squarekvadrat
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Počet ćemo sa kvadratom "jedan sa jedan"
03:58
and nextsledeći to that put anotherdrugi one-by-onejedan po jedan squarekvadrat.
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i pored njega ćemo staviti isti takav kvadrat.
04:02
TogetherZajedno, they formobrazac a one-by-twojedan od dva rectanglepravokutnik.
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Zajedno, oni formiraju "jedan sa dva" pravougaonik.
04:06
BeneathIspod that, I'll put a two-by-twodva po dva squarekvadrat,
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Ispod njega, stavit ću "dva sa dva",
04:08
and nextsledeći to that, a three-by-threetri po tri squarekvadrat,
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pored njega "tri sa tri" kvadrat,
04:11
beneathispod that, a five-by-fivepet puta pet squarekvadrat,
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ispod kvadrat "pet sa pet" ,
04:13
and then an eight-by-eightosam od osam squarekvadrat,
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a zatim "osam sa osam",
04:15
creatingstvaranje one giantgigant rectanglepravokutnik, right?
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kreirajući jedan veliki pravougaonik, zar ne?
04:18
Now let me askpitajte you a simplejednostavno questionpitanje:
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Sada dopustite da vam postavim jednostavno pitanje:
04:20
what is the areapodručje of the rectanglepravokutnik?
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šta predstavlja površinu ovog pravougaonika?
04:23
Well, on the one handruka,
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Pa, s jedne strane,
04:25
it's the sumsuma of the areasoblasti
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to je zbir površina
04:28
of the squareskvadrati insideunutra it, right?
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sadržanih kvadrata, je li tako?
04:30
Just as we createdstvoreno it.
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Baš kao što smo ih i kreirali.
04:31
It's one squaredkvadratna plusplus one squaredkvadratna
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To je jedan na kvadrat plus jedan na kvadrat,
04:33
plusplus two squaredkvadratna plusplus threetri squaredkvadratna
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sabrano sa kvadratom od dva i tri
04:35
plusplus fivepet squaredkvadratna plusplus eightosam squaredkvadratna. Right?
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te kvadratom od pet i osam. Jesam li u pravu?
04:38
That's the areapodručje.
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To je tražena površina.
04:40
On the other handruka, because it's a rectanglepravokutnik,
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S druge strane, s obzirom na to da se radi o pravougaoniku,
04:42
the areapodručje is equaljednak to its heightvisina timesputa its basebazu,
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površina je jednaka proizvodu dužine i širine,
04:46
and the heightvisina is clearlyjasno eightosam,
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širina je očito jednaka osam,
04:48
and the basebazu is fivepet plusplus eightosam,
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dok je dužina jednaka zbiru pet i osam,
04:51
whichšto is the nextsledeći FibonacciFibonacci numberbroj, 13. Right?
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koji predstavlja sljedeći Fibonačijev broj, 13. Je li tako?
04:55
So the areapodručje is alsotakođe eightosam timesputa 13.
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Dakle, površina je jednaka i proizvodu 8 i 13.
04:58
SinceOd we'vemi smo correctlyispravno calculatedizračunati the areapodručje
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Pošto smo tačno izračunali površinu
05:00
two differentdrugačiji waysnačina,
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na dva različita načina,
05:02
they have to be the sameisto numberbroj,
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ona mora biti jednaka,
05:04
and that's why the squareskvadrati of one,
one, two, threetri, fivepet and eightosam
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i zato je zbir kvadrata od jedan, jedan, dva, tri, pet i osam
05:08
adddodati up to eightosam timesputa 13.
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jednak proizvodu 8 i 13.
05:10
Now, if we continueNastavite this processproces,
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Ukoliko nastavimo sa ovim postupkom,
05:12
we'llmi ćemo generategenerirati rectanglespravokutnici of the formobrazac 13 by 21,
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kreira ćemo pravougaonike dimenzija 13 sa 21,
05:16
21 by 34, and so on.
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21 sa 34, itd.
05:19
Now checkproveri this out.
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Pogledajte sada ovo.
05:20
If you dividepodelite 13 by eightosam,
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Ako podijelimo 13 sa osam,
05:22
you get 1.625.
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dobijemo 1,625.
05:24
And if you dividepodelite the largerveće numberbroj
by the smallermanji numberbroj,
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Međutim, što veći broj dijelimo sa manjim brojem
05:28
then these ratiosodnosa get closerbliže and closerbliže
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ovaj se odnos sve više približava
05:31
to about 1.618,
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do otprilike 1,618,
05:33
knownpoznato to manymnogi people as the GoldenZlatni RatioOmjer,
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poznatog mnogima kao "zlatni rez",
05:37
a numberbroj whichšto has fascinatedfasciniran mathematiciansmatematičari,
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broja koji fascinira matematičare,
05:39
scientistsnaučnici and artistsumetnici for centuriesvekovima.
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naučnike i umjetnike već stoljećima.
05:42
Now, I showshow all this to you because,
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Pokazao sam vam sve ovo,
05:45
like so much of mathematicsmatematika,
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jer pored sve te matematike
05:47
there's a beautifulprelepo sidestrana to it
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postoji i lijepa strana
05:49
that I fearstrah does not get enoughdosta attentionpažnja
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kojoj se ne pridaje mnogo pažnje
05:51
in our schoolsškole.
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u našim školama.
05:52
We spendpotrošiti lots of time learningučenje about calculationproračun,
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Provodimo mnogo vremena baveći se računanjima,
05:55
but let's not forgetzaboraviti about applicationaplikacija,
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ali ne treba zaboraviti njihovu primjenu,
05:58
includinguključujući, perhapsmožda, the mostnajviše
importantbitan applicationaplikacija of all,
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uključujući najvažniju od svih,
06:01
learningučenje how to think.
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a to je da nas uče kako da razmišljamo.
06:03
If I could summarizesumirati this in one sentencekazna,
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Ako bih trebao sumirati sve navedeno u jednoj rečenici,
06:05
it would be this:
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to bi bila ova:
06:07
MathematicsMatematika is not just solvingrešavanje problema for x,
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Matematika nije samo rješavanje nepoznate x,
06:10
it's alsotakođe figuringfiguring out why.
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nego i shvatanje njene svrhe.
06:13
Thank you very much.
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Hvala vam.
06:15
(ApplausePljesak)
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(Aplauz)
Translated by Nejra Hodžić
Reviewed by Ema Bilbija Zulic

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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