TEDGlobal 2013
Arthur Benjamin: The magic of Fibonacci numbers
Arthur Benjamin: Magija Fibonačijevih brojeva
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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!).
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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Dakle, 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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primjena,
00:21
and last, and unfortunately least
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i posljednji, nažalost najmanje važan
00:24
in terms of the time we give it,
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u smislu vremena koji mu posvetimo,
00:26
inspiration.
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je inspiracija.
00:28
Mathematics is the science of patterns,
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Matematika je nauka o uzorcima
00:30
and we study it to learn how to think logically,
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i proučavamo je s ciljem da naučimo kako razmišljati 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 matematika koju učimo u školi
00:39
is not effectively motivated,
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uglavnom neuspješno motiviše
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 ovo učimo?"
00:44
then they often hear that they'll need it
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obično čuju da će im to zatrebati
00:46
in an upcoming math class or on a future test.
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na narednom času matematike ili na budućem ispitu.
00:50
But wouldn't it be great
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Međutim, zar ne bi bilo divno
00:51
if every once in a while we did mathematics
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kad bismo se s vremena na vrijeme bavili matematikom
00:54
simply because it was fun or beautiful
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jednostavno zato što je zabavna i lijepa
00:57
or because it excited the mind?
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ili možda zato što je uspjela uzbuditi um?
00:59
Now, I know many people have not
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Znam da mnogi nisu
01:01
had the opportunity to see how this can happen,
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uspjeli doživjeti to o čemu pričam,
01:03
so let me give you a quick example
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pa zato dopustite da vam dam jednostavan primjer
01:05
with my favorite collection of numbers,
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koristeći moju omiljenu kolekciju brojeva,
01:07
the Fibonacci numbers. (Applause)
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Fibonačijeve brojeve. (Aplauz)
01:10
Yeah! I already have Fibonacci fans 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 numbers can be appreciated
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Značaj ovih brojeva se ogleda
01:15
in many different ways.
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na više 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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jednostavno ih je razumjeti
01:22
as one plus one, which is two.
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kao što je i to da je jedan i jedan jednako dva.
01:24
Then one plus two is three,
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Zatim, jedan i dva je tri,
01:26
two plus three is five, three plus five is eight,
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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
Indeed, the person we call Fibonacci
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Zaista, osoba koju zovemo Fibonači
01:33
was actually named Leonardo of Pisa,
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se ustvari zvala Leonardo od Pise,
01:36
and these numbers appear in his book "Liber Abaci,"
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a ovi brojevi se spominju u njegovoj knjizi "Liber Abaci" ("Knjiga računanja"),
01:39
which taught the Western world
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koja je naučila zapadni svijet
01:41
the methods of arithmetic that we use today.
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metodama aritmetike koje koristimo danas.
01:44
In terms of applications,
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U smislu primjene,
01:45
Fibonacci numbers appear in nature
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Fibonačijevi brojevi se pojavljuju u prirodi
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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je obično 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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također teži da bude Fibonačijev broj.
02:00
In fact, there are many more
applications of Fibonacci numbers,
applications of Fibonacci numbers,
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Ustvari, postoje mnoge druge primjene Fibonačijevih brojeva,
02:03
but what I find most inspirational about them
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ali ono sto smatram najinspirativnijim
02:06
are the beautiful number patterns they display.
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su divni šabloni brojeva koje predstavljaju.
02:08
Let me show you one of my favorites.
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Sad ću vam pokazati jedan od mojih 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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a realno, ko 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 Fibonačijevih brojeva.
02:20
So one squared is one,
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Dakle, kvadrat broja jedan je jedan,
02:22
two squared is four, three squared is nine,
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kvadrat broja dva je četiri, tri na kvadrat je devet,
02:24
five squared is 25, and so on.
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pet na kvadrat je 25, itd.
02:27
Now, it's no surprise
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Nije nikakvo iznenađenje
02:29
that when you add consecutive Fibonacci numbers,
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da sabiranjem dva uzastopna Fibonačijeva broja,
02:32
you get the next Fibonacci number. Right?
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dobijemo sljedeći Fibonačijev broj, je li tako?
02:34
That's how they're created.
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Tako se oni i kreiraju.
02:35
But you wouldn't expect anything special
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Međutim, ne biste očekivali nista posebno
02:37
to happen when you add the squares together.
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da se dogodi u slučaju sabiranja njihovih kvadrata.
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 i jedan je dva,
02:44
and one plus four gives us five.
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a jedan i četiri je pet.
02:46
And four plus nine is 13,
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Četiri i devet je 13,
02:48
nine plus 25 is 34,
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devet i 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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Ustvari, evo jos jednog.
02:56
Suppose you wanted to look at
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Pretpostavimo da ste htjeli pokušati
02:58
adding the squares of
the first few Fibonacci numbers.
the first few Fibonacci numbers.
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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 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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Ako dodamo devet na to, dobit ćemo 15.
03:07
Add 25, we get 40.
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Dodavanjem 25, dobijamo 40.
03:09
Add 64, we get 104.
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Dodavanjem 64, dobijamo 104.
03:12
Now look at those numbers.
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Sada pogledajte ove brojeve.
03:14
Those are not Fibonacci numbers,
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Ovo nisu Fibonačijevi brojevi,
03:16
but if you look at them closely,
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ali ako ih bolje pogledate,
03:18
you'll see the Fibonacci numbers
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vidjet ćete Fibonačijeve brojeve
03:20
buried inside of them.
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unutar ovih brojeva.
03:22
Do you see it? I'll show it to you.
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Vidite li? Pokazat ću vam.
03:24
Six is two times three, 15 is three times five,
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Šest je dva pomnoženo sa tri, 15 je tri pomnoženo sa pet,
03:28
40 is five times eight,
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40 je pet pomnoženo sa osam,
03:30
two, three, five, eight, who do we appreciate?
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dva, tri, pet, osam, pogodi ko sam?
03:33
(Laughter)
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(Smijeh)
03:34
Fibonacci! Of course.
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Fibonači, naravno!
03:36
Now, as much fun as it is to discover these patterns,
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Koliko god da je zabavno otkriti ove šablone,
03:40
it's even more satisfying to understand
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još je bolje shvatiti
03:42
why they are true.
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zašto oni postoje.
03:44
Let's look at that last equation.
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Pogledajmo posljednju 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 zbir kvadrata od jedan, jedan, dva, tri, pet i osam
03:50
add up to eight times 13?
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bio jednak rezultatu proizvoda brojeva osam i 13?
03:53
I'll show you by drawing a simple picture.
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Pokazat ću vam pomoću jednostavne slike.
03:56
We'll start with a one-by-one square
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Počet ćemo sa kvadratom "jedan sa jedan"
03:58
and next to that put another one-by-one square.
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i pored njega ćemo staviti isti takav kvadrat.
04:02
Together, they form a one-by-two rectangle.
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Zajedno, oni formiraju "jedan sa dva" pravougaonik.
04:06
Beneath that, I'll put a two-by-two square,
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Ispod njega, stavit ću "dva sa dva",
04:08
and next to that, a three-by-three square,
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pored njega "tri sa tri" kvadrat,
04:11
beneath that, a five-by-five square,
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ispod kvadrat "pet sa pet" ,
04:13
and then an eight-by-eight square,
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a zatim "osam sa osam",
04:15
creating one giant rectangle, right?
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kreirajući jedan veliki pravougaonik, zar ne?
04:18
Now let me ask you a simple question:
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Sada dopustite da vam postavim jednostavno pitanje:
04:20
what is the area of the rectangle?
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šta predstavlja površinu ovog pravougaonika?
04:23
Well, on the one hand,
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Pa, s 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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sadržanih kvadrata, je li tako?
04:30
Just as we created it.
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Baš kao što smo ih i kreirali.
04:31
It's one squared plus one squared
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To je jedan na kvadrat plus jedan na kvadrat,
04:33
plus two squared plus three squared
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sabrano sa kvadratom od dva i tri
04:35
plus five squared plus eight squared. Right?
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te kvadratom od pet i osam. Jesam li u pravu?
04:38
That's the area.
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To je tražena površina.
04:40
On the other hand, because it's a rectangle,
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S druge strane, s obzirom na to da se radi o pravougaoniku,
04:42
the area is equal to its height times its base,
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površina je jednaka proizvodu dužine i širine,
04:46
and the height is clearly eight,
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širina je očito jednaka osam,
04:48
and the base is five plus eight,
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dok je dužina jednaka zbiru pet i osam,
04:51
which is the next Fibonacci number, 13. Right?
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koji predstavlja sljedeći Fibonačijev broj, 13. Je li tako?
04:55
So the area is also eight times 13.
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Dakle, površina je jednaka i proizvodu 8 i 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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ona mora biti jednaka,
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 zato je zbir kvadrata od jedan, jedan, dva, tri, pet i osam
05:08
add up to eight times 13.
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jednak proizvodu 8 i 13.
05:10
Now, if we continue this process,
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Ukoliko nastavimo sa ovim postupkom,
05:12
we'll generate rectangles of the form 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 check this out.
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Pogledajte sada ovo.
05:20
If you divide 13 by eight,
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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 divide the larger number
by the smaller number,
by the smaller number,
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Međutim, što veći broj dijelimo sa manjim brojem
05:28
then these ratios get closer and closer
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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
known to many people as the Golden Ratio,
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poznatog mnogima kao "zlatni rez",
05:37
a number which has fascinated mathematicians,
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broja koji fascinira matematičare,
05:39
scientists and artists for centuries.
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naučnike i umjetnike već stoljećima.
05:42
Now, I show all this to you because,
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Pokazao sam vam sve ovo,
05:45
like so much of mathematics,
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jer pored sve te matematike
05:47
there's a beautiful side to it
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postoji i lijepa strana
05:49
that I fear does not get enough attention
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kojoj se ne pridaje mnogo 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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Provodimo mnogo vremena baveći se računanjima,
05:55
but let's not forget about application,
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ali ne treba zaboraviti njihovu primjenu,
05:58
including, perhaps, the most
important application of all,
important application of all,
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uključujući najvažniju od svih,
06:01
learning how to think.
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a to je da nas uče kako da razmišljamo.
06:03
If I could summarize this in one sentence,
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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
Mathematics is not just solving for x,
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Matematika nije samo rješavanje nepoznate x,
06:10
it's also figuring 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
(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